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‎dev/.documenter-siteinfo.json‎

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{"documenter":{"julia_version":"1.12.5","generation_timestamp":"2026-02-11T13:01:46","documenter_version":"1.16.1"}}
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{"documenter":{"julia_version":"1.12.5","generation_timestamp":"2026-02-11T13:38:10","documenter_version":"1.16.1"}}

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‎dev/asymptotic/index.html‎

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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Asymptotics · IntU.jl</title><meta name="title" content="Asymptotics · IntU.jl"/><meta property="og:title" content="Asymptotics · IntU.jl"/><meta property="twitter:title" content="Asymptotics · IntU.jl"/><meta name="description" content="Documentation for IntU.jl."/><meta property="og:description" content="Documentation for IntU.jl."/><meta property="twitter:description" content="Documentation for IntU.jl."/><meta property="og:url" content="https://iitis.github.io/IntU.jl/asymptotic/"/><meta property="twitter:url" content="https://iitis.github.io/IntU.jl/asymptotic/"/><link rel="canonical" href="https://iitis.github.io/IntU.jl/asymptotic/"/><script data-outdated-warner src="../assets/warner.js"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/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><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="../">IntU.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><span class="tocitem">Manual</span><ul><li><a class="tocitem" href="../unitary_integration/">Unitary Integration</a></li><li><a class="tocitem" href="../su_integration/">Special Unitary Group</a></li><li><a class="tocitem" href="../diagonal_unitary/">Diagonal Unitaries</a></li><li><a class="tocitem" href="../unitary_designs/">Unitary Designs</a></li><li><a class="tocitem" href="../orthogonal_integration/">Orthogonal &amp; Symplectic</a></li><li><a class="tocitem" href="../permutation_integration/">Permutation Groups</a></li><li><a class="tocitem" href="../circular_ensembles/">Circular Ensembles</a></li><li><a class="tocitem" href="../gaussian_integration/">Gaussian Ensembles</a></li><li><a class="tocitem" href="../pure_states/">Pure States</a></li><li><a class="tocitem" href="../stiefel_manifold/">Stiefel Manifolds</a></li><li><a class="tocitem" href="../symbolic_trace/">Symbolic Traces</a></li><li class="is-active"><a class="tocitem" href>Asymptotics</a><ul class="internal"><li><a class="tocitem" href="#Usage"><span>Usage</span></a></li><li><a class="tocitem" href="#Example"><span>Example</span></a></li><li><a class="tocitem" href="#References"><span>References</span></a></li></ul></li><li><a class="tocitem" href="../qi_helpers/">QI Helpers</a></li><li><a class="tocitem" href="../itensors/">ITensors Integration</a></li></ul></li><li><a class="tocitem" href="../api/">API Reference</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><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Asymptotics</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Asymptotics</a></li></ul></nav><div class="docs-right"><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="Asymptotic-Expansions"><a class="docs-heading-anchor" href="#Asymptotic-Expansions">Asymptotic Expansions</a><a id="Asymptotic-Expansions-1"></a><a class="docs-heading-anchor-permalink" href="#Asymptotic-Expansions" title="Permalink"></a></h1><p>For large Hilbert space dimension <span>$d$</span>, exact Weingarten results can be complicated rational functions. IntU.jl provides utilities to expand these results as a Taylor series in <span>$1/d$</span>.</p><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><pre><code class="language-julia hljs">asymptotic(expr, measure, order=1)</code></pre><ul><li><strong>expr</strong>: The symbolic expression to integrate.</li><li><strong>measure</strong>: The integration measure (Haar, PureState, GinUE, etc.).</li><li><strong>order</strong>: The maximum power of <span>$1/d$</span> to retain (default 1).</li></ul><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><p>Evaluating the fourth moment of a matrix entry <span>$|U_{11}|^4$</span>:</p><pre><code class="language-julia hljs">@variables d
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@symbolic_dimension U[1:d, 1:d]
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res = asymptotic(abs(U[1,1])^4, dU(U), 4)
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# Output: 2/d^2 - 2/d^3 + 2/d^4</code></pre><p>This approximation is useful for checking convergence properties or leading-order behavior in high-dimensional quantum systems.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Collins, B. (2003). Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. <em>International Mathematics Research Notices</em>.</li><li>Puchała, Z., &amp; Miszczak, J. A. (2017). Symbolic integration with respect to the Haar measure on the unitary group. <em>Bulletin of the Polish Academy of Sciences: Technical Sciences</em>.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../symbolic_trace/">« Symbolic Traces</a><a class="docs-footer-nextpage" href="../qi_helpers/">QI Helpers »</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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 11 February 2026 13:01">Wednesday 11 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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# Output: 2/d^2 - 2/d^3 + 2/d^4</code></pre><p>This approximation is useful for checking convergence properties or leading-order behavior in high-dimensional quantum systems.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><ul><li>Collins, B. (2003). Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. <em>International Mathematics Research Notices</em>.</li><li>Puchała, Z., &amp; Miszczak, J. A. (2017). Symbolic integration with respect to the Haar measure on the unitary group. <em>Bulletin of the Polish Academy of Sciences: Technical Sciences</em>.</li></ul></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../symbolic_trace/">« Symbolic Traces</a><a class="docs-footer-nextpage" href="../qi_helpers/">QI Helpers »</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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 11 February 2026 13:38">Wednesday 11 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

‎dev/circular_ensembles/index.html‎

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measure_cue = dCUE(U, d)
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integrate(abs(U[1,1])^2, measure_cue)
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# Output: 1 / d</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../permutation_integration/">« Permutation Groups</a><a class="docs-footer-nextpage" href="../gaussian_integration/">Gaussian Ensembles »</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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 11 February 2026 13:01">Wednesday 11 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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# Output: 1 / d</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../permutation_integration/">« Permutation Groups</a><a class="docs-footer-nextpage" href="../gaussian_integration/">Gaussian Ensembles »</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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 11 February 2026 13:38">Wednesday 11 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

‎dev/diagonal_unitary/index.html‎

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integrate(V[1,1] * conj(V[2,2]), measure)
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# Non-diagonal entries are zero by definition
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integrate(abs(V[1,2])^2, measure) # Output: 0</code></pre><h2 id="Performance-Note"><a class="docs-heading-anchor" href="#Performance-Note">Performance Note</a><a id="Performance-Note-1"></a><a class="docs-heading-anchor-permalink" href="#Performance-Note" title="Permalink"></a></h2><p>Integration over the diagonal group is significantly faster than over the full unitary group <span>$U(d)$</span>, as it avoids the combinatorial complexity of Weingarten functions and reduces to simple multiset comparisons.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../su_integration/">« Special Unitary Group</a><a class="docs-footer-nextpage" href="../unitary_designs/">Unitary Designs »</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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 11 February 2026 13:01">Wednesday 11 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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integrate(abs(V[1,2])^2, measure) # Output: 0</code></pre><h2 id="Performance-Note"><a class="docs-heading-anchor" href="#Performance-Note">Performance Note</a><a id="Performance-Note-1"></a><a class="docs-heading-anchor-permalink" href="#Performance-Note" title="Permalink"></a></h2><p>Integration over the diagonal group is significantly faster than over the full unitary group <span>$U(d)$</span>, as it avoids the combinatorial complexity of Weingarten functions and reduces to simple multiset comparisons.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../su_integration/">« Special Unitary Group</a><a class="docs-footer-nextpage" href="../unitary_designs/">Unitary Designs »</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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.16.1 on <span class="colophon-date" title="Wednesday 11 February 2026 13:38">Wednesday 11 February 2026</span>. Using Julia version 1.12.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>

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