From 75cf80bf380945366359db917fac7bb2fde77d56 Mon Sep 17 00:00:00 2001
From: "Documenter.jl" <documenter@juliadocs.github.io>
Date: Tue, 7 May 2024 02:47:03 +0000
Subject: [PATCH] build based on 26303f3

---
 dev/.documenter-siteinfo.json          |    1 +
 dev/assets/documenter.js               | 1050 ++++++++++++++++++++++++
 dev/assets/themes/documenter-dark.css  |    7 +
 dev/assets/themes/documenter-light.css |    9 +
 dev/assets/themeswap.js                |   84 ++
 dev/assets/warner.js                   |   52 ++
 dev/checkmatrix/index.html             |   44 +
 dev/distance/index.html                |  167 ++++
 dev/img/bench_ref_optimized.jpg        |  Bin 0 -> 32942 bytes
 dev/img/bench_ref_vs_aarref.jpg        |  Bin 0 -> 74031 bytes
 dev/index.html                         |  155 ++++
 dev/objects.inv                        |  Bin 0 -> 1254 bytes
 dev/pauliops/index.html                |   52 ++
 dev/search_index.js                    |    3 +
 dev/siteinfo.js                        |    1 +
 index.html                             |    2 +
 versions.js                            |    5 +
 17 files changed, 1632 insertions(+)
 create mode 100644 dev/.documenter-siteinfo.json
 create mode 100644 dev/assets/documenter.js
 create mode 100644 dev/assets/themes/documenter-dark.css
 create mode 100644 dev/assets/themes/documenter-light.css
 create mode 100644 dev/assets/themeswap.js
 create mode 100644 dev/assets/warner.js
 create mode 100644 dev/checkmatrix/index.html
 create mode 100644 dev/distance/index.html
 create mode 100644 dev/img/bench_ref_optimized.jpg
 create mode 100644 dev/img/bench_ref_vs_aarref.jpg
 create mode 100644 dev/index.html
 create mode 100644 dev/objects.inv
 create mode 100644 dev/pauliops/index.html
 create mode 100644 dev/search_index.js
 create mode 100644 dev/siteinfo.js
 create mode 100644 index.html
 create mode 100644 versions.js

diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json
new file mode 100644
index 0000000..cd4068a
--- /dev/null
+++ b/dev/.documenter-siteinfo.json
@@ -0,0 +1 @@
+{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-07T02:47:00","documenter_version":"1.4.1"}}
\ No newline at end of file
diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js
new file mode 100644
index 0000000..c6562b5
--- /dev/null
+++ b/dev/assets/documenter.js
@@ -0,0 +1,1050 @@
+// Generated by Documenter.jl
+requirejs.config({
+  paths: {
+    'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min',
+    'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min',
+    'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min',
+    'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min',
+    'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min',
+    'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min',
+    'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min',
+    'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min',
+    'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min',
+  },
+  shim: {
+  "highlight-julia": {
+    "deps": [
+      "highlight"
+    ]
+  },
+  "katex-auto-render": {
+    "deps": [
+      "katex"
+    ]
+  },
+  "headroom-jquery": {
+    "deps": [
+      "jquery",
+      "headroom"
+    ]
+  },
+  "highlight-julia-repl": {
+    "deps": [
+      "highlight"
+    ]
+  }
+}
+});
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) {
+$(document).ready(function() {
+  renderMathInElement(
+    document.body,
+    {
+  "delimiters": [
+    {
+      "left": "$",
+      "right": "$",
+      "display": false
+    },
+    {
+      "left": "$$",
+      "right": "$$",
+      "display": true
+    },
+    {
+      "left": "\\[",
+      "right": "\\]",
+      "display": true
+    }
+  ]
+}
+
+  );
+})
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) {
+$(document).ready(function() {
+    hljs.highlightAll();
+})
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+let timer = 0;
+var isExpanded = true;
+
+$(document).on("click", ".docstring header", function () {
+  let articleToggleTitle = "Expand docstring";
+
+  debounce(() => {
+    if ($(this).siblings("section").is(":visible")) {
+      $(this)
+        .find(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-down")
+        .addClass("fa-chevron-right");
+    } else {
+      $(this)
+        .find(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-right")
+        .addClass("fa-chevron-down");
+
+      articleToggleTitle = "Collapse docstring";
+    }
+
+    $(this)
+      .find(".docstring-article-toggle-button")
+      .prop("title", articleToggleTitle);
+    $(this).siblings("section").slideToggle();
+  });
+});
+
+$(document).on("click", ".docs-article-toggle-button", function (event) {
+  let articleToggleTitle = "Expand docstring";
+  let navArticleToggleTitle = "Expand all docstrings";
+  let animationSpeed = event.noToggleAnimation ? 0 : 400;
+
+  debounce(() => {
+    if (isExpanded) {
+      $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down");
+      $(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-down")
+        .addClass("fa-chevron-right");
+
+      isExpanded = false;
+
+      $(".docstring section").slideUp(animationSpeed);
+    } else {
+      $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up");
+      $(".docstring-article-toggle-button")
+        .removeClass("fa-chevron-right")
+        .addClass("fa-chevron-down");
+
+      isExpanded = true;
+      articleToggleTitle = "Collapse docstring";
+      navArticleToggleTitle = "Collapse all docstrings";
+
+      $(".docstring section").slideDown(animationSpeed);
+    }
+
+    $(this).prop("title", navArticleToggleTitle);
+    $(".docstring-article-toggle-button").prop("title", articleToggleTitle);
+  });
+});
+
+function debounce(callback, timeout = 300) {
+  if (Date.now() - timer > timeout) {
+    callback();
+  }
+
+  clearTimeout(timer);
+
+  timer = Date.now();
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require([], function() {
+function addCopyButtonCallbacks() {
+  for (const el of document.getElementsByTagName("pre")) {
+    const button = document.createElement("button");
+    button.classList.add("copy-button", "fa-solid", "fa-copy");
+    button.setAttribute("aria-label", "Copy this code block");
+    button.setAttribute("title", "Copy");
+
+    el.appendChild(button);
+
+    const success = function () {
+      button.classList.add("success", "fa-check");
+      button.classList.remove("fa-copy");
+    };
+
+    const failure = function () {
+      button.classList.add("error", "fa-xmark");
+      button.classList.remove("fa-copy");
+    };
+
+    button.addEventListener("click", function () {
+      copyToClipboard(el.innerText).then(success, failure);
+
+      setTimeout(function () {
+        button.classList.add("fa-copy");
+        button.classList.remove("success", "fa-check", "fa-xmark");
+      }, 5000);
+    });
+  }
+}
+
+function copyToClipboard(text) {
+  // clipboard API is only available in secure contexts
+  if (window.navigator && window.navigator.clipboard) {
+    return window.navigator.clipboard.writeText(text);
+  } else {
+    return new Promise(function (resolve, reject) {
+      try {
+        const el = document.createElement("textarea");
+        el.textContent = text;
+        el.style.position = "fixed";
+        el.style.opacity = 0;
+        document.body.appendChild(el);
+        el.select();
+        document.execCommand("copy");
+
+        resolve();
+      } catch (err) {
+        reject(err);
+      } finally {
+        document.body.removeChild(el);
+      }
+    });
+  }
+}
+
+if (document.readyState === "loading") {
+  document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks);
+} else {
+  addCopyButtonCallbacks();
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) {
+
+// Manages the top navigation bar (hides it when the user starts scrolling down on the
+// mobile).
+window.Headroom = Headroom; // work around buggy module loading?
+$(document).ready(function () {
+  $("#documenter .docs-navbar").headroom({
+    tolerance: { up: 10, down: 10 },
+  });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+$(document).ready(function () {
+  let meta = $("div[data-docstringscollapsed]").data();
+
+  if (meta?.docstringscollapsed) {
+    $("#documenter-article-toggle-button").trigger({
+      type: "click",
+      noToggleAnimation: true,
+    });
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+/*
+To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc
+
+PSEUDOCODE:
+
+Searching happens automatically as the user types or adjusts the selected filters.
+To preserve responsiveness, as much as possible of the slow parts of the search are done
+in a web worker. Searching and result generation are done in the worker, and filtering and
+DOM updates are done in the main thread. The filters are in the main thread as they should
+be very quick to apply. This lets filters be changed without re-searching with minisearch
+(which is possible even if filtering is on the worker thread) and also lets filters be
+changed _while_ the worker is searching and without message passing (neither of which are
+possible if filtering is on the worker thread)
+
+SEARCH WORKER:
+
+Import minisearch
+
+Build index
+
+On message from main thread
+  run search
+  find the first 200 unique results from each category, and compute their divs for display
+    note that this is necessary and sufficient information for the main thread to find the
+    first 200 unique results from any given filter set
+  post results to main thread
+
+MAIN:
+
+Launch worker
+
+Declare nonconstant globals (worker_is_running,  last_search_text, unfiltered_results)
+
+On text update
+  if worker is not running, launch_search()
+
+launch_search
+  set worker_is_running to true, set last_search_text to the search text
+  post the search query to worker
+
+on message from worker
+  if last_search_text is not the same as the text in the search field,
+    the latest search result is not reflective of the latest search query, so update again
+    launch_search()
+  otherwise
+    set worker_is_running to false
+
+  regardless, display the new search results to the user
+  save the unfiltered_results as a global
+  update_search()
+
+on filter click
+  adjust the filter selection
+  update_search()
+
+update_search
+  apply search filters by looping through the unfiltered_results and finding the first 200
+    unique results that match the filters
+
+  Update the DOM
+*/
+
+/////// SEARCH WORKER ///////
+
+function worker_function(documenterSearchIndex, documenterBaseURL, filters) {
+  importScripts(
+    "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js"
+  );
+
+  let data = documenterSearchIndex.map((x, key) => {
+    x["id"] = key; // minisearch requires a unique for each object
+    return x;
+  });
+
+  // list below is the lunr 2.1.3 list minus the intersect with names(Base)
+  // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with)
+  // ideally we'd just filter the original list but it's not available as a variable
+  const stopWords = new Set([
+    "a",
+    "able",
+    "about",
+    "across",
+    "after",
+    "almost",
+    "also",
+    "am",
+    "among",
+    "an",
+    "and",
+    "are",
+    "as",
+    "at",
+    "be",
+    "because",
+    "been",
+    "but",
+    "by",
+    "can",
+    "cannot",
+    "could",
+    "dear",
+    "did",
+    "does",
+    "either",
+    "ever",
+    "every",
+    "from",
+    "got",
+    "had",
+    "has",
+    "have",
+    "he",
+    "her",
+    "hers",
+    "him",
+    "his",
+    "how",
+    "however",
+    "i",
+    "if",
+    "into",
+    "it",
+    "its",
+    "just",
+    "least",
+    "like",
+    "likely",
+    "may",
+    "me",
+    "might",
+    "most",
+    "must",
+    "my",
+    "neither",
+    "no",
+    "nor",
+    "not",
+    "of",
+    "off",
+    "often",
+    "on",
+    "or",
+    "other",
+    "our",
+    "own",
+    "rather",
+    "said",
+    "say",
+    "says",
+    "she",
+    "should",
+    "since",
+    "so",
+    "some",
+    "than",
+    "that",
+    "the",
+    "their",
+    "them",
+    "then",
+    "there",
+    "these",
+    "they",
+    "this",
+    "tis",
+    "to",
+    "too",
+    "twas",
+    "us",
+    "wants",
+    "was",
+    "we",
+    "were",
+    "what",
+    "when",
+    "who",
+    "whom",
+    "why",
+    "will",
+    "would",
+    "yet",
+    "you",
+    "your",
+  ]);
+
+  let index = new MiniSearch({
+    fields: ["title", "text"], // fields to index for full-text search
+    storeFields: ["location", "title", "text", "category", "page"], // fields to return with results
+    processTerm: (term) => {
+      let word = stopWords.has(term) ? null : term;
+      if (word) {
+        // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names
+        word = word
+          .replace(/^[^a-zA-Z0-9@!]+/, "")
+          .replace(/[^a-zA-Z0-9@!]+$/, "");
+
+        word = word.toLowerCase();
+      }
+
+      return word ?? null;
+    },
+    // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not
+    // find anything if searching for "add!", only for the entire qualification
+    tokenize: (string) => string.split(/[\s\-\.]+/),
+    // options which will be applied during the search
+    searchOptions: {
+      prefix: true,
+      boost: { title: 100 },
+      fuzzy: 2,
+    },
+  });
+
+  index.addAll(data);
+
+  /**
+   *  Used to map characters to HTML entities.
+   * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts
+   */
+  const htmlEscapes = {
+    "&": "&amp;",
+    "<": "&lt;",
+    ">": "&gt;",
+    '"': "&quot;",
+    "'": "&#39;",
+  };
+
+  /**
+   * Used to match HTML entities and HTML characters.
+   * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts
+   */
+  const reUnescapedHtml = /[&<>"']/g;
+  const reHasUnescapedHtml = RegExp(reUnescapedHtml.source);
+
+  /**
+   * Escape function from lodash
+   * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts
+   */
+  function escape(string) {
+    return string && reHasUnescapedHtml.test(string)
+      ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr])
+      : string || "";
+  }
+
+  /**
+   * Make the result component given a minisearch result data object and the value
+   * of the search input as queryString. To view the result object structure, refer:
+   * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult
+   *
+   * @param {object} result
+   * @param {string} querystring
+   * @returns string
+   */
+  function make_search_result(result, querystring) {
+    let search_divider = `<div class="search-divider w-100"></div>`;
+    let display_link =
+      result.location.slice(Math.max(0), Math.min(50, result.location.length)) +
+      (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div
+
+    if (result.page !== "") {
+      display_link += ` (${result.page})`;
+    }
+
+    let textindex = new RegExp(`${querystring}`, "i").exec(result.text);
+    let text =
+      textindex !== null
+        ? result.text.slice(
+            Math.max(textindex.index - 100, 0),
+            Math.min(
+              textindex.index + querystring.length + 100,
+              result.text.length
+            )
+          )
+        : ""; // cut-off text before and after from the match
+
+    text = text.length ? escape(text) : "";
+
+    let display_result = text.length
+      ? "..." +
+        text.replace(
+          new RegExp(`${escape(querystring)}`, "i"), // For first occurrence
+          '<span class="search-result-highlight py-1">$&</span>'
+        ) +
+        "..."
+      : ""; // highlights the match
+
+    let in_code = false;
+    if (!["page", "section"].includes(result.category.toLowerCase())) {
+      in_code = true;
+    }
+
+    // We encode the full url to escape some special characters which can lead to broken links
+    let result_div = `
+        <a href="${encodeURI(
+          documenterBaseURL + "/" + result.location
+        )}" class="search-result-link w-100 is-flex is-flex-direction-column gap-2 px-4 py-2">
+          <div class="w-100 is-flex is-flex-wrap-wrap is-justify-content-space-between is-align-items-flex-start">
+            <div class="search-result-title has-text-weight-bold ${
+              in_code ? "search-result-code-title" : ""
+            }">${escape(result.title)}</div>
+            <div class="property-search-result-badge">${result.category}</div>
+          </div>
+          <p>
+            ${display_result}
+          </p>
+          <div
+            class="has-text-left"
+            style="font-size: smaller;"
+            title="${result.location}"
+          >
+            <i class="fas fa-link"></i> ${display_link}
+          </div>
+        </a>
+        ${search_divider}
+      `;
+
+    return result_div;
+  }
+
+  self.onmessage = function (e) {
+    let query = e.data;
+    let results = index.search(query, {
+      filter: (result) => {
+        // Only return relevant results
+        return result.score >= 1;
+      },
+    });
+
+    // Pre-filter to deduplicate and limit to 200 per category to the extent
+    // possible without knowing what the filters are.
+    let filtered_results = [];
+    let counts = {};
+    for (let filter of filters) {
+      counts[filter] = 0;
+    }
+    let present = {};
+
+    for (let result of results) {
+      cat = result.category;
+      cnt = counts[cat];
+      if (cnt < 200) {
+        id = cat + "---" + result.location;
+        if (present[id]) {
+          continue;
+        }
+        present[id] = true;
+        filtered_results.push({
+          location: result.location,
+          category: cat,
+          div: make_search_result(result, query),
+        });
+      }
+    }
+
+    postMessage(filtered_results);
+  };
+}
+
+// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript!
+const filters = [
+  ...new Set(documenterSearchIndex["docs"].map((x) => x.category)),
+];
+const worker_str =
+  "(" +
+  worker_function.toString() +
+  ")(" +
+  JSON.stringify(documenterSearchIndex["docs"]) +
+  "," +
+  JSON.stringify(documenterBaseURL) +
+  "," +
+  JSON.stringify(filters) +
+  ")";
+const worker_blob = new Blob([worker_str], { type: "text/javascript" });
+const worker = new Worker(URL.createObjectURL(worker_blob));
+
+/////// SEARCH MAIN ///////
+
+// Whether the worker is currently handling a search. This is a boolean
+// as the worker only ever handles 1 or 0 searches at a time.
+var worker_is_running = false;
+
+// The last search text that was sent to the worker. This is used to determine
+// if the worker should be launched again when it reports back results.
+var last_search_text = "";
+
+// The results of the last search. This, in combination with the state of the filters
+// in the DOM, is used compute the results to display on calls to update_search.
