Local $ε$-uniform mixing in continuous quantum walks
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911534782873600 |
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| author | Monterde, Hermie |
| author_facet | Monterde, Hermie |
| contents | Let $X$ be a weighted graph and $M$ be its adjacency, Laplacian or signless Laplacian matrix. In a continuous quantum walk on $X$, local $ε$-uniform mixing occurs at vertex $u$ if the $u$th column of the matrix $U(t)=e^{itM}$ can be made arbitrarily close to a vector whose all entries have equal magnitude. Using the spectral and combinatorial properties of $X$, we derive necessary conditions for local $ε$-uniform mixing to occur in $X$. This includes an inequality involving all entries of each eigenvector of $M$, as well as an upper bound on the degree of vertex $u$ when $M$ is the Laplacian or signless Laplacian matrix. We use these necessary conditions to rule out local $ε$-uniform mixing in numerous classes of graphs, most of which are non-regular. We also show that almost all planar graphs (resp., trees) contain a vertex that does not admit local $ε$-uniform mixing for any assignment of edge weights. Furthermore, we prove if $X$ has $n$ vertices and admits local $ε$-uniform mixing at a vertex contained in a subgraph with a twin, then the number of vertices of this twin subgraph must be at least $\sqrt{n}$. In particular, we establish that a graph on $n\geq 5$ vertices does not admit local $ε$-uniform mixing at a vertex with a twin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20977 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local $ε$-uniform mixing in continuous quantum walks Monterde, Hermie Combinatorics 05C50, 81P45 Let $X$ be a weighted graph and $M$ be its adjacency, Laplacian or signless Laplacian matrix. In a continuous quantum walk on $X$, local $ε$-uniform mixing occurs at vertex $u$ if the $u$th column of the matrix $U(t)=e^{itM}$ can be made arbitrarily close to a vector whose all entries have equal magnitude. Using the spectral and combinatorial properties of $X$, we derive necessary conditions for local $ε$-uniform mixing to occur in $X$. This includes an inequality involving all entries of each eigenvector of $M$, as well as an upper bound on the degree of vertex $u$ when $M$ is the Laplacian or signless Laplacian matrix. We use these necessary conditions to rule out local $ε$-uniform mixing in numerous classes of graphs, most of which are non-regular. We also show that almost all planar graphs (resp., trees) contain a vertex that does not admit local $ε$-uniform mixing for any assignment of edge weights. Furthermore, we prove if $X$ has $n$ vertices and admits local $ε$-uniform mixing at a vertex contained in a subgraph with a twin, then the number of vertices of this twin subgraph must be at least $\sqrt{n}$. In particular, we establish that a graph on $n\geq 5$ vertices does not admit local $ε$-uniform mixing at a vertex with a twin. |
| title | Local $ε$-uniform mixing in continuous quantum walks |
| topic | Combinatorics 05C50, 81P45 |
| url | https://arxiv.org/abs/2603.20977 |