We study whether the probability distribution of a discrete quantum walk can get arbitrarily close to uniform, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We establish a characterization of this phenomenon on regular non-bipartite graphs in terms of their adjacency eigenvalues and eigenprojections. Using theory from association schemes, we show this phenomenon happens on a strongly regular graph X if and only if X or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1431_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\hspace{1.111pt}\overline{\hspace{-1.111pt}X}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="1.111pt" /> <mover> <mrow> <mspace width="-1.111pt" /> <mi>X</mi> </mrow> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> has parameters <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1431_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="244" /> </InlineMediaObject> <EquationSource Format="TEX">$(4m^2, 2m^2\pm m, m^2\pm m, m^2\pm m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>,</mo> <mn>2</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>±</mo> <mi>m</mi> <mo>,</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>±</mo> <mi>m</mi> <mo>,</mo> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>±</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10801_2025_1431_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

$\epsilon $-Uniform mixing in discrete quantum walks

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发表日期：

2025年6月18日

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Hanmeng Zhan

摘要

We study whether the probability distribution of a discrete quantum walk can get arbitrarily close to uniform, given that the walk starts with a uniform superposition of the outgoing arcs of some vertex. We establish a characterization of this phenomenon on regular non-bipartite graphs in terms of their adjacency eigenvalues and eigenprojections. Using theory from association schemes, we show this phenomenon happens on a strongly regular graph X if and only if X or ${\hspace{1.111pt}\overline{\hspace{-1.111pt}X}}$ $\bar{X}$ has parameters $(4m^2, 2m^2\pm m, m^2\pm m, m^2\pm m)$ $(4 m^{2}, 2 m^{2} \pm m, m^{2} \pm m, m^{2} \pm m)$ where $m\ge 2$ $m \geq 2$ .

\(\epsilon \)-Uniform mixing in discrete quantum walks

摘要