+var unfiltered_results = [];
+
+// Which filter is currently selected
+var selected_filter = "";
+
+$(document).on("input", ".documenter-search-input", function (event) {
+  if (!worker_is_running) {
+    launch_search();
+  }
+});
+
+function launch_search() {
+  worker_is_running = true;
+  last_search_text = $(".documenter-search-input").val();
+  worker.postMessage(last_search_text);
+}
+
+worker.onmessage = function (e) {
+  if (last_search_text !== $(".documenter-search-input").val()) {
+    launch_search();
+  } else {
+    worker_is_running = false;
+  }
+
+  unfiltered_results = e.data;
+  update_search();
+};
+
+$(document).on("click", ".search-filter", function () {
+  if ($(this).hasClass("search-filter-selected")) {
+    selected_filter = "";
+  } else {
+    selected_filter = $(this).text().toLowerCase();
+  }
+
+  // This updates search results and toggles classes for UI:
+  update_search();
+});
+
+/**
+ * Make/Update the search component
+ */
+function update_search() {
+  let querystring = $(".documenter-search-input").val();
+
+  if (querystring.trim()) {
+    if (selected_filter == "") {
+      results = unfiltered_results;
+    } else {
+      results = unfiltered_results.filter((result) => {
+        return selected_filter == result.category.toLowerCase();
+      });
+    }
+
+    let search_result_container = ``;
+    let modal_filters = make_modal_body_filters();
+    let search_divider = `<div class="search-divider w-100"></div>`;
+
+    if (results.length) {
+      let links = [];
+      let count = 0;
+      let search_results = "";
+
+      for (var i = 0, n = results.length; i < n && count < 200; ++i) {
+        let result = results[i];
+        if (result.location && !links.includes(result.location)) {
+          search_results += result.div;
+          count++;
+          links.push(result.location);
+        }
+      }
+
+      if (count == 1) {
+        count_str = "1 result";
+      } else if (count == 200) {
+        count_str = "200+ results";
+      } else {
+        count_str = count + " results";
+      }
+      let result_count = `<div class="is-size-6">${count_str}</div>`;
+
+      search_result_container = `
+            <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
+                ${modal_filters}
+                ${search_divider}
+                ${result_count}
+                <div class="is-clipped w-100 is-flex is-flex-direction-column gap-2 is-align-items-flex-start has-text-justified mt-1">
+                  ${search_results}
+                </div>
+            </div>
+        `;
+    } else {
+      search_result_container = `
+           <div class="is-flex is-flex-direction-column gap-2 is-align-items-flex-start">
+               ${modal_filters}
+               ${search_divider}
+               <div class="is-size-6">0 result(s)</div>
+            </div>
+            <div class="has-text-centered my-5 py-5">No result found!</div>
+       `;
+    }
+
+    if ($(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").removeClass("is-justify-content-center");
+    }
+
+    $(".search-modal-card-body").html(search_result_container);
+  } else {
+    if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").addClass("is-justify-content-center");
+    }
+
+    $(".search-modal-card-body").html(`
+      <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+    `);
+  }
+}
+
+/**
+ * Make the modal filter html
+ *
+ * @returns string
+ */
+function make_modal_body_filters() {
+  let str = filters
+    .map((val) => {
+      if (selected_filter == val.toLowerCase()) {
+        return `<a href="javascript:;" class="search-filter search-filter-selected"><span>${val}</span></a>`;
+      } else {
+        return `<a href="javascript:;" class="search-filter"><span>${val}</span></a>`;
+      }
+    })
+    .join("");
+
+  return `
+        <div class="is-flex gap-2 is-flex-wrap-wrap is-justify-content-flex-start is-align-items-center search-filters">
+            <span class="is-size-6">Filters:</span>
+            ${str}
+        </div>`;
+}
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Modal settings dialog
+$(document).ready(function () {
+  var settings = $("#documenter-settings");
+  $("#documenter-settings-button").click(function () {
+    settings.toggleClass("is-active");
+  });
+  // Close the dialog if X is clicked
+  $("#documenter-settings button.delete").click(function () {
+    settings.removeClass("is-active");
+  });
+  // Close dialog if ESC is pressed
+  $(document).keyup(function (e) {
+    if (e.keyCode == 27) settings.removeClass("is-active");
+  });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+$(document).ready(function () {
+  let search_modal_header = `
+    <header class="modal-card-head gap-2 is-align-items-center is-justify-content-space-between w-100 px-3">
+      <div class="field mb-0 w-100">
+        <p class="control has-icons-right">
+          <input class="input documenter-search-input" type="text" placeholder="Search" />
+          <span class="icon is-small is-right has-text-primary-dark">
+            <i class="fas fa-magnifying-glass"></i>
+          </span>
+        </p>
+      </div>
+      <div class="icon is-size-4 is-clickable close-search-modal">
+        <i class="fas fa-times"></i>
+      </div>
+    </header>
+  `;
+
+  let initial_search_body = `
+    <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+  `;
+
+  let search_modal_footer = `
+    <footer class="modal-card-foot">
+      <span>
+        <kbd class="search-modal-key-hints">Ctrl</kbd> +
+        <kbd class="search-modal-key-hints">/</kbd> to search
+      </span>
+      <span class="ml-3"> <kbd class="search-modal-key-hints">esc</kbd> to close </span>
+    </footer>
+  `;
+
+  $(document.body).append(
+    `
+      <div class="modal" id="search-modal">
+        <div class="modal-background"></div>
+        <div class="modal-card search-min-width-50 search-min-height-100 is-justify-content-center">
+          ${search_modal_header}
+          <section class="modal-card-body is-flex is-flex-direction-column is-justify-content-center gap-4 search-modal-card-body">
+            ${initial_search_body}
+          </section>
+          ${search_modal_footer}
+        </div>
+      </div>
+    `
+  );
+
+  document.querySelector(".docs-search-query").addEventListener("click", () => {
+    openModal();
+  });
+
+  document
+    .querySelector(".close-search-modal")
+    .addEventListener("click", () => {
+      closeModal();
+    });
+
+  $(document).on("click", ".search-result-link", function () {
+    closeModal();
+  });
+
+  document.addEventListener("keydown", (event) => {
+    if ((event.ctrlKey || event.metaKey) && event.key === "/") {
+      openModal();
+    } else if (event.key === "Escape") {
+      closeModal();
+    }
+
+    return false;
+  });
+
+  // Functions to open and close a modal
+  function openModal() {
+    let searchModal = document.querySelector("#search-modal");
+
+    searchModal.classList.add("is-active");
+    document.querySelector(".documenter-search-input").focus();
+  }
+
+  function closeModal() {
+    let searchModal = document.querySelector("#search-modal");
+    let initial_search_body = `
+      <div class="has-text-centered my-5 py-5">Type something to get started!</div>
+    `;
+
+    searchModal.classList.remove("is-active");
+    document.querySelector(".documenter-search-input").blur();
+
+    if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) {
+      $(".search-modal-card-body").addClass("is-justify-content-center");
+    }
+
+    $(".documenter-search-input").val("");
+    $(".search-modal-card-body").html(initial_search_body);
+  }
+
+  document
+    .querySelector("#search-modal .modal-background")
+    .addEventListener("click", () => {
+      closeModal();
+    });
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Manages the showing and hiding of the sidebar.
+$(document).ready(function () {
+  var sidebar = $("#documenter > .docs-sidebar");
+  var sidebar_button = $("#documenter-sidebar-button");
+  sidebar_button.click(function (ev) {
+    ev.preventDefault();
+    sidebar.toggleClass("visible");
+    if (sidebar.hasClass("visible")) {
+      // Makes sure that the current menu item is visible in the sidebar.
+      $("#documenter .docs-menu a.is-active").focus();
+    }
+  });
+  $("#documenter > .docs-main").bind("click", function (ev) {
+    if ($(ev.target).is(sidebar_button)) {
+      return;
+    }
+    if (sidebar.hasClass("visible")) {
+      sidebar.removeClass("visible");
+    }
+  });
+});
+
+// Resizes the package name / sitename in the sidebar if it is too wide.
+// Inspired by: https://github.com/davatron5000/FitText.js
+$(document).ready(function () {
+  e = $("#documenter .docs-autofit");
+  function resize() {
+    var L = parseInt(e.css("max-width"), 10);
+    var L0 = e.width();
+    if (L0 > L) {
+      var h0 = parseInt(e.css("font-size"), 10);
+      e.css("font-size", (L * h0) / L0);
+      // TODO: make sure it survives resizes?
+    }
+  }
+  // call once and then register events
+  resize();
+  $(window).resize(resize);
+  $(window).on("orientationchange", resize);
+});
+
+// Scroll the navigation bar to the currently selected menu item
+$(document).ready(function () {
+  var sidebar = $("#documenter .docs-menu").get(0);
+  var active = $("#documenter .docs-menu .is-active").get(0);
+  if (typeof active !== "undefined") {
+    sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15;
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// Theme picker setup
+$(document).ready(function () {
+  // onchange callback
+  $("#documenter-themepicker").change(function themepick_callback(ev) {
+    var themename = $("#documenter-themepicker option:selected").attr("value");
+    if (themename === "auto") {
+      // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light');
+      window.localStorage.removeItem("documenter-theme");
+    } else {
+      // set_theme(themename);
+      window.localStorage.setItem("documenter-theme", themename);
+    }
+    // We re-use the global function from themeswap.js to actually do the swapping.
+    set_theme_from_local_storage();
+  });
+
+  // Make sure that the themepicker displays the correct theme when the theme is retrieved
+  // from localStorage
+  if (typeof window.localStorage !== "undefined") {
+    var theme = window.localStorage.getItem("documenter-theme");
+    if (theme !== null) {
+      $("#documenter-themepicker option").each(function (i, e) {
+        e.selected = e.value === theme;
+      });
+    }
+  }
+});
+
+})
+////////////////////////////////////////////////////////////////////////////////
+require(['jquery'], function($) {
+
+// update the version selector with info from the siteinfo.js and ../versions.js files
+$(document).ready(function () {
+  // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the
+  // siteinfo.js file, we just return immediately and not display the version selector.
+  if (
+    typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" &&
+    DOCUMENTER_VERSION_SELECTOR_DISABLED
+  ) {
+    return;
+  }
+
+  var version_selector = $("#documenter .docs-version-selector");
+  var version_selector_select = $("#documenter .docs-version-selector select");
+
+  version_selector_select.change(function (x) {
+    target_href = version_selector_select
+      .children("option:selected")
+      .get(0).value;
+    window.location.href = target_href;
+  });
+
+  // add the current version to the selector based on siteinfo.js, but only if the selector is empty
+  if (
+    typeof DOCUMENTER_CURRENT_VERSION !== "undefined" &&
+    $("#version-selector > option").length == 0
+  ) {
+    var option = $(
+      "<option value='#' selected='selected'>" +
+        DOCUMENTER_CURRENT_VERSION +
+        "</option>"
+    );
+    version_selector_select.append(option);
+  }
+
+  if (typeof DOC_VERSIONS !== "undefined") {
+    var existing_versions = version_selector_select.children("option");
+    var existing_versions_texts = existing_versions.map(function (i, x) {
+      return x.text;
+    });
+    DOC_VERSIONS.forEach(function (each) {
+      var version_url = documenterBaseURL + "/../" + each + "/";
+      var existing_id = $.inArray(each, existing_versions_texts);
+      // if not already in the version selector, add it as a new option,
+      // otherwise update the old option with the URL and enable it
+      if (existing_id == -1) {
+        var option = $(
+          "<option value='" + version_url + "'>" + each + "</option>"
+        );
+        version_selector_select.append(option);
+      } else {
+        var option = existing_versions[existing_id];
+        option.value = version_url;
+        option.disabled = false;
+      }
+    });
+  }
+
+  // only show the version selector if the selector has been populated
+  if (version_selector_select.children("option").length > 0) {
+    version_selector.toggleClass("visible");
+  }
+});
+
+})
diff --git a/dev/assets/themes/documenter-dark.css b/dev/assets/themes/documenter-dark.css
new file mode 100644
index 0000000..1d71701
--- /dev/null
+++ b/dev/assets/themes/documenter-dark.css
@@ -0,0 +1,7 @@
+html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark .file-cta[disabled],html.theme--documenter-dark .file-name[disabled],html.theme--documenter-dark .select select[disabled],html.theme--documenter-dark .textarea[disabled],html.theme--documenter-dark .input[disabled],html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input[disabled],html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark fieldset[disabled] .pagination-previous,fieldset[disabled] html.theme--documenter-dark .pagination-next,html.theme--documenter-dark fieldset[disabled] .pagination-next,fieldset[disabled] html.theme--documenter-dark .pagination-link,html.theme--documenter-dark fieldset[disabled] .pagination-link,fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis,fieldset[disabled] html.theme--documenter-dark .file-cta,html.theme--documenter-dark fieldset[disabled] .file-cta,fieldset[disabled] html.theme--documenter-dark .file-name,html.theme--documenter-dark fieldset[disabled] .file-name,fieldset[disabled] html.theme--documenter-dark .select select,fieldset[disabled] html.theme--documenter-dark .textarea,fieldset[disabled] html.theme--documenter-dark .input,fieldset[disabled] html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark fieldset[disabled] .select select,html.theme--documenter-dark .select fieldset[disabled] select,html.theme--documenter-dark fieldset[disabled] .textarea,html.theme--documenter-dark fieldset[disabled] .input,html.theme--documenter-dark fieldset[disabled] #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark #documenter .docs-sidebar fieldset[disabled] form.docs-search>input,fieldset[disabled] html.theme--documenter-dark .button,html.theme--documenter-dark fieldset[disabled] .button{cursor:not-allowed}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" ";display:block;height:0.625em;margin-top:-0.4375em;pointer-events:none;position:absolute;top:50%;transform:rotate(-45deg);transform-origin:center;width:0.625em}html.theme--documenter-dark .admonition:not(:last-child),html.theme--documenter-dark .tabs:not(:last-child),html.theme--documenter-dark .pagination:not(:last-child),html.theme--documenter-dark .message:not(:last-child),html.theme--documenter-dark .level:not(:last-child),html.theme--documenter-dark .breadcrumb:not(:last-child),html.theme--documenter-dark .block:not(:last-child),html.theme--documenter-dark .title:not(:last-child),html.theme--documenter-dark .subtitle:not(:last-child),html.theme--documenter-dark .table-container:not(:last-child),html.theme--documenter-dark .table:not(:last-child),html.theme--documenter-dark .progress:not(:last-child),html.theme--documenter-dark .notification:not(:last-child),html.theme--documenter-dark .content:not(:last-child),html.theme--documenter-dark .box:not(:last-child){margin-bottom:1.5rem}html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;-moz-appearance:none;-webkit-appearance:none;background-color:rgba(10,10,10,0.2);border:none;border-radius:9999px;cursor:pointer;pointer-events:auto;display:inline-block;flex-grow:0;flex-shrink:0;font-size:0;height:20px;max-height:20px;max-width:20px;min-height:20px;min-width:20px;outline:none;position:relative;vertical-align:top;width:20px}html.theme--documenter-dark .modal-close::before,html.theme--documenter-dark .delete::before,html.theme--documenter-dark .modal-close::after,html.theme--documenter-dark .delete::after{background-color:#fff;content:"";display:block;left:50%;position:absolute;top:50%;transform:translateX(-50%) translateY(-50%) rotate(45deg);transform-origin:center center}html.theme--documenter-dark .modal-close::before,html.theme--documenter-dark .delete::before{height:2px;width:50%}html.theme--documenter-dark .modal-close::after,html.theme--documenter-dark .delete::after{height:50%;width:2px}html.theme--documenter-dark .modal-close:hover,html.theme--documenter-dark .delete:hover,html.theme--documenter-dark .modal-close:focus,html.theme--documenter-dark .delete:focus{background-color:rgba(10,10,10,0.3)}html.theme--documenter-dark .modal-close:active,html.theme--documenter-dark .delete:active{background-color:rgba(10,10,10,0.4)}html.theme--documenter-dark .is-small.modal-close,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.modal-close,html.theme--documenter-dark .is-small.delete,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.delete{height:16px;max-height:16px;max-width:16px;min-height:16px;min-width:16px;width:16px}html.theme--documenter-dark .is-medium.modal-close,html.theme--documenter-dark .is-medium.delete{height:24px;max-height:24px;max-width:24px;min-height:24px;min-width:24px;width:24px}html.theme--documenter-dark .is-large.modal-close,html.theme--documenter-dark .is-large.delete{height:32px;max-height:32px;max-width:32px;min-height:32px;min-width:32px;width:32px}html.theme--documenter-dark .control.is-loading::after,html.theme--documenter-dark .select.is-loading::after,html.theme--documenter-dark .loader,html.theme--documenter-dark .button.is-loading::after{animation:spinAround 500ms infinite linear;border:2px solid 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+  Author: @ericwbailey
+  Maintainer: @ericwbailey
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.button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter 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#documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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diff --git a/dev/assets/themes/documenter-light.css b/dev/assets/themes/documenter-light.css
new file mode 100644
index 0000000..07f9d08
--- /dev/null
+++ b/dev/assets/themes/documenter-light.css
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.filter-tabs::-webkit-scrollbar-thumb{background-color:gray;border-radius:1rem}#search-modal .modal-card-body::-webkit-scrollbar-track,#search-modal .filter-tabs::-webkit-scrollbar-track{-webkit-box-shadow:inset 0 0 6px rgba(0,0,0,0.6);background-color:transparent}.w-100{width:100%}.gap-2{gap:0.5rem}.gap-4{gap:1rem}.gap-8{gap:2rem}.ansi span.sgr1{font-weight:bolder}.ansi span.sgr2{font-weight:lighter}.ansi span.sgr3{font-style:italic}.ansi span.sgr4{text-decoration:underline}.ansi span.sgr7{color:#fff;background-color:#222}.ansi span.sgr8{color:transparent}.ansi span.sgr8 span{color:transparent}.ansi span.sgr9{text-decoration:line-through}.ansi span.sgr30{color:#242424}.ansi span.sgr31{color:#a7201f}.ansi span.sgr32{color:#066f00}.ansi span.sgr33{color:#856b00}.ansi span.sgr34{color:#2149b0}.ansi span.sgr35{color:#7d4498}.ansi span.sgr36{color:#007989}.ansi span.sgr37{color:gray}.ansi span.sgr40{background-color:#242424}.ansi span.sgr41{background-color:#a7201f}.ansi span.sgr42{background-color:#066f00}.ansi span.sgr43{background-color:#856b00}.ansi span.sgr44{background-color:#2149b0}.ansi span.sgr45{background-color:#7d4498}.ansi span.sgr46{background-color:#007989}.ansi span.sgr47{background-color:gray}.ansi span.sgr90{color:#616161}.ansi span.sgr91{color:#cb3c33}.ansi span.sgr92{color:#0e8300}.ansi span.sgr93{color:#a98800}.ansi span.sgr94{color:#3c5dcd}.ansi span.sgr95{color:#9256af}.ansi span.sgr96{color:#008fa3}.ansi span.sgr97{color:#f5f5f5}.ansi span.sgr100{background-color:#616161}.ansi span.sgr101{background-color:#cb3c33}.ansi span.sgr102{background-color:#0e8300}.ansi span.sgr103{background-color:#a98800}.ansi span.sgr104{background-color:#3c5dcd}.ansi span.sgr105{background-color:#9256af}.ansi span.sgr106{background-color:#008fa3}.ansi span.sgr107{background-color:#f5f5f5}code.language-julia-repl>span.hljs-meta{color:#066f00;font-weight:bolder}/*!
