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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>How to calculate code distance from the state. · QuantumLegos.jl</title><meta name="title" content="How to calculate code distance from the state. · QuantumLegos.jl"/><meta property="og:title" content="How to calculate code distance from the state. · QuantumLegos.jl"/><meta property="twitter:title" content="How to calculate code distance from the state. · 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script data-outdated-warner src="../assets/warner.js"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 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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'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'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's construct <span>$[[6, 0]]$</span> state on QuantumLegos.jl.</p><pre><code class="language-julia-repl hljs">julia> using QuantumLegos
julia> stab_513 = pauliop.(["IXZZXI", "IIXZZX", "IXIXZZ", "IZXIXZ", "XXXXXX", "ZZZZZZ"])
6-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
pauliop("IXZZXI")
pauliop("IIXZZX")
pauliop("IXIXZZ")
pauliop("IZXIXZ")
pauliop("XXXXXX")
pauliop("ZZZZZZ")
julia> lego_513 = Lego(stab_513)
Lego{6}(6, StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}[pauliop("IXZZXI"), pauliop("IIXZZX"), pauliop("IXIXZZ"), pauliop("IZXIXZ"), pauliop("XXXXXX"), pauliop("ZZZZZZ")])
julia> state_513 = State([lego_513], edge.([]))
State(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))
</code></pre><p>Then collect generators of the state.</p><pre><code class="language-julia-repl hljs">julia> normalizers = state_513.cmat |> generators |> GeneratedPauliGroup |> collect
64-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
pauliop("IIIIII")
pauliop("IXZZXI")
pauliop("IIXZZX")
pauliop("IXYIYX")
pauliop("IXIXZZ")
pauliop("IIZYYZ")
pauliop("IXXYIY")
pauliop("IIYXXY")
pauliop("IZXIXZ")
pauliop("IYYZIZ")
⋮
pauliop("YYIZZI")
pauliop("YXZYZX")
pauliop("YIIXYX")
pauliop("YXYXII")
pauliop("YIXYXI")
pauliop("YIZZIY")
pauliop("YXIIXY")
pauliop("YIYIZZ")
pauliop("YXXZYZ")
</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> stabs = filter(x -> x[1] == PauliOps.I, normalizers)
16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
pauliop("IIIIII")
pauliop("IXZZXI")
pauliop("IIXZZX")
pauliop("IXYIYX")
pauliop("IXIXZZ")
pauliop("IIZYYZ")
pauliop("IXXYIY")
pauliop("IIYXXY")
pauliop("IZXIXZ")
pauliop("IYYZIZ")
pauliop("IZIZYY")
pauliop("IYZIZY")
pauliop("IYXXYI")
pauliop("IZYYZI")
pauliop("IYIYXX")
pauliop("IZZXIX")
julia> norm_x = filter(x -> x[1] == PauliOps.X, normalizers)
16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
pauliop("XXXXXX")
pauliop("XIYYIX")
pauliop("XXIYYI")
pauliop("XIZXZI")
pauliop("XIXIYY")
pauliop("XXYZZY")
pauliop("XIIZXZ")
pauliop("XXZIIZ")
pauliop("XYIXIY")
pauliop("XZZYXY")
pauliop("XYXYZZ")
pauliop("XZYXYZ")
pauliop("XZIIZX")
pauliop("XYZZYX")
pauliop("XZXZII")
pauliop("XYYIXI")
julia> norm_y = filter(x -> x[1] == PauliOps.Y, normalizers)
16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
pauliop("YYYYYY")
pauliop("YZXXZY")
pauliop("YYZXXZ")
pauliop("YZIYIZ")
pauliop("YZYZXX")
pauliop("YYXIIX")
pauliop("YZZIYI")
pauliop("YYIZZI")
pauliop("YXZYZX")
pauliop("YIIXYX")
pauliop("YXYXII")
pauliop("YIXYXI")
pauliop("YIZZIY")
pauliop("YXIIXY")
pauliop("YIYIZZ")
pauliop("YXXZYZ")
julia> norm_z = filter(x -> x[1] == PauliOps.Z, normalizers)
16-element Vector{StaticArraysCore.SVector{6, QuantumLegos.PauliOps.SinglePauliOp}}:
pauliop("ZZZZZZ")
pauliop("ZYIIYZ")
pauliop("ZZYIIY")
pauliop("ZYXZXY")
pauliop("ZYZYII")
pauliop("ZZIXXI")
pauliop("ZYYXZX")
pauliop("ZZXYYX")
pauliop("ZIYZYI")
pauliop("ZXXIZI")
pauliop("ZIZIXX")
pauliop("ZXIZIX")
pauliop("ZXYYXZ")
pauliop("ZIXXIZ")
pauliop("ZXZXYY")
pauliop("ZIIYZY")
</code></pre><p>These normalizers are generated from one logical operator and stabilizers.</p><pre><code class="language-julia-repl hljs">julia> map(x -> x .* pauliop("XXXXXX"), stabs) |> Set == Set(norm_x)
true
julia> map(x -> x .* pauliop("ZZZZZZ"), stabs) |> Set == Set(norm_x)
false
julia> map(x -> x .* pauliop("ZZZZZZ"), stabs) |> Set == Set(norm_z)
true
julia> map(x -> x .* pauliop("YYYYYY"), stabs) |> Set == Set(norm_y)
true
julia> using IterTools
julia> groupby(x -> x[1], normalizers) .|> 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> function weight(x, i = 1)
count(x[i:end] .!= PauliOps.I)
end
weight (generic function with 2 methods)
julia> weight(pauliop("XIXIXI"))
3
julia> weight(pauliop("XIXIXI"), 2)
2
</code></pre><p>Calculate coefficients of enumerator polynomial.</p><pre><code class="language-julia-repl hljs">julia> using DataStructures
julia> stabs .|> weight |> counter
Accumulator{Int64, Int64} with 2 entries:
0 => 1
4 => 15
julia> function weight(i::Integer)
Base.Fix2(weight, i)
end
weight (generic function with 3 methods)
julia> normalizers .|> weight(2) |> counter
Accumulator{Int64, Int64} with 4 entries:
0 => 1
4 => 15
5 => 18
3 => 30
julia> [norm_x..., norm_y..., norm_z...] .|> weight(2) |> counter
Accumulator{Int64, Int64} with 2 entries:
5 => 18
3 => 30
julia> [norm_x..., norm_y..., norm_z...] .|> weight(2) |> counter |> keys |> minimum
3
</code></pre><p>Code distance of the encoding is the minimum degree of the non-zero term in the normalizer's polynomial(<span>$B$</span>) and not in the stabilizer'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 07:29">Tuesday 7 May 2024</span>. Using Julia version 1.10.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>