Quantum walks on blow-up graphs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917566961680384 |
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| author | Bhattacharjya, Bikash Monterde, Hermie Pal, Hiranmoy |
| author_facet | Bhattacharjya, Bikash Monterde, Hermie Pal, Hiranmoy |
| contents | A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$ obtained by replacing every vertex of $G$ by an independent set of size $n$, where the copies of vertices in $G$ are adjacent in the blow-up if and only if the vertices adjacent in $G$. Our goal is to investigate the existence of quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the adjacency matrix is taken to be the time-independent Hamiltonian of the quantum system represented by $\overset{n}\uplus~G$. In particular, we establish necessary and sufficient conditions for vertices in a blow-up graph to exhibit strong cospectrality and various types of high probability quantum transport, such as periodicity, perfect state transfer (PST) and pretty good state transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST, then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix, then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique vertex to exhibit strong cospectrality. We then apply our results to determine infinite families of graphs whose blow-ups admit PST and PGST. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13887 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum walks on blow-up graphs Bhattacharjya, Bikash Monterde, Hermie Pal, Hiranmoy Quantum Physics Combinatorics 05C50, 05C76, 15A16, 15A18, 81P45 A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$ obtained by replacing every vertex of $G$ by an independent set of size $n$, where the copies of vertices in $G$ are adjacent in the blow-up if and only if the vertices adjacent in $G$. Our goal is to investigate the existence of quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the adjacency matrix is taken to be the time-independent Hamiltonian of the quantum system represented by $\overset{n}\uplus~G$. In particular, we establish necessary and sufficient conditions for vertices in a blow-up graph to exhibit strong cospectrality and various types of high probability quantum transport, such as periodicity, perfect state transfer (PST) and pretty good state transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST, then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix, then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique vertex to exhibit strong cospectrality. We then apply our results to determine infinite families of graphs whose blow-ups admit PST and PGST. |
| title | Quantum walks on blow-up graphs |
| topic | Quantum Physics Combinatorics 05C50, 05C76, 15A16, 15A18, 81P45 |
| url | https://arxiv.org/abs/2308.13887 |