+  Theme: Default
+  Description: Original highlight.js style
+  Author: (c) Ivan Sagalaev <maniac@softwaremaniacs.org>
+  Maintainer: @highlightjs/core-team
+  Website: https://highlightjs.org/
+  License: see project LICENSE
+  Touched: 2021
+*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem}
diff --git a/dev/assets/themeswap.js b/dev/assets/themeswap.js
new file mode 100644
index 0000000..9f5eebe
--- /dev/null
+++ b/dev/assets/themeswap.js
@@ -0,0 +1,84 @@
+// Small function to quickly swap out themes. Gets put into the <head> tag..
+function set_theme_from_local_storage() {
+  // Initialize the theme to null, which means default
+  var theme = null;
+  // If the browser supports the localstorage and is not disabled then try to get the
+  // documenter theme
+  if (window.localStorage != null) {
+    // Get the user-picked theme from localStorage. May be `null`, which means the default
+    // theme.
+    theme = window.localStorage.getItem("documenter-theme");
+  }
+  // Check if the users preference is for dark color scheme
+  var darkPreference =
+    window.matchMedia("(prefers-color-scheme: dark)").matches === true;
+  // Initialize a few variables for the loop:
+  //
+  //  - active: will contain the index of the theme that should be active. Note that there
+  //    is no guarantee that localStorage contains sane values. If `active` stays `null`
+  //    we either could not find the theme or it is the default (primary) theme anyway.
+  //    Either way, we then need to stick to the primary theme.
+  //
+  //  - disabled: style sheets that should be disabled (i.e. all the theme style sheets
+  //    that are not the currently active theme)
+  var active = null;
+  var disabled = [];
+  var primaryLightTheme = null;
+  var primaryDarkTheme = null;
+  for (var i = 0; i < document.styleSheets.length; i++) {
+    var ss = document.styleSheets[i];
+    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
+    // which must contain the name of the theme. The names in localStorage much match this.
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    // attribute not set => non-theme stylesheet => ignore
+    if (themename === null) continue;
+    // To distinguish the default (primary) theme, it needs to have the data-theme-primary
+    // attribute set.
+    if (ss.ownerNode.getAttribute("data-theme-primary") !== null) {
+      primaryLightTheme = themename;
+    }
+    // Check if the theme is primary dark theme so that we could store its name in darkTheme
+    if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) {
+      primaryDarkTheme = themename;
+    }
+    // If we find a matching theme (and it's not the default), we'll set active to non-null
+    if (themename === theme) active = i;
+    // Store the style sheets of inactive themes so that we could disable them
+    if (themename !== theme) disabled.push(ss);
+  }
+  var activeTheme = null;
+  if (active !== null) {
+    // If we did find an active theme, we'll (1) add the theme--$(theme) class to <html>
+    document.getElementsByTagName("html")[0].className = "theme--" + theme;
+    activeTheme = theme;
+  } else {
+    // If we did _not_ find an active theme, then we need to fall back to the primary theme
+    // which can either be dark or light, depending on the user's OS preference.
+    var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme;
+    // In case it somehow happens that the relevant primary theme was not found in the
+    // preceding loop, we abort without doing anything.
+    if (activeTheme === null) {
+      console.error("Unable to determine primary theme.");
+      return;
+    }
+    // When switching to the primary light theme, then we must not have a class name
+    // for the <html> tag. That's only for non-primary or the primary dark theme.
+    if (darkPreference) {
+      document.getElementsByTagName("html")[0].className =
+        "theme--" + activeTheme;
+    } else {
+      document.getElementsByTagName("html")[0].className = "";
+    }
+  }
+  for (var i = 0; i < document.styleSheets.length; i++) {
+    var ss = document.styleSheets[i];
+    // The <link> tag of each style sheet is expected to have a data-theme-name attribute
+    // which must contain the name of the theme. The names in localStorage much match this.
+    var themename = ss.ownerNode.getAttribute("data-theme-name");
+    // attribute not set => non-theme stylesheet => ignore
+    if (themename === null) continue;
+    // we'll disable all the stylesheets, except for the active one
+    ss.disabled = !(themename == activeTheme);
+  }
+}
+set_theme_from_local_storage();
diff --git a/dev/assets/warner.js b/dev/assets/warner.js
new file mode 100644
index 0000000..3f6f5d0
--- /dev/null
+++ b/dev/assets/warner.js
@@ -0,0 +1,52 @@
+function maybeAddWarning() {
+  // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE
+  // in siteinfo.js.
+  // If either of these are undefined something went horribly wrong, so we abort.
+  if (
+    window.DOCUMENTER_NEWEST === undefined ||
+    window.DOCUMENTER_CURRENT_VERSION === undefined ||
+    window.DOCUMENTER_STABLE === undefined
+  ) {
+    return;
+  }
+
+  // Current version is not a version number, so we can't tell if it's the newest version. Abort.
+  if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) {
+    return;
+  }
+
+  // Current version is newest version, so no need to add a warning.
+  if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) {
+    return;
+  }
+
+  // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs.
+  if (document.body.querySelector('meta[name="robots"]') === null) {
+    const meta = document.createElement("meta");
+    meta.name = "robots";
+    meta.content = "noindex";
+
+    document.getElementsByTagName("head")[0].appendChild(meta);
+  }
+
+  const div = document.createElement("div");
+  div.classList.add("outdated-warning-overlay");
+  const closer = document.createElement("button");
+  closer.classList.add("outdated-warning-closer", "delete");
+  closer.addEventListener("click", function () {
+    document.body.removeChild(div);
+  });
+  const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE;
+  div.innerHTML =
+    'This documentation is not for the latest stable release, but for either the development version or an older release.<br><a href="' +
+    href +
+    '">Click here to go to the documentation for the latest stable release.</a>';
+  div.appendChild(closer);
+  document.body.appendChild(div);
+}
+
+if (document.readyState === "loading") {
+  document.addEventListener("DOMContentLoaded", maybeAddWarning);
+} else {
+  maybeAddWarning();
+}
diff --git a/dev/checkmatrix/index.html b/dev/checkmatrix/index.html
new file mode 100644
index 0000000..5bfbe3d
--- /dev/null
+++ b/dev/checkmatrix/index.html
@@ -0,0 +1,44 @@
+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Details on check matrix operations · QuantumLegos.jl</title><meta name="title" content="Details on check matrix operations · QuantumLegos.jl"/><meta property="og:title" content="Details on check matrix operations · QuantumLegos.jl"/><meta property="twitter:title" content="Details on check matrix operations · QuantumLegos.jl"/><meta name="description" content="Documentation for QuantumLegos.jl."/><meta property="og:description" content="Documentation for QuantumLegos.jl."/><meta property="twitter:description" content="Documentation for QuantumLegos.jl."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link 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type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">QuantumLegos.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../pauliops/">PauliOps</a></li><li class="is-active"><a class="tocitem" href>Details on check matrix operations</a><ul class="internal"><li><a class="tocitem" href="#Basic-features-of-check-matrix"><span>Basic features of check matrix</span></a></li><li><a class="tocitem" href="#Construction-of-check-matrices-and-retrieving-features"><span>Construction of check matrices and retrieving features</span></a></li><li><a class="tocitem" href="#Direct-sum"><span>Direct sum</span></a></li><li><a class="tocitem" href="#Self-tracing"><span>Self-tracing</span></a></li></ul></li><li><a class="tocitem" href="../distance/">How to calculate code distance from the state.</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Details on check matrix operations</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Details on check matrix operations</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl/blob/main/docs/src/checkmatrix.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Details-on-check-matrix-operations"><a class="docs-heading-anchor" href="#Details-on-check-matrix-operations">Details on check matrix operations</a><a id="Details-on-check-matrix-operations-1"></a><a class="docs-heading-anchor-permalink" href="#Details-on-check-matrix-operations" title="Permalink"></a></h1><h2 id="Basic-features-of-check-matrix"><a class="docs-heading-anchor" href="#Basic-features-of-check-matrix">Basic features of check matrix</a><a id="Basic-features-of-check-matrix-1"></a><a class="docs-heading-anchor-permalink" href="#Basic-features-of-check-matrix" title="Permalink"></a></h2><ul><li>All action on state can be represented as operations on check matrix.</li><li>QuantumLegos which don&#39;t connect each other are represented by block diagonal check matrix.</li></ul><p>Thus, one needs only one check matrix to represent state.</p><ul><li>Generated group from generators represented by check matrix is invariant under row-swap and row-multiplying transforms on the check matrix.</li></ul><p>Thus, one can freely perform these row operations on check matrices.</p><h2 id="Construction-of-check-matrices-and-retrieving-features"><a class="docs-heading-anchor" href="#Construction-of-check-matrices-and-retrieving-features">Construction of check matrices and retrieving features</a><a id="Construction-of-check-matrices-and-retrieving-features-1"></a><a class="docs-heading-anchor-permalink" href="#Construction-of-check-matrices-and-retrieving-features" title="Permalink"></a></h2><p>Use <a href="#QuantumLegos.CheckMatrix"><code>CheckMatrix</code></a> from <code>Matrix{Bool}</code> and <a href="#QuantumLegos.checkmatrix"><code>checkmatrix</code></a> from <code>Vector{PauliOp}</code>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.CheckMatrix" href="#QuantumLegos.CheckMatrix"><code>QuantumLegos.CheckMatrix</code></a> — <span class="docstring-category">Type</span></header><section><div><p>CheckMatrix</p><p><strong>Fields</strong></p><ul><li><code>cmat</code>: matrix itself (matrix of <code>Bool</code> since we consider only pauli ops.)</li><li><code>nlegs</code>: number of columns ÷ 2</li><li><code>ngens</code>: number of rows</li></ul><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">CheckMatrix(cmat::AbstractMatrix{Bool})</code></pre><p>See also <a href="#QuantumLegos.checkmatrix"><code>checkmatrix</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L1-L12">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.checkmatrix" href="#QuantumLegos.checkmatrix"><code>QuantumLegos.checkmatrix</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">checkmatrix(stbgens::AbstractVector{T})::CheckMatrix where {T &lt;: PauliOp}</code></pre><p>Get <a href="#QuantumLegos.CheckMatrix"><code>CheckMatrix</code></a> from stabilizer generator.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L78-L82">source</a></section></article><p>Get xpart and zpart of check matrix.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.xpart" href="#QuantumLegos.xpart"><code>QuantumLegos.xpart</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">xpart(cmat::CheckMatrix)</code></pre><p>Get X part (left half) of CheckMatrix.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L110-L114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.zpart" href="#QuantumLegos.zpart"><code>QuantumLegos.zpart</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">zpart(cmat::CheckMatrix)</code></pre><p>Get Z part (right half) of CheckMatrix.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L119-L123">source</a></section></article><p>Get generators from check matrix with <a href="#QuantumLegos.generators"><code>generators</code></a>.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.generators" href="#QuantumLegos.generators"><code>QuantumLegos.generators</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">generators(cmat::CheckMatrix)::Vector{PauliOp}</code></pre><p>Get generators from <a href="#QuantumLegos.CheckMatrix"><code>CheckMatrix</code></a>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L132-L136">source</a></section></article><p>See also, <a href="../pauliops/#QuantumLegos.PauliOps.GeneratedPauliGroup"><code>GeneratedPauliGroup</code></a> and <a href="https://docs.julialang.org/en/v1/base/collections/#Base.Set"><code>Base.Set</code></a> to get group generated by generators.</p><h2 id="Direct-sum"><a class="docs-heading-anchor" href="#Direct-sum">Direct sum</a><a id="Direct-sum-1"></a><a class="docs-heading-anchor-permalink" href="#Direct-sum" title="Permalink"></a></h2><p>Used when adding lego without edge.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.direct_sum" href="#QuantumLegos.direct_sum"><code>QuantumLegos.direct_sum</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">direct_sum(cmat_1::T, cmat_2::T)::T where {T &lt;: CheckMatrix}</code></pre><p>Returns block diagonal <code>CheckMatrix</code> consists of <code>cmat_1</code> and <code>cmat_2</code>.</p><p><strong>CheckMatrix transform</strong></p><p>When adding a lego <span>$l$</span> with check matrix <span>$H_l$</span> to a state with check matrix <span>$H_s$</span>, the resulting check matrix of the state will be</p><p class="math-container">\[\begin{pmatrix}
+    H_s &amp; O   \\
+    O   &amp; H_l \\
+\end{pmatrix}\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L151-L165">source</a></section></article><h2 id="Self-tracing"><a class="docs-heading-anchor" href="#Self-tracing">Self-tracing</a><a id="Self-tracing-1"></a><a class="docs-heading-anchor-permalink" href="#Self-tracing" title="Permalink"></a></h2><h3 id="self-tracing"><a class="docs-heading-anchor" href="#self-tracing">self-tracing</a><a id="self-tracing-1"></a><a class="docs-heading-anchor-permalink" href="#self-tracing" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.self_trace!" href="#QuantumLegos.self_trace!"><code>QuantumLegos.self_trace!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">self_trace!(cmat::CheckMatrix, col_1::Integer, col_2::Integer)</code></pre><p>Take a self-trace of checkmatrix <code>cmat</code> with column <code>col_1</code> and <code>col_2</code>.</p><p><strong>Example</strong></p><p>TODO</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L424-L431">source</a></section></article><p>During self-tracing, pre-formatting and post-formatting described below is performed.</p><h3 id="pre-formatting"><a class="docs-heading-anchor" href="#pre-formatting">pre-formatting</a><a id="pre-formatting-1"></a><a class="docs-heading-anchor-permalink" href="#pre-formatting" title="Permalink"></a></h3><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.eliminate_column!" href="#QuantumLegos.eliminate_column!"><code>QuantumLegos.eliminate_column!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eliminate_column!(
+    cmat::CheckMatrix,
+    col::Integer,
+    avoid_row::AbstractVector{T},
+)::Union{Nothing, Int64} where {T &lt;: Integer}</code></pre><p>Perform Gauss Elimination on <code>col</code>. Keep only one <code>1</code> on <code>col</code> and remove from other rows by performing multiplication. Choose row not in <code>avoid_row</code>. If all rows with 1 on <code>col</code> are in <code>avoid_row</code>, then the last row of them is chose to keep 1. If all rows on <code>col</code> is 0, return nothing.</p><p><strong>Return</strong></p><p>Row index which have 1 on <code>col</code>. If all row on <code>col</code> is 0, return nothing.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; ex_cmat = CheckMatrix(Bool[1 0 1 0; 1 1 1 1])
+CheckMatrix with 2 generators, 2 legs:
+  1  0 | 1  0
+  1  1 | 1  1
+
+julia&gt; QuantumLegos.eliminate_column!(ex_cmat, 1, Int64[])
+1
+
+julia&gt; ex_cmat
+CheckMatrix with 2 generators, 2 legs:
+  1  0 | 1  0
+  0  1 | 0  1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L175-L207">source</a></section><section><div><pre><code class="language-julia hljs">eliminate_column!(cmat::CheckMatrix, col::Integer,
+avoid_row::AbstractVector{T}) where {T &lt;: Union{Nothing, Integer}}</code></pre><p><code>avoid_row</code> can include <code>Nothing</code>, which is ignored in actual evaluation.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L246-L251">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.align_row!" href="#QuantumLegos.align_row!"><code>QuantumLegos.align_row!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">align_row!(m::AbstractMatrix, row::Integer, occupied::Vector{Union{Nothing, Integer}}) -&gt; Integer
+align_row!(m::AbstractMatrix, row::Nothing, occupied::Vector{Union{Nothing, Integer}}) -&gt; Nothing</code></pre><p>Swap row at <code>row</code> in <code>m</code> and row at next to the maximum in <code>occupied</code>. <code>occupied</code> is supposed to be a list of returns from <a href="#QuantumLegos.eliminate_column!"><code>eliminate_column!</code></a>. If <code>row</code> is in <code>occupied</code>, do nothing and returns <code>row</code>. If <code>row</code> is <code>nothing</code>, return <code>nothing</code>.</p><p><strong>Arguments</strong></p><ul><li><code>m::AbstractMatrix</code>: mutated</li><li><code>row::Union{Nothing, Integer}</code>: row to be aligned</li><li><code>occupied::Vector{Union{Nothing, Integer}}</code>: indices of already occupied rows. <code>row</code> will be next to these rows.</li></ul><p><strong>Return</strong></p><ul><li>Row index where <code>row</code> is moved.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L265-L281">source</a></section></article><p><a href="#QuantumLegos.align_row!"><code>QuantumLegos.align_row!</code></a></p><p>By this process, all columns to be traced have <code>1</code>s on only 1st to 4th rows. So during <a href="#QuantumLegos.self_trace!"><code>QuantumLegos.self_trace!</code></a>, all stabilizers generated from top three rows are calculated and perform operator matching.</p><h3 id="post-formatting"><a class="docs-heading-anchor" href="#post-formatting">post-formatting</a><a id="post-formatting-1"></a><a class="docs-heading-anchor-permalink" href="#post-formatting" title="Permalink"></a></h3><p>Remove rows which are linear combinations of other rows.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.ref!" href="#QuantumLegos.ref!"><code>QuantumLegos.ref!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">ref!(cmat::CheckMatrix) -&gt; Int</code></pre><p>Convert <code>cmat</code> to row echelon form.</p><p>Returns rank of check matrix.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; cmat = CheckMatrix(Bool[
+           1 0 1 0 1 1 0 1
+           0 1 0 0 0 1 0 0
+           1 1 1 0 1 0 1 1
+           0 1 0 0 0 1 0 0
+       ])
+CheckMatrix with 4 generators, 4 legs:
+  1  0  1  0 | 1  1  0  1
+  0  1  0  0 | 0  1  0  0
+  1  1  1  0 | 1  0  1  1
+  0  1  0  0 | 0  1  0  0
+
+
+julia&gt; QuantumLegos.ref!(cmat)
+3
+
+julia&gt; cmat
+CheckMatrix with 4 generators, 4 legs:
+  1  0  1  0 | 1  1  0  1
+  0  1  0  0 | 0  1  0  0
+  0  0  0  0 | 0  0  1  0
+  0  0  0  0 | 0  0  0  0
+</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L319-L352">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.eliminate_dependent_row!" href="#QuantumLegos.eliminate_dependent_row!"><code>QuantumLegos.eliminate_dependent_row!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">eliminate_dependent_row!(cmat::CheckMatrix) -&gt; CheckMatrix</code></pre><p>Remove dependent rows to keep only independent generators.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/checkmatrix.jl#L407-L411">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../pauliops/">« PauliOps</a><a class="docs-footer-nextpage" href="../distance/">How to calculate code distance from the state. »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Tuesday 7 May 2024 02:47">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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of code distance.</span></a></li><li><a class="tocitem" href="#Classification-of-the-stabilizers."><span>Classification of the stabilizers.</span></a></li><li><a class="tocitem" href="#Calculating-code-distance-from-the-check-matrix."><span>Calculating code distance from the check matrix.</span></a></li><li><a class="tocitem" href="#Examples"><span>Examples</span></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>How to calculate code distance from the state.</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>How to calculate code distance from the state.</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl/blob/main/docs/src/distance.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="How-to-calculate-code-distance-from-the-state."><a class="docs-heading-anchor" href="#How-to-calculate-code-distance-from-the-state.">How to calculate code distance from the state.</a><a id="How-to-calculate-code-distance-from-the-state.-1"></a><a class="docs-heading-anchor-permalink" href="#How-to-calculate-code-distance-from-the-state." title="Permalink"></a></h1><div class="admonition is-warning"><header class="admonition-header">WIP</header><div class="admonition-body"><p>This document is not fully completed.</p></div></div><h2 id="Definition-of-code-distance."><a class="docs-heading-anchor" href="#Definition-of-code-distance.">Definition of code distance.</a><a id="Definition-of-code-distance.-1"></a><a class="docs-heading-anchor-permalink" href="#Definition-of-code-distance." title="Permalink"></a></h2><p>Let&#39;s consider encoding circuit with <span>$1$</span> logical bit and <span>$k$</span> physical qubits<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>. Then this encoding has two physical basis, <span>$\ket{0}_L$</span> and <span>$\ket{1}_L$</span>. The <strong>distance</strong> of this encoding is the minimum bit flip required to convert between <span>$\ket{0}_L$</span> and <span>$\ket{1}_L$</span>.</p><h2 id="Classification-of-the-stabilizers."><a class="docs-heading-anchor" href="#Classification-of-the-stabilizers.">Classification of the stabilizers.</a><a id="Classification-of-the-stabilizers.-1"></a><a class="docs-heading-anchor-permalink" href="#Classification-of-the-stabilizers." title="Permalink"></a></h2><p>When treating <code>State</code>, the logical leg is not assigned and one can treat all stabilizers equally. However, if logical leg is assigned to the state, these stabilizers can be classified to <span>$4$</span> groups.</p><ol><li>stabilizers on physical qubits</li><li>​<span>$\bar{X}$</span>, which corresponds to logical <span>$X$</span></li><li>​<span>$\bar{Z}$</span>, which corresponds to logical <span>$Z$</span></li><li>​<span>$\bar{Y}$</span>, which corresponds to logical <span>$Y$</span></li></ol><p>Let <span>$\ket{V}$</span> is the dual state of the channel or encoding map <span>$[[n, 1, d]]$</span>,</p><p class="math-container">\[distance = \min_{S \in stabilizers} \#\left\{ i \mid \bar{Z}_i ≠ S_i \right\}\]</p><h2 id="Calculating-code-distance-from-the-check-matrix."><a class="docs-heading-anchor" href="#Calculating-code-distance-from-the-check-matrix.">Calculating code distance from the check matrix.</a><a id="Calculating-code-distance-from-the-check-matrix.-1"></a><a class="docs-heading-anchor-permalink" href="#Calculating-code-distance-from-the-check-matrix." title="Permalink"></a></h2><p>TODO: nor required if the performance doesn&#39;t matter.</p><h2 id="Examples"><a class="docs-heading-anchor" href="#Examples">Examples</a><a id="Examples-1"></a><a class="docs-heading-anchor-permalink" href="#Examples" title="Permalink"></a></h2><h3 id="[[5,-1,-3]]-code"><a class="docs-heading-anchor" href="#[[5,-1,-3]]-code"><span>$[[5, 1, 3]]$</span> code</a><a id="[[5,-1,-3]]-code-1"></a><a class="docs-heading-anchor-permalink" href="#[[5,-1,-3]]-code" title="Permalink"></a></h3><p>​<span>$[[5, 1, 3]]$</span> code has <span>$4$</span> stabilizers generators, <span>$XZZXI, IXZZX, XIXZZ, ZXIXZ$</span> and <span>$2$</span> logical operators, <span>$\bar{X} = XXXXX$</span> and <span>$\bar{Z} = ZZZZZ$</span>. Therefore, stabilizer generators for the corresponding state <span>$[[6, 0]]$</span> is <span>$IXZZXI, IIXZZX, IXIXZZ, IZXIXZ, XXXXXX, ZZZZZZ$</span>.</p><p>Let&#39;s construct <span>$[[6, 0]]$</span> state on QuantumLegos.jl.</p><pre><code class="language-julia-repl hljs">julia&gt; using QuantumLegos
+
+julia&gt; stab_513 = pauliop.([&quot;IXZZXI&quot;, &quot;IIXZZX&quot;, &quot;IXIXZZ&quot;, &quot;IZXIXZ&quot;, &quot;XXXXXX&quot;, &quot;ZZZZZZ&quot;])
+6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IXZZXI&quot;)
+ pauliop(&quot;IIXZZX&quot;)
+ pauliop(&quot;IXIXZZ&quot;)
+ pauliop(&quot;IZXIXZ&quot;)
+ pauliop(&quot;XXXXXX&quot;)
+ pauliop(&quot;ZZZZZZ&quot;)
+
+julia&gt; lego_513 = Lego(stab_513)
+Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IXZZXI&quot;), pauliop(&quot;IIXZZX&quot;), pauliop(&quot;IXIXZZ&quot;), pauliop(&quot;IZXIXZ&quot;), pauliop(&quot;XXXXXX&quot;), pauliop(&quot;ZZZZZZ&quot;)])
+
+julia&gt; state_513 = State([lego_513], edge.([]))
+State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IXZZXI&quot;), pauliop(&quot;IIXZZX&quot;), pauliop(&quot;IXIXZZ&quot;), pauliop(&quot;IZXIXZ&quot;), pauliop(&quot;XXXXXX&quot;), pauliop(&quot;ZZZZZZ&quot;)])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 1 … 0 0; 0 0 … 1 0; … ; 1 1 … 0 0; 0 0 … 1 1], 6, 6))
+</code></pre><p>Then collect generators of the state.</p><pre><code class="language-julia-repl hljs">julia&gt; normalizers = state_513.cmat |&gt; generators |&gt; GeneratedPauliGroup |&gt; collect
+64-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIIIII&quot;)
+ pauliop(&quot;IXZZXI&quot;)
+ pauliop(&quot;IIXZZX&quot;)
+ pauliop(&quot;IXYIYX&quot;)
+ pauliop(&quot;IXIXZZ&quot;)
+ pauliop(&quot;IIZYYZ&quot;)
+ pauliop(&quot;IXXYIY&quot;)
+ pauliop(&quot;IIYXXY&quot;)
+ pauliop(&quot;IZXIXZ&quot;)
+ pauliop(&quot;IYYZIZ&quot;)
+ ⋮
+ pauliop(&quot;YYIZZI&quot;)
+ pauliop(&quot;YXZYZX&quot;)
+ pauliop(&quot;YIIXYX&quot;)
+ pauliop(&quot;YXYXII&quot;)
+ pauliop(&quot;YIXYXI&quot;)
+ pauliop(&quot;YIZZIY&quot;)
+ pauliop(&quot;YXIIXY&quot;)
+ pauliop(&quot;YIYIZZ&quot;)
+ pauliop(&quot;YXXZYZ&quot;)
+</code></pre><p>Get stabilizer and normalizers of the <span>$[[5, 1, 3]]$</span> code by assigning the first leg as logical.</p><pre><code class="language-julia-repl hljs">julia&gt; stabs = filter(x -&gt; x[1] == PauliOps.I, normalizers)
+16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIIIII&quot;)
+ pauliop(&quot;IXZZXI&quot;)
+ pauliop(&quot;IIXZZX&quot;)
+ pauliop(&quot;IXYIYX&quot;)
+ pauliop(&quot;IXIXZZ&quot;)
+ pauliop(&quot;IIZYYZ&quot;)
+ pauliop(&quot;IXXYIY&quot;)
+ pauliop(&quot;IIYXXY&quot;)
+ pauliop(&quot;IZXIXZ&quot;)
+ pauliop(&quot;IYYZIZ&quot;)
+ pauliop(&quot;IZIZYY&quot;)
+ pauliop(&quot;IYZIZY&quot;)
+ pauliop(&quot;IYXXYI&quot;)
+ pauliop(&quot;IZYYZI&quot;)
+ pauliop(&quot;IYIYXX&quot;)
+ pauliop(&quot;IZZXIX&quot;)
+
+julia&gt; norm_x = filter(x -&gt; x[1] == PauliOps.X, normalizers)
+16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;XXXXXX&quot;)
+ pauliop(&quot;XIYYIX&quot;)
+ pauliop(&quot;XXIYYI&quot;)
+ pauliop(&quot;XIZXZI&quot;)
+ pauliop(&quot;XIXIYY&quot;)
+ pauliop(&quot;XXYZZY&quot;)
+ pauliop(&quot;XIIZXZ&quot;)
+ pauliop(&quot;XXZIIZ&quot;)
+ pauliop(&quot;XYIXIY&quot;)
+ pauliop(&quot;XZZYXY&quot;)
+ pauliop(&quot;XYXYZZ&quot;)
+ pauliop(&quot;XZYXYZ&quot;)
+ pauliop(&quot;XZIIZX&quot;)
+ pauliop(&quot;XYZZYX&quot;)
+ pauliop(&quot;XZXZII&quot;)
+ pauliop(&quot;XYYIXI&quot;)
+
+julia&gt; norm_y = filter(x -&gt; x[1] == PauliOps.Y, normalizers)
+16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;YYYYYY&quot;)
+ pauliop(&quot;YZXXZY&quot;)
+ pauliop(&quot;YYZXXZ&quot;)
+ pauliop(&quot;YZIYIZ&quot;)
+ pauliop(&quot;YZYZXX&quot;)
+ pauliop(&quot;YYXIIX&quot;)
+ pauliop(&quot;YZZIYI&quot;)
+ pauliop(&quot;YYIZZI&quot;)
+ pauliop(&quot;YXZYZX&quot;)
+ pauliop(&quot;YIIXYX&quot;)
+ pauliop(&quot;YXYXII&quot;)
+ pauliop(&quot;YIXYXI&quot;)
+ pauliop(&quot;YIZZIY&quot;)
+ pauliop(&quot;YXIIXY&quot;)
+ pauliop(&quot;YIYIZZ&quot;)
+ pauliop(&quot;YXXZYZ&quot;)
+
+julia&gt; norm_z = filter(x -&gt; x[1] == PauliOps.Z, normalizers)
+16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;ZZZZZZ&quot;)
+ pauliop(&quot;ZYIIYZ&quot;)
+ pauliop(&quot;ZZYIIY&quot;)
+ pauliop(&quot;ZYXZXY&quot;)
+ pauliop(&quot;ZYZYII&quot;)
+ pauliop(&quot;ZZIXXI&quot;)
+ pauliop(&quot;ZYYXZX&quot;)
+ pauliop(&quot;ZZXYYX&quot;)
+ pauliop(&quot;ZIYZYI&quot;)
+ pauliop(&quot;ZXXIZI&quot;)
+ pauliop(&quot;ZIZIXX&quot;)
+ pauliop(&quot;ZXIZIX&quot;)
+ pauliop(&quot;ZXYYXZ&quot;)
+ pauliop(&quot;ZIXXIZ&quot;)
+ pauliop(&quot;ZXZXYY&quot;)
+ pauliop(&quot;ZIIYZY&quot;)
+</code></pre><p>These normalizers are generated from one logical operator and stabilizers.</p><pre><code class="language-julia-repl hljs">julia&gt; map(x -&gt; x .* pauliop(&quot;XXXXXX&quot;), stabs) |&gt; Set == Set(norm_x)
+true
+
+julia&gt; map(x -&gt; x .* pauliop(&quot;ZZZZZZ&quot;), stabs) |&gt; Set == Set(norm_x)
+false
+
+julia&gt; map(x -&gt; x .* pauliop(&quot;ZZZZZZ&quot;), stabs) |&gt; Set == Set(norm_z)
+true
+
+julia&gt; map(x -&gt; x .* pauliop(&quot;YYYYYY&quot;), stabs) |&gt; Set == Set(norm_y)
+true
+
+julia&gt; using IterTools
+
+julia&gt; groupby(x -&gt; x[1], normalizers) .|&gt; Set == Set.([stabs, norm_x, norm_z, norm_y])
+true
+</code></pre><p>Define a function to get weight of the operator.</p><pre><code class="language-julia-repl hljs">julia&gt; function weight(x, i = 1)
+           count(x[i:end] .!= PauliOps.I)
+       end
+weight (generic function with 2 methods)
+
+julia&gt; weight(pauliop(&quot;XIXIXI&quot;))
+3
+
+julia&gt; weight(pauliop(&quot;XIXIXI&quot;), 2)
+2
+</code></pre><p>Calculate coefficients of enumerator polynomial.</p><pre><code class="language-julia-repl hljs">julia&gt; using DataStructures
+
+julia&gt; stabs .|&gt; weight |&gt; counter
+Accumulator{Int64, Int64} with 2 entries:
+  0 =&gt; 1
+  4 =&gt; 15
+
+julia&gt; function weight(i::Integer)
+           Base.Fix2(weight, i)
+       end
+weight (generic function with 3 methods)
+
+julia&gt; normalizers .|&gt; weight(2) |&gt; counter
+Accumulator{Int64, Int64} with 4 entries:
+  0 =&gt; 1
+  4 =&gt; 15
+  5 =&gt; 18
+  3 =&gt; 30
+
+julia&gt; [norm_x..., norm_y..., norm_z...] .|&gt; weight(2) |&gt; counter
+Accumulator{Int64, Int64} with 2 entries:
+  5 =&gt; 18
+  3 =&gt; 30
+
+julia&gt; [norm_x..., norm_y..., norm_z...] .|&gt; weight(2) |&gt; counter |&gt; keys |&gt; minimum
+3
+</code></pre><p>Code distance of the encoding is the minimum degree of the non-zero term in the normalizer&#39;s polynomial(<span>$B$</span>) and not in the stabilizer&#39;s polynomial(<span>$A$</span>). So the code distance of this encoding is <span>$3$</span>.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Not all state can be formalized like this. TODO</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../checkmatrix/">« Details on check matrix operations</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Tuesday 7 May 2024 02:47">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · QuantumLegos.jl</title><meta name="title" content="Home · QuantumLegos.jl"/><meta property="og:title" content="Home · QuantumLegos.jl"/><meta property="twitter:title" content="Home · QuantumLegos.jl"/><meta name="description" content="Documentation for QuantumLegos.jl."/><meta property="og:description" content="Documentation for QuantumLegos.jl."/><meta property="twitter:description" content="Documentation for QuantumLegos.jl."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" 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src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>QuantumLegos.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li class="toplevel"><a class="tocitem" href="#Example"><span>Example</span></a></li><li><a class="tocitem" href="#CheckMatrix-and-defining-Lego"><span>CheckMatrix and defining Lego</span></a></li><li><a class="tocitem" href="#Defining-and-Updating-State"><span>Defining and Updating State</span></a></li><li><a class="tocitem" href="#2-Lego-1-edge-state"><span>2 Lego 1 edge state</span></a></li><li class="toplevel"><a class="tocitem" href="#Internal(how-it-works)"><span>Internal(how it works)</span></a></li><li><a class="tocitem" href="#Notes-on-Overall-flow"><span>Notes on Overall flow</span></a></li><li class="toplevel"><a class="tocitem" href="#API"><span>API</span></a></li></ul></li><li><a class="tocitem" href="pauliops/">PauliOps</a></li><li><a class="tocitem" href="checkmatrix/">Details on check matrix operations</a></li><li><a class="tocitem" href="distance/">How to calculate code distance from the state.</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl/blob/main/docs/src/index.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="QuantumLegos"><a class="docs-heading-anchor" href="#QuantumLegos">QuantumLegos</a><a id="QuantumLegos-1"></a><a class="docs-heading-anchor-permalink" href="#QuantumLegos" title="Permalink"></a></h1><p>Documentation for QuantumLegos.</p><p>All contents:</p><ul><li><a href="checkmatrix/#Details-on-check-matrix-operations">Details on check matrix operations</a></li><li class="no-marker"><ul><li><a href="checkmatrix/#Basic-features-of-check-matrix">Basic features of check matrix</a></li><li><a href="checkmatrix/#Construction-of-check-matrices-and-retrieving-features">Construction of check matrices and retrieving features</a></li><li><a href="checkmatrix/#Direct-sum">Direct sum</a></li><li><a href="checkmatrix/#Self-tracing">Self-tracing</a></li></ul></li><li><a href="distance/#How-to-calculate-code-distance-from-the-state.">How to calculate code distance from the state.</a></li><li class="no-marker"><ul><li><a href="distance/#Definition-of-code-distance.">Definition of code distance.</a></li><li><a href="distance/#Classification-of-the-stabilizers.">Classification of the stabilizers.</a></li><li><a href="distance/#Calculating-code-distance-from-the-check-matrix.">Calculating code distance from the check matrix.</a></li><li><a href="distance/#Examples">Examples</a></li></ul></li><li><a href="#QuantumLegos">QuantumLegos</a></li><li><a href="#Example">Example</a></li><li class="no-marker"><ul><li><a href="#CheckMatrix-and-defining-Lego">CheckMatrix and defining Lego</a></li><li><a href="#Defining-and-Updating-State">Defining and Updating State</a></li><li><a href="#2-Lego-1-edge-state">2 Lego 1 edge state</a></li></ul></li><li><a href="#Internal(how-it-works)">Internal(how it works)</a></li><li class="no-marker"><ul><li><a href="#Notes-on-Overall-flow">Notes on Overall flow</a></li></ul></li><li><a href="#API">API</a></li><li><a href="pauliops/#PauliOps">PauliOps</a></li><li class="no-marker"><ul><li><a href="pauliops/#Example">Example</a></li><li><a href="pauliops/#API">API</a></li></ul></li></ul><h1 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h1><h2 id="CheckMatrix-and-defining-Lego"><a class="docs-heading-anchor" href="#CheckMatrix-and-defining-Lego">CheckMatrix and defining Lego</a><a id="CheckMatrix-and-defining-Lego-1"></a><a class="docs-heading-anchor-permalink" href="#CheckMatrix-and-defining-Lego" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; using QuantumLegos
+
+julia&gt; stabilizers = pauliop.([&quot;IIXXXX&quot;, &quot;IIZZZZ&quot;, &quot;ZIZZII&quot;, &quot;IZZIZI&quot;, &quot;IXXXII&quot;, &quot;XIXIXI&quot;])
+6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIXXXX&quot;)
+ pauliop(&quot;IIZZZZ&quot;)
+ pauliop(&quot;ZIZZII&quot;)
+ pauliop(&quot;IZZIZI&quot;)
+ pauliop(&quot;IXXXII&quot;)
+ pauliop(&quot;XIXIXI&quot;)
+
+julia&gt; cmat = checkmatrix(stabilizers)
+CheckMatrix with 6 generators, 6 legs:
+  0  0  1  1  1  1 | 0  0  0  0  0  0
+  0  0  0  0  0  0 | 0  0  1  1  1  1
+  0  0  0  0  0  0 | 1  0  1  1  0  0
+  0  0  0  0  0  0 | 0  1  1  0  1  0
+  0  1  1  1  0  0 | 0  0  0  0  0  0
+  1  0  1  0  1  0 | 0  0  0  0  0  0
+
+
+julia&gt; cmat.nlegs
+6
+
+julia&gt; cmat.ngens
+6
+
+julia&gt; cmat.cmat
+6×12 Matrix{Bool}:
+ 0  0  1  1  1  1  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  1  1  1  1
+ 0  0  0  0  0  0  1  0  1  1  0  0
+ 0  0  0  0  0  0  0  1  1  0  1  0
+ 0  1  1  1  0  0  0  0  0  0  0  0
+ 1  0  1  0  1  0  0  0  0  0  0  0
+
+julia&gt; # define lego
+
+julia&gt; lego = Lego(stabilizers)
+Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])
+
+julia&gt; lego.stabgens |&gt; checkmatrix
+CheckMatrix with 6 generators, 6 legs:
+  0  0  1  1  1  1 | 0  0  0  0  0  0
+  0  0  0  0  0  0 | 0  0  1  1  1  1
+  0  0  0  0  0  0 | 1  0  1  1  0  0
+  0  0  0  0  0  0 | 0  1  1  0  1  0
+  0  1  1  1  0  0 | 0  0  0  0  0  0
+  1  0  1  0  1  0 | 0  0  0  0  0  0
+</code></pre><ul><li><a href="pauliops/#QuantumLegos.PauliOps.pauliop-Tuple{AbstractString}"><code>pauliop</code></a></li><li><a href="checkmatrix/#QuantumLegos.checkmatrix"><code>checkmatrix</code></a> and <a href="checkmatrix/#QuantumLegos.CheckMatrix"><code>CheckMatrix</code></a></li><li><a href="#QuantumLegos.Lego"><code>Lego</code></a></li></ul><h2 id="Defining-and-Updating-State"><a class="docs-heading-anchor" href="#Defining-and-Updating-State">Defining and Updating State</a><a id="Defining-and-Updating-State-1"></a><a class="docs-heading-anchor-permalink" href="#Defining-and-Updating-State" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; using QuantumLegos
+
+julia&gt; stabilizers = pauliop.([&quot;IIXXXX&quot;, &quot;IIZZZZ&quot;, &quot;ZIZZII&quot;, &quot;IZZIZI&quot;, &quot;IXXXII&quot;, &quot;XIXIXI&quot;])
+6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIXXXX&quot;)
+ pauliop(&quot;IIZZZZ&quot;)
+ pauliop(&quot;ZIZZII&quot;)
+ pauliop(&quot;IZZIZI&quot;)
+ pauliop(&quot;IXXXII&quot;)
+ pauliop(&quot;XIXIXI&quot;)
+
+julia&gt; lego = Lego(stabilizers)
+Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])
+
+julia&gt; # state with 1 lego, 0 leg
+
+julia&gt; st = State([lego, ], Tuple{LegoLeg, LegoLeg}[])
+State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 1 1; … ; 0 1 … 0 0; 1 0 … 0 0], 6, 6))
+
+julia&gt; st.cmat.cmat
+6×12 Matrix{Bool}:
+ 0  0  1  1  1  1  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  1  1  1  1
+ 0  0  0  0  0  0  1  0  1  1  0  0
+ 0  0  0  0  0  0  0  1  1  0  1  0
+ 0  1  1  1  0  0  0  0  0  0  0  0
+ 1  0  1  0  1  0  0  0  0  0  0  0
+
+julia&gt; add_lego!(st, lego)
+State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], 12, 12))
+
+julia&gt; st.cmat.cmat
+12×24 Matrix{Bool}:
+ 0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  1  0  0  0  0  0  0  0
+ 0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
+ 1  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0
+ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1
+ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0
+ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  1  0
+ 0  0  0  0  0  0  0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
+ 0  0  0  0  0  0  1  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0
+
+julia&gt; # state with 2 legos, 0 leg
+
+julia&gt; st2 = State([lego, lego], Tuple{LegoLeg, LegoLeg}[])
+State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], 12, 12))
+
+julia&gt; st == st2
+true</code></pre><h2 id="2-Lego-1-edge-state"><a class="docs-heading-anchor" href="#2-Lego-1-edge-state">2 Lego 1 edge state</a><a id="2-Lego-1-edge-state-1"></a><a class="docs-heading-anchor-permalink" href="#2-Lego-1-edge-state" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; using QuantumLegos
+
+julia&gt; stabilizers = pauliop.([&quot;IIXXXX&quot;, &quot;IIZZZZ&quot;, &quot;ZIZZII&quot;, &quot;IZZIZI&quot;, &quot;IXXXII&quot;, &quot;XIXIXI&quot;])
+6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIXXXX&quot;)
+ pauliop(&quot;IIZZZZ&quot;)
+ pauliop(&quot;ZIZZII&quot;)
+ pauliop(&quot;IZZIZI&quot;)
+ pauliop(&quot;IXXXII&quot;)
+ pauliop(&quot;XIXIXI&quot;)
+
+julia&gt; lego = Lego(stabilizers)
+Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])
+
+julia&gt; state = State([lego, lego], edge.([((1, 3), (2, 3))]))
+State(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXXXX&quot;), pauliop(&quot;IIZZZZ&quot;), pauliop(&quot;ZIZZII&quot;), pauliop(&quot;IZZIZI&quot;), pauliop(&quot;IXXXII&quot;), pauliop(&quot;XIXIXI&quot;)])], Tuple{LegoLeg, LegoLeg}[(LegoLeg(1, 3), LegoLeg(2, 3))], CheckMatrix(Bool[1 0 … 0 0; 0 1 … 0 0; … ; 0 0 … 1 1; 0 0 … 0 1], 10, 10))
+
+julia&gt; state.cmat
+CheckMatrix with 10 generators, 10 legs:
+  1  0  1  0  1  0  0  0  0  0 | 0  0  0  0  0  0  0  0  0  0
+  0  1  0  1  1  0  0  0  0  0 | 0  0  0  0  0  0  0  0  0  0
+  0  0  1  1  1  0  0  1  1  1 | 0  0  0  0  0  0  0  0  0  0
+  0  0  0  0  0  1  0  1  0  1 | 0  0  0  0  0  0  0  0  0  0
+  0  0  0  0  0  0  1  0  1  1 | 0  0  0  0  0  0  0  0  0  0
+  0  0  0  0  0  0  0  0  0  0 | 1  0  0  1  1  0  0  0  0  0
+  0  0  0  0  0  0  0  0  0  0 | 0  1  1  0  1  0  0  0  0  0
+  0  0  0  0  0  0  0  0  0  0 | 0  0  1  1  1  0  0  1  1  1
+  0  0  0  0  0  0  0  0  0  0 | 0  0  0  0  0  1  0  0  1  1
+  0  0  0  0  0  0  0  0  0  0 | 0  0  0  0  0  0  1  1  0  1
+
+
+julia&gt; pg = state.cmat |&gt; generators |&gt; GeneratedPauliGroup
+GeneratedPauliGroup{10}(StaticArraysCore.SVector{10, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;XIXIXIIIII&quot;), pauliop(&quot;IXIXXIIIII&quot;), pauliop(&quot;IIXXXIIXXX&quot;), pauliop(&quot;IIIIIXIXIX&quot;), pauliop(&quot;IIIIIIXIXX&quot;), pauliop(&quot;ZIIZZIIIII&quot;), pauliop(&quot;IZZIZIIIII&quot;), pauliop(&quot;IIZZZIIZZZ&quot;), pauliop(&quot;IIIIIZIIZZ&quot;), pauliop(&quot;IIIIIIZZIZ&quot;)], IterTools.Subsets{Vector{StaticArraysCore.SVector{N, QuantumLegos.PauliOps.SinglePauliOp} where N}}(StaticArraysCore.SVector{N, QuantumLegos.PauliOps.SinglePauliOp} where N[pauliop(&quot;XIXIXIIIII&quot;), pauliop(&quot;IXIXXIIIII&quot;), pauliop(&quot;IIXXXIIXXX&quot;), pauliop(&quot;IIIIIXIXIX&quot;), pauliop(&quot;IIIIIIXIXX&quot;), pauliop(&quot;ZIIZZIIIII&quot;), pauliop(&quot;IZZIZIIIII&quot;), pauliop(&quot;IIZZZIIZZZ&quot;), pauliop(&quot;IIIIIZIIZZ&quot;), pauliop(&quot;IIIIIIZZIZ&quot;)]))
+
+julia&gt; pauliop(&quot;XIIXIXIIXI&quot;) in pg
+true
+</code></pre><h1 id="Internal(how-it-works)"><a class="docs-heading-anchor" href="#Internal(how-it-works)">Internal(how it works)</a><a id="Internal(how-it-works)-1"></a><a class="docs-heading-anchor-permalink" href="#Internal(how-it-works)" title="Permalink"></a></h1><h2 id="Notes-on-Overall-flow"><a class="docs-heading-anchor" href="#Notes-on-Overall-flow">Notes on Overall flow</a><a id="Notes-on-Overall-flow-1"></a><a class="docs-heading-anchor-permalink" href="#Notes-on-Overall-flow" title="Permalink"></a></h2><p>Details on <sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup></p><ul><li>state is translated to a single check matrix<ul><li>the size is ≤ <span>$N \times 2N$</span> where <span>$N$</span> is maximum number of lego logs.</li></ul></li><li>any contraction can be performed on this single check matrix</li><li>if the check matrix can be represented as direct sum of matrices with <span>$k N$</span> columns where <span>$k ∈ ℕ$</span>, then they are not contracted</li></ul><h3 id="Construction-of-State"><a class="docs-heading-anchor" href="#Construction-of-State">Construction of State</a><a id="Construction-of-State-1"></a><a class="docs-heading-anchor-permalink" href="#Construction-of-State" title="Permalink"></a></h3><p>Construction of <code>State</code> is completed by calling <code>State</code> constructor recursively.</p><ol><li>Construct <code>State</code> without edge. Just adding legos. Checkmatrix is just a direct sum of each lego&#39;s checkmatrix</li><li>Concatenate each edges. During this operation, self tracing of checkmatrix is evaluated.</li></ol><p>Each constructor calls action function (which is a map from <code>State</code> to <code>State</code>). Therefore, action functions can be used both for direct construction of <code>State</code> and action application to <code>State</code> during the game.</p><h1 id="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h1><ul><li><a href="pauliops/#QuantumLegos.PauliOps"><code>QuantumLegos.PauliOps</code></a></li><li><a href="checkmatrix/#QuantumLegos.CheckMatrix"><code>QuantumLegos.CheckMatrix</code></a></li><li><a href="#QuantumLegos.Lego"><code>QuantumLegos.Lego</code></a></li><li><a href="#QuantumLegos.LegoLeg"><code>QuantumLegos.LegoLeg</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.GeneratedPauliGroup"><code>QuantumLegos.PauliOps.GeneratedPauliGroup</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.PauliOp"><code>QuantumLegos.PauliOps.PauliOp</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.SinglePauliOp"><code>QuantumLegos.PauliOps.SinglePauliOp</code></a></li><li><a href="#QuantumLegos.State"><code>QuantumLegos.State</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.pauliop-Tuple{AbstractString}"><code>QuantumLegos.PauliOps.pauliop</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.single_pauliop-Tuple{Char}"><code>QuantumLegos.PauliOps.single_pauliop</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.weight"><code>QuantumLegos.PauliOps.weight</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.xweight"><code>QuantumLegos.PauliOps.xweight</code></a></li><li><a href="pauliops/#QuantumLegos.PauliOps.zweight"><code>QuantumLegos.PauliOps.zweight</code></a></li><li><a href="#QuantumLegos._naive_distance-Tuple{State}"><code>QuantumLegos._naive_distance</code></a></li><li><a href="#QuantumLegos.add_edge!-Tuple{State, LegoLeg, LegoLeg}"><code>QuantumLegos.add_edge!</code></a></li><li><a href="#QuantumLegos.add_lego!-Tuple{State, Lego}"><code>QuantumLegos.add_lego!</code></a></li><li><a href="checkmatrix/#QuantumLegos.align_row!"><code>QuantumLegos.align_row!</code></a></li><li><a href="checkmatrix/#QuantumLegos.checkmatrix"><code>QuantumLegos.checkmatrix</code></a></li><li><a href="#QuantumLegos.cmat_index-Tuple{State, LegoLeg}"><code>QuantumLegos.cmat_index</code></a></li><li><a href="checkmatrix/#QuantumLegos.direct_sum"><code>QuantumLegos.direct_sum</code></a></li><li><a href="#QuantumLegos.distance-Tuple{State}"><code>QuantumLegos.distance</code></a></li><li><a href="#QuantumLegos.edge-Union{Tuple{T}, NTuple{4, T}} where T&lt;:Integer"><code>QuantumLegos.edge</code></a></li><li><a href="#QuantumLegos.edge-Union{Tuple{Tuple{T, T}}, Tuple{T}} where T&lt;:Tuple{Integer, Integer}"><code>QuantumLegos.edge</code></a></li><li><a href="#QuantumLegos.edge-Union{Tuple{NTuple{4, T}}, Tuple{T}} where T&lt;:Integer"><code>QuantumLegos.edge</code></a></li><li><a href="#QuantumLegos.edge"><code>QuantumLegos.edge</code></a></li><li><a href="checkmatrix/#QuantumLegos.eliminate_column!"><code>QuantumLegos.eliminate_column!</code></a></li><li><a href="checkmatrix/#QuantumLegos.eliminate_dependent_row!"><code>QuantumLegos.eliminate_dependent_row!</code></a></li><li><a href="checkmatrix/#QuantumLegos.generators"><code>QuantumLegos.generators</code></a></li><li><a href="#QuantumLegos.is_connected_to_firstlego-Tuple{State}"><code>QuantumLegos.is_connected_to_firstlego</code></a></li><li><a href="checkmatrix/#QuantumLegos.ref!"><code>QuantumLegos.ref!</code></a></li><li><a href="checkmatrix/#QuantumLegos.self_trace!"><code>QuantumLegos.self_trace!</code></a></li><li><a href="checkmatrix/#QuantumLegos.xpart"><code>QuantumLegos.xpart</code></a></li><li><a href="checkmatrix/#QuantumLegos.zpart"><code>QuantumLegos.zpart</code></a></li></ul><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.Lego" href="#QuantumLegos.Lego"><code>QuantumLegos.Lego</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Quantum lego with <code>N</code> legs.</p><p><strong>Fields</strong></p><ul><li><code>nlegs::Int64</code>: number of legs, equals <code>N</code></li><li><code>stabgens::SVector{N, PauliOp{N}}</code>: stabilizer generators. vector of <a href="pauliops/#QuantumLegos.PauliOps.PauliOp"><code>PauliOp</code></a></li></ul><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">Lego([nlegs::Integer], stabgens::AbstractVector{PauliOp{N}})</code></pre><p>Constructor for <a href="#QuantumLegos.Lego"><code>Lego</code></a>. <code>nlegs</code> is optional (default is length of the first stabilizer generator).</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; stabgens = pauliop.([&quot;II&quot;, &quot;XX&quot;])
+2-element Vector{StaticArraysCore.SVector{2, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;II&quot;)
+ pauliop(&quot;XX&quot;)
+
+julia&gt; Lego(stabgens)
+Lego{2}(2, StaticArraysCore.SVector{2, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;II&quot;), pauliop(&quot;XX&quot;)])</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L8-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.LegoLeg" href="#QuantumLegos.LegoLeg"><code>QuantumLegos.LegoLeg</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct State</code></pre><p>To be used in <a href="#QuantumLegos.State"><code>State</code></a>.</p><p><strong>Fields</strong></p><ul><li><code>lego_id::Int64</code>: index in <code>legos</code> in <code>State</code></li><li><code>edge_id::Int64</code>: index in <code>Lego</code> in <code>legos</code> in <code>State</code>. No validation check included in <code>LegoLeg</code>.</li></ul><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia&gt; x = LegoLeg.([(2, 1), (1, 1), (1, 0)])
+3-element Vector{LegoLeg}:
+ LegoLeg(2, 1)
+ LegoLeg(1, 1)
+ LegoLeg(1, 0)
+
+julia&gt; sort(x)
+3-element Vector{LegoLeg}:
+ LegoLeg(1, 0)
+ LegoLeg(1, 1)
+ LegoLeg(2, 1)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L50-L73">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.State" href="#QuantumLegos.State"><code>QuantumLegos.State</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">mutable struct State</code></pre><p>State (in p.4)</p><p><strong>Fields</strong></p><ul><li><code>legos</code>: <code>Vector{Lego}</code></li><li><code>edges</code>: Vector of ((<code>lego_i, leg_n</code>), (<code>lego_j, leg_m</code>)).   Each element is sorted (i.e. <code>lego_i &lt; lego_j</code> or <code>lego_i == lego_j &amp;&amp; leg_n &lt; leg_m</code>).   This feature is used in <a href="#QuantumLegos.is_connected_to_firstlego-Tuple{State}"><code>is_connected_to_firstlego</code></a>.</li><li><code>cmat::CheckMatrix</code>: CheckMatrix</li></ul><p><strong>Constructor</strong></p><pre><code class="nohighlight hljs">State(legos::Vector{Lego{N}}, edges::Vector{Tuple{LegoLeg, LegoLeg}})</code></pre><p><strong>Methods with</strong></p><ul><li><a href="#QuantumLegos.add_lego!-Tuple{State, Lego}"><code>add_lego!</code></a></li><li><a href="#QuantumLegos.add_edge!-Tuple{State, LegoLeg, LegoLeg}"><code>add_edge!</code></a></li></ul><p><strong>Example</strong></p><p>TODO</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L115-L136">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos._naive_distance-Tuple{State}" href="#QuantumLegos._naive_distance-Tuple{State}"><code>QuantumLegos._naive_distance</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Calculate distance. Use minimum distance of all generated normalizers.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L261-L264">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.add_edge!-Tuple{State, LegoLeg, LegoLeg}" href="#QuantumLegos.add_edge!-Tuple{State, LegoLeg, LegoLeg}"><code>QuantumLegos.add_edge!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">add_edge!(state::State, leg_1::LegoLeg, leg_2::LegoLeg)::State</code></pre><p>Add a new edge between <code>leg_1</code> and <code>leg_2</code>, updating <code>state</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L209-L213">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.add_lego!-Tuple{State, Lego}" href="#QuantumLegos.add_lego!-Tuple{State, Lego}"><code>QuantumLegos.add_lego!</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">add_lego!(state::State, lego::Lego) -&gt; State</code></pre><p>Add a new lego, updating <code>state</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L171-L175">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.cmat_index-Tuple{State, LegoLeg}" href="#QuantumLegos.cmat_index-Tuple{State, LegoLeg}"><code>QuantumLegos.cmat_index</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">cmat_index(state::State, leg::LegoLeg)::Int64</code></pre><p>Get column index corresponds to <code>leg</code> in check matrix of <code>state</code>. If given <code>leg</code> is already connected, it throws <code>ArgumentError</code>. If given <code>lego_id</code> of <code>leg</code> is out of <code>state.legos</code>, throws <code>ArgumentError</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L183-L189">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.distance-Tuple{State}" href="#QuantumLegos.distance-Tuple{State}"><code>QuantumLegos.distance</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">distance(state::State) -&gt; Int</code></pre><p>Calculate code distance when the first leg of <code>state</code> is assigned as logical.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L253-L257">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.edge" href="#QuantumLegos.edge"><code>QuantumLegos.edge</code></a> — <span class="docstring-category">Function</span></header><section><div><p>Helper function to create <code>Tuple{LegoLeg, LegoLeg}</code> to represent edge.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L81-L83">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.edge-Union{Tuple{NTuple{4, T}}, Tuple{T}} where T&lt;:Integer" href="#QuantumLegos.edge-Union{Tuple{NTuple{4, T}}, Tuple{T}} where T&lt;:Integer"><code>QuantumLegos.edge</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">edge(t::Tuple{T, T, T, T}) where {T &lt;: Integer}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L93-L95">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.edge-Union{Tuple{Tuple{T, T}}, Tuple{T}} where T&lt;:Tuple{Integer, Integer}" href="#QuantumLegos.edge-Union{Tuple{Tuple{T, T}}, Tuple{T}} where T&lt;:Tuple{Integer, Integer}"><code>QuantumLegos.edge</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">edge(t::Tuple{T, T}) where {T &lt;: Tuple{Integer, Integer}}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L86-L88">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.edge-Union{Tuple{T}, NTuple{4, T}} where T&lt;:Integer" href="#QuantumLegos.edge-Union{Tuple{T}, NTuple{4, T}} where T&lt;:Integer"><code>QuantumLegos.edge</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">edge(x::T, y::T, z::T, w::T) where {T &lt;: Integer}</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L100-L102">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.is_connected_to_firstlego-Tuple{State}" href="#QuantumLegos.is_connected_to_firstlego-Tuple{State}"><code>QuantumLegos.is_connected_to_firstlego</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">is_connected_to_firstlego(state::State)::BitVector</code></pre><p>Returns vector which stores whether each lego is connected to the first lego.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/game.jl#L228-L232">source</a></section></article><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a><a href="https://doi.org/10.1007/JHEP05(2021)127">C. Cao and B. Lackey, ‘Approximate Bacon-Shor code and holography’, J. High Energ. Phys., vol. 2021, no. 5, p. 127, May 2021, doi: 10.1007/JHEP05(2021)127.</a></li></ul></section></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="pauliops/">PauliOps »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Tuesday 7 May 2024 02:47">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+<!DOCTYPE html>
+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>PauliOps · QuantumLegos.jl</title><meta name="title" content="PauliOps · QuantumLegos.jl"/><meta property="og:title" content="PauliOps · QuantumLegos.jl"/><meta property="twitter:title" content="PauliOps · QuantumLegos.jl"/><meta name="description" content="Documentation for QuantumLegos.jl."/><meta property="og:description" content="Documentation for QuantumLegos.jl."/><meta property="twitter:description" content="Documentation for QuantumLegos.jl."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">QuantumLegos.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li class="is-active"><a class="tocitem" href>PauliOps</a><ul class="internal"><li><a class="tocitem" href="#Example"><span>Example</span></a></li><li><a class="tocitem" href="#API"><span>API</span></a></li></ul></li><li><a class="tocitem" href="../checkmatrix/">Details on check matrix operations</a></li><li><a class="tocitem" href="../distance/">How to calculate code distance from the state.</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>PauliOps</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>PauliOps</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/qwjyh/QuantumLegos.jl/blob/main/docs/src/pauliops.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="PauliOps"><a class="docs-heading-anchor" href="#PauliOps">PauliOps</a><a id="PauliOps-1"></a><a class="docs-heading-anchor-permalink" href="#PauliOps" title="Permalink"></a></h1><p>submodule</p><h2 id="Example"><a class="docs-heading-anchor" href="#Example">Example</a><a id="Example-1"></a><a class="docs-heading-anchor-permalink" href="#Example" title="Permalink"></a></h2><pre><code class="language-julia-repl hljs">julia&gt; using QuantumLegos
+
+julia&gt; p = PauliOps.single_pauliop(&#39;I&#39;)
+I::SinglePauliOp = 0
+
+julia&gt; typeof(p)
+Enum QuantumLegos.PauliOps.SinglePauliOp:
+I = 0
+X = 1
+Y = 2
+Z = 3
+
+julia&gt; pauliop(&quot;IXYZ&quot;)
+4-element PauliOp:
+ I::SinglePauliOp = 0
+ X::SinglePauliOp = 1
+ Y::SinglePauliOp = 2
+ Z::SinglePauliOp = 3
+
+julia&gt; typeof(ans)
+SVector{4, SinglePauliOp} (alias for StaticArraysCore.SArray{Tuple{4}, QuantumLegos.PauliOps.SinglePauliOp, 1, 4})
+
+julia&gt; PauliOps.I * PauliOps.X
+X::SinglePauliOp = 1
+
+julia&gt; PauliOps.X * PauliOps.Z
+Y::SinglePauliOp = 2
+
+julia&gt; pauliop(&quot;IIX&quot;) .* pauliop(&quot;XIY&quot;)
+3-element PauliOp:
+ X::SinglePauliOp = 1
+ I::SinglePauliOp = 0
+ Z::SinglePauliOp = 3</code></pre><h2 id="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps" href="#QuantumLegos.PauliOps"><code>QuantumLegos.PauliOps</code></a> — <span class="docstring-category">Module</span></header><section><div><p>Pauli operator</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L1-L3">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.GeneratedPauliGroup" href="#QuantumLegos.PauliOps.GeneratedPauliGroup"><code>QuantumLegos.PauliOps.GeneratedPauliGroup</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">struct GeneratedPauliGroup</code></pre><p>Iterator for group generated from <code>gens</code>.</p><pre><code class="nohighlight hljs">GeneratedPauliGroup(gens::AbstractVector{T}) where {T &lt;: PauliOp}</code></pre><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia&gt; gens = pauliop.([&quot;IIXX&quot;, &quot;IZZI&quot;])
+2-element Vector{StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIXX&quot;)
+ pauliop(&quot;IZZI&quot;)
+
+julia&gt; g = PauliOps.GeneratedPauliGroup(gens)
+GeneratedPauliGroup{4}(StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXX&quot;), pauliop(&quot;IZZI&quot;)], IterTools.Subsets{Vector{StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}}}(StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(&quot;IIXX&quot;), pauliop(&quot;IZZI&quot;)]))
+
+julia&gt; collect(g)
+4-element Vector{StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}}:
+ pauliop(&quot;IIII&quot;)
+ pauliop(&quot;IIXX&quot;)
+ pauliop(&quot;IZZI&quot;)
+ pauliop(&quot;IZYX&quot;)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L147-L171">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.PauliOp" href="#QuantumLegos.PauliOps.PauliOp"><code>QuantumLegos.PauliOps.PauliOp</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">PauliOp{N}</code></pre><p>Pauli operator on multiple qubits.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L31-L35">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.SinglePauliOp" href="#QuantumLegos.PauliOps.SinglePauliOp"><code>QuantumLegos.PauliOps.SinglePauliOp</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">@enum SinglePauliOp begin
+    I
+    X
+    Y
+    Z
+end</code></pre><p>Pauli Operator on a single qubit.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L14-L23">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.pauliop-Tuple{AbstractString}" href="#QuantumLegos.PauliOps.pauliop-Tuple{AbstractString}"><code>QuantumLegos.PauliOps.pauliop</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">pauliop(str::AbstractString)::PauliOp</code></pre><p>Convert <code>str</code> to <code>PauliOp</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L75-L79">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.single_pauliop-Tuple{Char}" href="#QuantumLegos.PauliOps.single_pauliop-Tuple{Char}"><code>QuantumLegos.PauliOps.single_pauliop</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">single_pauliop(char::Char)::SinglePauliOp</code></pre><p>Convert <code>char</code> to <code>SinglePauliOp</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L38-L42">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.weight" href="#QuantumLegos.PauliOps.weight"><code>QuantumLegos.PauliOps.weight</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">weight(p::PauliOp, [init = 1])</code></pre><p>Weight of the operator <code>p</code>, i.e. non <span>$I$</span> operator.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L96-L100">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.xweight" href="#QuantumLegos.PauliOps.xweight"><code>QuantumLegos.PauliOps.xweight</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">xweight(p::PauliOp, [init = 1])</code></pre><p>Number of <span>$X, Y$</span> in <code>p</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L113-L117">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="QuantumLegos.PauliOps.zweight" href="#QuantumLegos.PauliOps.zweight"><code>QuantumLegos.PauliOps.zweight</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">zweight(p::PauliOp, [init = 1])</code></pre><p>Number of <span>$Z, Y$</span> in <code>p</code>. a</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/qwjyh/QuantumLegos.jl/blob/26303f34915c28c47cbb7c7550cd03634b962b94/src/PauliOps/PauliOps.jl#L129-L133">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../checkmatrix/">Details on check matrix operations »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Tuesday 7 May 2024 02:47">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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+var documenterSearchIndex = {"docs":
+[{"location":"checkmatrix/#Details-on-check-matrix-operations","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"","category":"section"},{"location":"checkmatrix/#Basic-features-of-check-matrix","page":"Details on check matrix operations","title":"Basic features of check matrix","text":"","category":"section"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"All action on state can be represented as operations on check matrix.\nQuantumLegos which don't connect each other are represented by block diagonal check matrix.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Thus, one needs only one check matrix to represent state.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Generated group from generators represented by check matrix is invariant under row-swap and row-multiplying transforms on the check matrix.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Thus, one can freely perform these row operations on check matrices.","category":"page"},{"location":"checkmatrix/#Construction-of-check-matrices-and-retrieving-features","page":"Details on check matrix operations","title":"Construction of check matrices and retrieving features","text":"","category":"section"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Use CheckMatrix from Matrix{Bool} and checkmatrix from Vector{PauliOp}.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"CheckMatrix\ncheckmatrix","category":"page"},{"location":"checkmatrix/#QuantumLegos.CheckMatrix","page":"Details on check matrix operations","title":"QuantumLegos.CheckMatrix","text":"CheckMatrix\n\nFields\n\ncmat: matrix itself (matrix of Bool since we consider only pauli ops.)\nnlegs: number of columns ÷ 2\nngens: number of rows\n\nConstructor\n\nCheckMatrix(cmat::AbstractMatrix{Bool})\n\nSee also checkmatrix.\n\n\n\n\n\n","category":"type"},{"location":"checkmatrix/#QuantumLegos.checkmatrix","page":"Details on check matrix operations","title":"QuantumLegos.checkmatrix","text":"checkmatrix(stbgens::AbstractVector{T})::CheckMatrix where {T <: PauliOp}\n\nGet CheckMatrix from stabilizer generator.\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Get xpart and zpart of check matrix.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.xpart\nQuantumLegos.zpart","category":"page"},{"location":"checkmatrix/#QuantumLegos.xpart","page":"Details on check matrix operations","title":"QuantumLegos.xpart","text":"xpart(cmat::CheckMatrix)\n\nGet X part (left half) of CheckMatrix.\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/#QuantumLegos.zpart","page":"Details on check matrix operations","title":"QuantumLegos.zpart","text":"zpart(cmat::CheckMatrix)\n\nGet Z part (right half) of CheckMatrix.\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Get generators from check matrix with generators.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"generators","category":"page"},{"location":"checkmatrix/#QuantumLegos.generators","page":"Details on check matrix operations","title":"QuantumLegos.generators","text":"generators(cmat::CheckMatrix)::Vector{PauliOp}\n\nGet generators from CheckMatrix.\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"See also, GeneratedPauliGroup and Base.Set to get group generated by generators.","category":"page"},{"location":"checkmatrix/#Direct-sum","page":"Details on check matrix operations","title":"Direct sum","text":"","category":"section"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Used when adding lego without edge.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.direct_sum","category":"page"},{"location":"checkmatrix/#QuantumLegos.direct_sum","page":"Details on check matrix operations","title":"QuantumLegos.direct_sum","text":"direct_sum(cmat_1::T, cmat_2::T)::T where {T <: CheckMatrix}\n\nReturns block diagonal CheckMatrix consists of cmat_1 and cmat_2.\n\nCheckMatrix transform\n\nWhen adding a lego l with check matrix H_l to a state with check matrix H_s, the resulting check matrix of the state will be\n\nbeginpmatrix\n    H_s  O   \n    O    H_l \nendpmatrix\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/#Self-tracing","page":"Details on check matrix operations","title":"Self-tracing","text":"","category":"section"},{"location":"checkmatrix/#self-tracing","page":"Details on check matrix operations","title":"self-tracing","text":"","category":"section"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.self_trace!","category":"page"},{"location":"checkmatrix/#QuantumLegos.self_trace!","page":"Details on check matrix operations","title":"QuantumLegos.self_trace!","text":"self_trace!(cmat::CheckMatrix, col_1::Integer, col_2::Integer)\n\nTake a self-trace of checkmatrix cmat with column col_1 and col_2.\n\nExample\n\nTODO\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"During self-tracing, pre-formatting and post-formatting described below is performed.","category":"page"},{"location":"checkmatrix/#pre-formatting","page":"Details on check matrix operations","title":"pre-formatting","text":"","category":"section"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.eliminate_column!","category":"page"},{"location":"checkmatrix/#QuantumLegos.eliminate_column!","page":"Details on check matrix operations","title":"QuantumLegos.eliminate_column!","text":"eliminate_column!(\n    cmat::CheckMatrix,\n    col::Integer,\n    avoid_row::AbstractVector{T},\n)::Union{Nothing, Int64} where {T <: Integer}\n\nPerform Gauss Elimination on col. Keep only one 1 on col and remove from other rows by performing multiplication. Choose row not in avoid_row. If all rows with 1 on col are in avoid_row, then the last row of them is chose to keep 1. If all rows on col is 0, return nothing.\n\nReturn\n\nRow index which have 1 on col. If all row on col is 0, return nothing.\n\nExample\n\njulia> ex_cmat = CheckMatrix(Bool[1 0 1 0; 1 1 1 1])\nCheckMatrix with 2 generators, 2 legs:\n  1  0 | 1  0\n  1  1 | 1  1\n\njulia> QuantumLegos.eliminate_column!(ex_cmat, 1, Int64[])\n1\n\njulia> ex_cmat\nCheckMatrix with 2 generators, 2 legs:\n  1  0 | 1  0\n  0  1 | 0  1\n\n\n\n\n\neliminate_column!(cmat::CheckMatrix, col::Integer,\navoid_row::AbstractVector{T}) where {T <: Union{Nothing, Integer}}\n\navoid_row can include Nothing, which is ignored in actual evaluation.\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.align_row!","category":"page"},{"location":"checkmatrix/#QuantumLegos.align_row!","page":"Details on check matrix operations","title":"QuantumLegos.align_row!","text":"align_row!(m::AbstractMatrix, row::Integer, occupied::Vector{Union{Nothing, Integer}}) -> Integer\nalign_row!(m::AbstractMatrix, row::Nothing, occupied::Vector{Union{Nothing, Integer}}) -> Nothing\n\nSwap row at row in m and row at next to the maximum in occupied. occupied is supposed to be a list of returns from eliminate_column!. If row is in occupied, do nothing and returns row. If row is nothing, return nothing.\n\nArguments\n\nm::AbstractMatrix: mutated\nrow::Union{Nothing, Integer}: row to be aligned\noccupied::Vector{Union{Nothing, Integer}}: indices of already occupied rows. row will be next to these rows.\n\nReturn\n\nRow index where row is moved.\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.align_row!","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"By this process, all columns to be traced have 1s on only 1st to 4th rows. So during QuantumLegos.self_trace!, all stabilizers generated from top three rows are calculated and perform operator matching.","category":"page"},{"location":"checkmatrix/#post-formatting","page":"Details on check matrix operations","title":"post-formatting","text":"","category":"section"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"Remove rows which are linear combinations of other rows.","category":"page"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.ref!","category":"page"},{"location":"checkmatrix/#QuantumLegos.ref!","page":"Details on check matrix operations","title":"QuantumLegos.ref!","text":"ref!(cmat::CheckMatrix) -> Int\n\nConvert cmat to row echelon form.\n\nReturns rank of check matrix.\n\nExamples\n\njulia> cmat = CheckMatrix(Bool[\n           1 0 1 0 1 1 0 1\n           0 1 0 0 0 1 0 0\n           1 1 1 0 1 0 1 1\n           0 1 0 0 0 1 0 0\n       ])\nCheckMatrix with 4 generators, 4 legs:\n  1  0  1  0 | 1  1  0  1\n  0  1  0  0 | 0  1  0  0\n  1  1  1  0 | 1  0  1  1\n  0  1  0  0 | 0  1  0  0\n\n\njulia> QuantumLegos.ref!(cmat)\n3\n\njulia> cmat\nCheckMatrix with 4 generators, 4 legs:\n  1  0  1  0 | 1  1  0  1\n  0  1  0  0 | 0  1  0  0\n  0  0  0  0 | 0  0  1  0\n  0  0  0  0 | 0  0  0  0\n\n\n\n\n\n\n","category":"function"},{"location":"checkmatrix/","page":"Details on check matrix operations","title":"Details on check matrix operations","text":"QuantumLegos.eliminate_dependent_row!","category":"page"},{"location":"checkmatrix/#QuantumLegos.eliminate_dependent_row!","page":"Details on check matrix operations","title":"QuantumLegos.eliminate_dependent_row!","text":"eliminate_dependent_row!(cmat::CheckMatrix) -> CheckMatrix\n\nRemove dependent rows to keep only independent generators.\n\n\n\n\n\n","category":"function"},{"location":"distance/#How-to-calculate-code-distance-from-the-state.","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"","category":"section"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"warning: WIP\nThis document is not fully completed.","category":"page"},{"location":"distance/#Definition-of-code-distance.","page":"How to calculate code distance from the state.","title":"Definition of code distance.","text":"","category":"section"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Let's consider encoding circuit with 1 logical bit and k physical qubits[1]. Then this encoding has two physical basis, ket0_L and ket1_L. The distance of this encoding is the minimum bit flip required to convert between ket0_L and ket1_L.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"[1]: Not all state can be formalized like this. TODO","category":"page"},{"location":"distance/#Classification-of-the-stabilizers.","page":"How to calculate code distance from the state.","title":"Classification of the stabilizers.","text":"","category":"section"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"When treating State, the logical leg is not assigned and one can treat all stabilizers equally. However, if logical leg is assigned to the state, these stabilizers can be classified to 4 groups.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"stabilizers on physical qubits\n​barX, which corresponds to logical X\n​barZ, which corresponds to logical Z\n​barY, which corresponds to logical Y","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Let ketV is the dual state of the channel or encoding map n 1 d,","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"distance = min_S in stabilizers left i mid barZ_i  S_i right","category":"page"},{"location":"distance/#Calculating-code-distance-from-the-check-matrix.","page":"How to calculate code distance from the state.","title":"Calculating code distance from the check matrix.","text":"","category":"section"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"TODO: nor required if the performance doesn't matter.","category":"page"},{"location":"distance/#Examples","page":"How to calculate code distance from the state.","title":"Examples","text":"","category":"section"},{"location":"distance/#[[5,-1,-3]]-code","page":"How to calculate code distance from the state.","title":"5 1 3 code","text":"","category":"section"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"​5 1 3 code has 4 stabilizers generators, XZZXI IXZZX XIXZZ ZXIXZ and 2 logical operators, barX = XXXXX and barZ = ZZZZZ. Therefore, stabilizer generators for the corresponding state 6 0 is IXZZXI IIXZZX IXIXZZ IZXIXZ XXXXXX ZZZZZZ.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Let's construct 6 0 state on QuantumLegos.jl.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"julia> using QuantumLegos\n\njulia> stab_513 = pauliop.([\"IXZZXI\", \"IIXZZX\", \"IXIXZZ\", \"IZXIXZ\", \"XXXXXX\", \"ZZZZZZ\"])\n6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IXZZXI\")\n pauliop(\"IIXZZX\")\n pauliop(\"IXIXZZ\")\n pauliop(\"IZXIXZ\")\n pauliop(\"XXXXXX\")\n pauliop(\"ZZZZZZ\")\n\njulia> lego_513 = Lego(stab_513)\nLego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IXZZXI\"), pauliop(\"IIXZZX\"), pauliop(\"IXIXZZ\"), pauliop(\"IZXIXZ\"), pauliop(\"XXXXXX\"), pauliop(\"ZZZZZZ\")])\n\njulia> state_513 = State([lego_513], edge.([]))\nState(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IXZZXI\"), pauliop(\"IIXZZX\"), pauliop(\"IXIXZZ\"), pauliop(\"IZXIXZ\"), pauliop(\"XXXXXX\"), pauliop(\"ZZZZZZ\")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 1 … 0 0; 0 0 … 1 0; … ; 1 1 … 0 0; 0 0 … 1 1], 6, 6))\n","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Then collect generators of the state.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"julia> normalizers = state_513.cmat |> generators |> GeneratedPauliGroup |> collect\n64-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIIIII\")\n pauliop(\"IXZZXI\")\n pauliop(\"IIXZZX\")\n pauliop(\"IXYIYX\")\n pauliop(\"IXIXZZ\")\n pauliop(\"IIZYYZ\")\n pauliop(\"IXXYIY\")\n pauliop(\"IIYXXY\")\n pauliop(\"IZXIXZ\")\n pauliop(\"IYYZIZ\")\n ⋮\n pauliop(\"YYIZZI\")\n pauliop(\"YXZYZX\")\n pauliop(\"YIIXYX\")\n pauliop(\"YXYXII\")\n pauliop(\"YIXYXI\")\n pauliop(\"YIZZIY\")\n pauliop(\"YXIIXY\")\n pauliop(\"YIYIZZ\")\n pauliop(\"YXXZYZ\")\n","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Get stabilizer and normalizers of the 5 1 3 code by assigning the first leg as logical.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"julia> stabs = filter(x -> x[1] == PauliOps.I, normalizers)\n16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIIIII\")\n pauliop(\"IXZZXI\")\n pauliop(\"IIXZZX\")\n pauliop(\"IXYIYX\")\n pauliop(\"IXIXZZ\")\n pauliop(\"IIZYYZ\")\n pauliop(\"IXXYIY\")\n pauliop(\"IIYXXY\")\n pauliop(\"IZXIXZ\")\n pauliop(\"IYYZIZ\")\n pauliop(\"IZIZYY\")\n pauliop(\"IYZIZY\")\n pauliop(\"IYXXYI\")\n pauliop(\"IZYYZI\")\n pauliop(\"IYIYXX\")\n pauliop(\"IZZXIX\")\n\njulia> norm_x = filter(x -> x[1] == PauliOps.X, normalizers)\n16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"XXXXXX\")\n pauliop(\"XIYYIX\")\n pauliop(\"XXIYYI\")\n pauliop(\"XIZXZI\")\n pauliop(\"XIXIYY\")\n pauliop(\"XXYZZY\")\n pauliop(\"XIIZXZ\")\n pauliop(\"XXZIIZ\")\n pauliop(\"XYIXIY\")\n pauliop(\"XZZYXY\")\n pauliop(\"XYXYZZ\")\n pauliop(\"XZYXYZ\")\n pauliop(\"XZIIZX\")\n pauliop(\"XYZZYX\")\n pauliop(\"XZXZII\")\n pauliop(\"XYYIXI\")\n\njulia> norm_y = filter(x -> x[1] == PauliOps.Y, normalizers)\n16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"YYYYYY\")\n pauliop(\"YZXXZY\")\n pauliop(\"YYZXXZ\")\n pauliop(\"YZIYIZ\")\n pauliop(\"YZYZXX\")\n pauliop(\"YYXIIX\")\n pauliop(\"YZZIYI\")\n pauliop(\"YYIZZI\")\n pauliop(\"YXZYZX\")\n pauliop(\"YIIXYX\")\n pauliop(\"YXYXII\")\n pauliop(\"YIXYXI\")\n pauliop(\"YIZZIY\")\n pauliop(\"YXIIXY\")\n pauliop(\"YIYIZZ\")\n pauliop(\"YXXZYZ\")\n\njulia> norm_z = filter(x -> x[1] == PauliOps.Z, normalizers)\n16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"ZZZZZZ\")\n pauliop(\"ZYIIYZ\")\n pauliop(\"ZZYIIY\")\n pauliop(\"ZYXZXY\")\n pauliop(\"ZYZYII\")\n pauliop(\"ZZIXXI\")\n pauliop(\"ZYYXZX\")\n pauliop(\"ZZXYYX\")\n pauliop(\"ZIYZYI\")\n pauliop(\"ZXXIZI\")\n pauliop(\"ZIZIXX\")\n pauliop(\"ZXIZIX\")\n pauliop(\"ZXYYXZ\")\n pauliop(\"ZIXXIZ\")\n pauliop(\"ZXZXYY\")\n pauliop(\"ZIIYZY\")\n","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"These normalizers are generated from one logical operator and stabilizers.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"julia> map(x -> x .* pauliop(\"XXXXXX\"), stabs) |> Set == Set(norm_x)\ntrue\n\njulia> map(x -> x .* pauliop(\"ZZZZZZ\"), stabs) |> Set == Set(norm_x)\nfalse\n\njulia> map(x -> x .* pauliop(\"ZZZZZZ\"), stabs) |> Set == Set(norm_z)\ntrue\n\njulia> map(x -> x .* pauliop(\"YYYYYY\"), stabs) |> Set == Set(norm_y)\ntrue\n\njulia> using IterTools\n\njulia> groupby(x -> x[1], normalizers) .|> Set == Set.([stabs, norm_x, norm_z, norm_y])\ntrue\n","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Define a function to get weight of the operator.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"julia> function weight(x, i = 1)\n           count(x[i:end] .!= PauliOps.I)\n       end\nweight (generic function with 2 methods)\n\njulia> weight(pauliop(\"XIXIXI\"))\n3\n\njulia> weight(pauliop(\"XIXIXI\"), 2)\n2\n","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Calculate coefficients of enumerator polynomial.","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"julia> using DataStructures\n\njulia> stabs .|> weight |> counter\nAccumulator{Int64, Int64} with 2 entries:\n  0 => 1\n  4 => 15\n\njulia> function weight(i::Integer)\n           Base.Fix2(weight, i)\n       end\nweight (generic function with 3 methods)\n\njulia> normalizers .|> weight(2) |> counter\nAccumulator{Int64, Int64} with 4 entries:\n  0 => 1\n  4 => 15\n  5 => 18\n  3 => 30\n\njulia> [norm_x..., norm_y..., norm_z...] .|> weight(2) |> counter\nAccumulator{Int64, Int64} with 2 entries:\n  5 => 18\n  3 => 30\n\njulia> [norm_x..., norm_y..., norm_z...] .|> weight(2) |> counter |> keys |> minimum\n3\n","category":"page"},{"location":"distance/","page":"How to calculate code distance from the state.","title":"How to calculate code distance from the state.","text":"Code distance of the encoding is the minimum degree of the non-zero term in the normalizer's polynomial(B) and not in the stabilizer's polynomial(A). So the code distance of this encoding is 3.","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = QuantumLegos","category":"page"},{"location":"#QuantumLegos","page":"Home","title":"QuantumLegos","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Documentation for QuantumLegos.","category":"page"},{"location":"","page":"Home","title":"Home","text":"All contents:","category":"page"},{"location":"","page":"Home","title":"Home","text":"","category":"page"},{"location":"#Example","page":"Home","title":"Example","text":"","category":"section"},{"location":"#CheckMatrix-and-defining-Lego","page":"Home","title":"CheckMatrix and defining Lego","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"julia> using QuantumLegos\n\njulia> stabilizers = pauliop.([\"IIXXXX\", \"IIZZZZ\", \"ZIZZII\", \"IZZIZI\", \"IXXXII\", \"XIXIXI\"])\n6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIXXXX\")\n pauliop(\"IIZZZZ\")\n pauliop(\"ZIZZII\")\n pauliop(\"IZZIZI\")\n pauliop(\"IXXXII\")\n pauliop(\"XIXIXI\")\n\njulia> cmat = checkmatrix(stabilizers)\nCheckMatrix with 6 generators, 6 legs:\n  0  0  1  1  1  1 | 0  0  0  0  0  0\n  0  0  0  0  0  0 | 0  0  1  1  1  1\n  0  0  0  0  0  0 | 1  0  1  1  0  0\n  0  0  0  0  0  0 | 0  1  1  0  1  0\n  0  1  1  1  0  0 | 0  0  0  0  0  0\n  1  0  1  0  1  0 | 0  0  0  0  0  0\n\n\njulia> cmat.nlegs\n6\n\njulia> cmat.ngens\n6\n\njulia> cmat.cmat\n6×12 Matrix{Bool}:\n 0  0  1  1  1  1  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  1  1  1  1\n 0  0  0  0  0  0  1  0  1  1  0  0\n 0  0  0  0  0  0  0  1  1  0  1  0\n 0  1  1  1  0  0  0  0  0  0  0  0\n 1  0  1  0  1  0  0  0  0  0  0  0\n\njulia> # define lego\n\njulia> lego = Lego(stabilizers)\nLego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])\n\njulia> lego.stabgens |> checkmatrix\nCheckMatrix with 6 generators, 6 legs:\n  0  0  1  1  1  1 | 0  0  0  0  0  0\n  0  0  0  0  0  0 | 0  0  1  1  1  1\n  0  0  0  0  0  0 | 1  0  1  1  0  0\n  0  0  0  0  0  0 | 0  1  1  0  1  0\n  0  1  1  1  0  0 | 0  0  0  0  0  0\n  1  0  1  0  1  0 | 0  0  0  0  0  0\n","category":"page"},{"location":"","page":"Home","title":"Home","text":"pauliop\ncheckmatrix and CheckMatrix\nLego","category":"page"},{"location":"#Defining-and-Updating-State","page":"Home","title":"Defining and Updating State","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"julia> using QuantumLegos\n\njulia> stabilizers = pauliop.([\"IIXXXX\", \"IIZZZZ\", \"ZIZZII\", \"IZZIZI\", \"IXXXII\", \"XIXIXI\"])\n6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIXXXX\")\n pauliop(\"IIZZZZ\")\n pauliop(\"ZIZZII\")\n pauliop(\"IZZIZI\")\n pauliop(\"IXXXII\")\n pauliop(\"XIXIXI\")\n\njulia> lego = Lego(stabilizers)\nLego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])\n\njulia> # state with 1 lego, 0 leg\n\njulia> st = State([lego, ], Tuple{LegoLeg, LegoLeg}[])\nState(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 1 1; … ; 0 1 … 0 0; 1 0 … 0 0], 6, 6))\n\njulia> st.cmat.cmat\n6×12 Matrix{Bool}:\n 0  0  1  1  1  1  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  1  1  1  1\n 0  0  0  0  0  0  1  0  1  1  0  0\n 0  0  0  0  0  0  0  1  1  0  1  0\n 0  1  1  1  0  0  0  0  0  0  0  0\n 1  0  1  0  1  0  0  0  0  0  0  0\n\njulia> add_lego!(st, lego)\nState(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], 12, 12))\n\njulia> st.cmat.cmat\n12×24 Matrix{Bool}:\n 0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  1  0  0  0  0  0  0  0\n 0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n 1  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0\n 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1  1\n 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1  0  0\n 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  1  0\n 0  0  0  0  0  0  0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0\n 0  0  0  0  0  0  1  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0\n\njulia> # state with 2 legos, 0 leg\n\njulia> st2 = State([lego, lego], Tuple{LegoLeg, LegoLeg}[])\nState(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])], Tuple{LegoLeg, LegoLeg}[], CheckMatrix(Bool[0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0], 12, 12))\n\njulia> st == st2\ntrue","category":"page"},{"location":"#2-Lego-1-edge-state","page":"Home","title":"2 Lego 1 edge state","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"julia> using QuantumLegos\n\njulia> stabilizers = pauliop.([\"IIXXXX\", \"IIZZZZ\", \"ZIZZII\", \"IZZIZI\", \"IXXXII\", \"XIXIXI\"])\n6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIXXXX\")\n pauliop(\"IIZZZZ\")\n pauliop(\"ZIZZII\")\n pauliop(\"IZZIZI\")\n pauliop(\"IXXXII\")\n pauliop(\"XIXIXI\")\n\njulia> lego = Lego(stabilizers)\nLego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])\n\njulia> state = State([lego, lego], edge.([((1, 3), (2, 3))]))\nState(Lego[Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")]), Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXXXX\"), pauliop(\"IIZZZZ\"), pauliop(\"ZIZZII\"), pauliop(\"IZZIZI\"), pauliop(\"IXXXII\"), pauliop(\"XIXIXI\")])], Tuple{LegoLeg, LegoLeg}[(LegoLeg(1, 3), LegoLeg(2, 3))], CheckMatrix(Bool[1 0 … 0 0; 0 1 … 0 0; … ; 0 0 … 1 1; 0 0 … 0 1], 10, 10))\n\njulia> state.cmat\nCheckMatrix with 10 generators, 10 legs:\n  1  0  1  0  1  0  0  0  0  0 | 0  0  0  0  0  0  0  0  0  0\n  0  1  0  1  1  0  0  0  0  0 | 0  0  0  0  0  0  0  0  0  0\n  0  0  1  1  1  0  0  1  1  1 | 0  0  0  0  0  0  0  0  0  0\n  0  0  0  0  0  1  0  1  0  1 | 0  0  0  0  0  0  0  0  0  0\n  0  0  0  0  0  0  1  0  1  1 | 0  0  0  0  0  0  0  0  0  0\n  0  0  0  0  0  0  0  0  0  0 | 1  0  0  1  1  0  0  0  0  0\n  0  0  0  0  0  0  0  0  0  0 | 0  1  1  0  1  0  0  0  0  0\n  0  0  0  0  0  0  0  0  0  0 | 0  0  1  1  1  0  0  1  1  1\n  0  0  0  0  0  0  0  0  0  0 | 0  0  0  0  0  1  0  0  1  1\n  0  0  0  0  0  0  0  0  0  0 | 0  0  0  0  0  0  1  1  0  1\n\n\njulia> pg = state.cmat |> generators |> GeneratedPauliGroup\nGeneratedPauliGroup{10}(StaticArraysCore.SVector{10, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"XIXIXIIIII\"), pauliop(\"IXIXXIIIII\"), pauliop(\"IIXXXIIXXX\"), pauliop(\"IIIIIXIXIX\"), pauliop(\"IIIIIIXIXX\"), pauliop(\"ZIIZZIIIII\"), pauliop(\"IZZIZIIIII\"), pauliop(\"IIZZZIIZZZ\"), pauliop(\"IIIIIZIIZZ\"), pauliop(\"IIIIIIZZIZ\")], IterTools.Subsets{Vector{StaticArraysCore.SVector{N, QuantumLegos.PauliOps.SinglePauliOp} where N}}(StaticArraysCore.SVector{N, QuantumLegos.PauliOps.SinglePauliOp} where N[pauliop(\"XIXIXIIIII\"), pauliop(\"IXIXXIIIII\"), pauliop(\"IIXXXIIXXX\"), pauliop(\"IIIIIXIXIX\"), pauliop(\"IIIIIIXIXX\"), pauliop(\"ZIIZZIIIII\"), pauliop(\"IZZIZIIIII\"), pauliop(\"IIZZZIIZZZ\"), pauliop(\"IIIIIZIIZZ\"), pauliop(\"IIIIIIZZIZ\")]))\n\njulia> pauliop(\"XIIXIXIIXI\") in pg\ntrue\n","category":"page"},{"location":"#Internal(how-it-works)","page":"Home","title":"Internal(how it works)","text":"","category":"section"},{"location":"#Notes-on-Overall-flow","page":"Home","title":"Notes on Overall flow","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Details on [1]","category":"page"},{"location":"","page":"Home","title":"Home","text":"state is translated to a single check matrix\nthe size is ≤ N times 2N where N is maximum number of lego logs.\nany contraction can be performed on this single check matrix\nif the check matrix can be represented as direct sum of matrices with k N columns where k  ℕ, then they are not contracted","category":"page"},{"location":"#Construction-of-State","page":"Home","title":"Construction of State","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Construction of State is completed by calling State constructor recursively.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Construct State without edge. Just adding legos. Checkmatrix is just a direct sum of each lego's checkmatrix\nConcatenate each edges. During this operation, self tracing of checkmatrix is evaluated.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Each constructor calls action function (which is a map from State to State). Therefore, action functions can be used both for direct construction of State and action application to State during the game.","category":"page"},{"location":"#API","page":"Home","title":"API","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"","category":"page"},{"location":"","page":"Home","title":"Home","text":"Modules = [QuantumLegos]\nPages   = [\"game.jl\"]","category":"page"},{"location":"#QuantumLegos.Lego","page":"Home","title":"QuantumLegos.Lego","text":"Quantum lego with N legs.\n\nFields\n\nnlegs::Int64: number of legs, equals N\nstabgens::SVector{N, PauliOp{N}}: stabilizer generators. vector of PauliOp\n\nConstructor\n\nLego([nlegs::Integer], stabgens::AbstractVector{PauliOp{N}})\n\nConstructor for Lego. nlegs is optional (default is length of the first stabilizer generator).\n\nExample\n\njulia> stabgens = pauliop.([\"II\", \"XX\"])\n2-element Vector{StaticArraysCore.SVector{2, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"II\")\n pauliop(\"XX\")\n\njulia> Lego(stabgens)\nLego{2}(2, StaticArraysCore.SVector{2, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"II\"), pauliop(\"XX\")])\n\n\n\n\n\n","category":"type"},{"location":"#QuantumLegos.LegoLeg","page":"Home","title":"QuantumLegos.LegoLeg","text":"mutable struct State\n\nTo be used in State.\n\nFields\n\nlego_id::Int64: index in legos in State\nedge_id::Int64: index in Lego in legos in State. No validation check included in LegoLeg.\n\nExample\n\njulia> x = LegoLeg.([(2, 1), (1, 1), (1, 0)])\n3-element Vector{LegoLeg}:\n LegoLeg(2, 1)\n LegoLeg(1, 1)\n LegoLeg(1, 0)\n\njulia> sort(x)\n3-element Vector{LegoLeg}:\n LegoLeg(1, 0)\n LegoLeg(1, 1)\n LegoLeg(2, 1)\n\n\n\n\n\n","category":"type"},{"location":"#QuantumLegos.State","page":"Home","title":"QuantumLegos.State","text":"mutable struct State\n\nState (in p.4)\n\nFields\n\nlegos: Vector{Lego}\nedges: Vector of ((lego_i, leg_n), (lego_j, leg_m)).   Each element is sorted (i.e. lego_i < lego_j or lego_i == lego_j && leg_n < leg_m).   This feature is used in is_connected_to_firstlego.\ncmat::CheckMatrix: CheckMatrix\n\nConstructor\n\nState(legos::Vector{Lego{N}}, edges::Vector{Tuple{LegoLeg, LegoLeg}})\n\nMethods with\n\nadd_lego!\nadd_edge!\n\nExample\n\nTODO\n\n\n\n\n\n","category":"type"},{"location":"#QuantumLegos._naive_distance-Tuple{State}","page":"Home","title":"QuantumLegos._naive_distance","text":"Calculate distance. Use minimum distance of all generated normalizers.\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.add_edge!-Tuple{State, LegoLeg, LegoLeg}","page":"Home","title":"QuantumLegos.add_edge!","text":"add_edge!(state::State, leg_1::LegoLeg, leg_2::LegoLeg)::State\n\nAdd a new edge between leg_1 and leg_2, updating state.\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.add_lego!-Tuple{State, Lego}","page":"Home","title":"QuantumLegos.add_lego!","text":"add_lego!(state::State, lego::Lego) -> State\n\nAdd a new lego, updating state.\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.cmat_index-Tuple{State, LegoLeg}","page":"Home","title":"QuantumLegos.cmat_index","text":"cmat_index(state::State, leg::LegoLeg)::Int64\n\nGet column index corresponds to leg in check matrix of state. If given leg is already connected, it throws ArgumentError. If given lego_id of leg is out of state.legos, throws ArgumentError.\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.distance-Tuple{State}","page":"Home","title":"QuantumLegos.distance","text":"distance(state::State) -> Int\n\nCalculate code distance when the first leg of state is assigned as logical.\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.edge","page":"Home","title":"QuantumLegos.edge","text":"Helper function to create Tuple{LegoLeg, LegoLeg} to represent edge.\n\n\n\n\n\n","category":"function"},{"location":"#QuantumLegos.edge-Union{Tuple{NTuple{4, T}}, Tuple{T}} where T<:Integer","page":"Home","title":"QuantumLegos.edge","text":"edge(t::Tuple{T, T, T, T}) where {T <: Integer}\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.edge-Union{Tuple{Tuple{T, T}}, Tuple{T}} where T<:Tuple{Integer, Integer}","page":"Home","title":"QuantumLegos.edge","text":"edge(t::Tuple{T, T}) where {T <: Tuple{Integer, Integer}}\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.edge-Union{Tuple{T}, NTuple{4, T}} where T<:Integer","page":"Home","title":"QuantumLegos.edge","text":"edge(x::T, y::T, z::T, w::T) where {T <: Integer}\n\n\n\n\n\n","category":"method"},{"location":"#QuantumLegos.is_connected_to_firstlego-Tuple{State}","page":"Home","title":"QuantumLegos.is_connected_to_firstlego","text":"is_connected_to_firstlego(state::State)::BitVector\n\nReturns vector which stores whether each lego is connected to the first lego.\n\n\n\n\n\n","category":"method"},{"location":"","page":"Home","title":"Home","text":"[1]: C. Cao and B. Lackey, ‘Approximate Bacon-Shor code and holography’, J. High Energ. Phys., vol. 2021, no. 5, p. 127, May 2021, doi: 10.1007/JHEP05(2021)127.","category":"page"},{"location":"pauliops/#PauliOps","page":"PauliOps","title":"PauliOps","text":"","category":"section"},{"location":"pauliops/","page":"PauliOps","title":"PauliOps","text":"submodule","category":"page"},{"location":"pauliops/#Example","page":"PauliOps","title":"Example","text":"","category":"section"},{"location":"pauliops/","page":"PauliOps","title":"PauliOps","text":"julia> using QuantumLegos\n\njulia> p = PauliOps.single_pauliop('I')\nI::SinglePauliOp = 0\n\njulia> typeof(p)\nEnum QuantumLegos.PauliOps.SinglePauliOp:\nI = 0\nX = 1\nY = 2\nZ = 3\n\njulia> pauliop(\"IXYZ\")\n4-element PauliOp:\n I::SinglePauliOp = 0\n X::SinglePauliOp = 1\n Y::SinglePauliOp = 2\n Z::SinglePauliOp = 3\n\njulia> typeof(ans)\nSVector{4, SinglePauliOp} (alias for StaticArraysCore.SArray{Tuple{4}, QuantumLegos.PauliOps.SinglePauliOp, 1, 4})\n\njulia> PauliOps.I * PauliOps.X\nX::SinglePauliOp = 1\n\njulia> PauliOps.X * PauliOps.Z\nY::SinglePauliOp = 2\n\njulia> pauliop(\"IIX\") .* pauliop(\"XIY\")\n3-element PauliOp:\n X::SinglePauliOp = 1\n I::SinglePauliOp = 0\n Z::SinglePauliOp = 3","category":"page"},{"location":"pauliops/#API","page":"PauliOps","title":"API","text":"","category":"section"},{"location":"pauliops/","page":"PauliOps","title":"PauliOps","text":"Modules = [PauliOps]","category":"page"},{"location":"pauliops/#QuantumLegos.PauliOps","page":"PauliOps","title":"QuantumLegos.PauliOps","text":"Pauli operator\n\n\n\n\n\n","category":"module"},{"location":"pauliops/#QuantumLegos.PauliOps.GeneratedPauliGroup","page":"PauliOps","title":"QuantumLegos.PauliOps.GeneratedPauliGroup","text":"struct GeneratedPauliGroup\n\nIterator for group generated from gens.\n\nGeneratedPauliGroup(gens::AbstractVector{T}) where {T <: PauliOp}\n\nExamples\n\njulia> gens = pauliop.([\"IIXX\", \"IZZI\"])\n2-element Vector{StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIXX\")\n pauliop(\"IZZI\")\n\njulia> g = PauliOps.GeneratedPauliGroup(gens)\nGeneratedPauliGroup{4}(StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXX\"), pauliop(\"IZZI\")], IterTools.Subsets{Vector{StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}}}(StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}[pauliop(\"IIXX\"), pauliop(\"IZZI\")]))\n\njulia> collect(g)\n4-element Vector{StaticArraysCore.SVector{4, QuantumLegos.PauliOps.SinglePauliOp}}:\n pauliop(\"IIII\")\n pauliop(\"IIXX\")\n pauliop(\"IZZI\")\n pauliop(\"IZYX\")\n\n\n\n\n\n","category":"type"},{"location":"pauliops/#QuantumLegos.PauliOps.PauliOp","page":"PauliOps","title":"QuantumLegos.PauliOps.PauliOp","text":"PauliOp{N}\n\nPauli operator on multiple qubits.\n\n\n\n\n\n","category":"type"},{"location":"pauliops/#QuantumLegos.PauliOps.SinglePauliOp","page":"PauliOps","title":"QuantumLegos.PauliOps.SinglePauliOp","text":"@enum SinglePauliOp begin\n    I\n    X\n    Y\n    Z\nend\n\nPauli Operator on a single qubit.\n\n\n\n\n\n","category":"type"},{"location":"pauliops/#QuantumLegos.PauliOps.pauliop-Tuple{AbstractString}","page":"PauliOps","title":"QuantumLegos.PauliOps.pauliop","text":"pauliop(str::AbstractString)::PauliOp\n\nConvert str to PauliOp.\n\n\n\n\n\n","category":"method"},{"location":"pauliops/#QuantumLegos.PauliOps.single_pauliop-Tuple{Char}","page":"PauliOps","title":"QuantumLegos.PauliOps.single_pauliop","text":"single_pauliop(char::Char)::SinglePauliOp\n\nConvert char to SinglePauliOp.\n\n\n\n\n\n","category":"method"},{"location":"pauliops/#QuantumLegos.PauliOps.weight","page":"PauliOps","title":"QuantumLegos.PauliOps.weight","text":"weight(p::PauliOp, [init = 1])\n\nWeight of the operator p, i.e. non I operator.\n\n\n\n\n\n","category":"function"},{"location":"pauliops/#QuantumLegos.PauliOps.xweight","page":"PauliOps","title":"QuantumLegos.PauliOps.xweight","text":"xweight(p::PauliOp, [init = 1])\n\nNumber of X Y in p.\n\n\n\n\n\n","category":"function"},{"location":"pauliops/#QuantumLegos.PauliOps.zweight","page":"PauliOps","title":"QuantumLegos.PauliOps.zweight","text":"zweight(p::PauliOp, [init = 1])\n\nNumber of Z Y in p. a\n\n\n\n\n\n","category":"function"}]
+}
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diff --git a/versions.js b/versions.js
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