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arXiv:1104.1825

Characterization of circulant graphs having perfect state transfer

Milan Bašić

Published Apr 11, 2011• 12 citations
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Abstract

In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph $G$, which is characterized by its circulant adjacency matrix $A$. Formally, we say that there exists a {\it perfect state transfer} (PST) between vertices $a,b\in V(G)$ if $|F(τ)_{ab}|=1$, for some positive real number $τ$, where $F(t)=\exp(ıAt)$. Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417--430) proved that $|F(τ)_{aa}|=1$ for some $a\in V(G)$ and $τ\in \R^+$ if and only if all eigenvalues of $G$ are integer (that is, the graph is integral). The integral circulant graph $\ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, ..., n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d

DOI
10.48550/arxiv.1104.1825

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In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph GGG, which is characterized by its circulant adjacency matrix AAA. Formally, we say that there exists a {\it perfect state transfer} (PST) between vertices a,b∈V(G)a,b\in V(G)a,b∈V(G) if ∣F(τ)ab∣=1|F(τ)_{ab}|=1∣F(τ)ab​∣=1, for some positive real number τττ, where F(t)=exp⁡(ıAt)F(t)=\exp(ıAt)F(t)=exp(ıAt). Saxena, Severini and Shparlinski ({\it International Journal of Quantum Information} 5 (2007), 417--430) proved that ∣F(τ)aa∣=1|F(τ)_{aa}|=1∣F(τ)aa​∣=1 for some a∈V(G)a\in V(G)a∈V(G) and τ∈R+τ\in \R^+τ∈R+ if and only if all eigenvalues of GGG are integer (that is, the graph is integral). The integral circulant graph \ICGn(D)\ICG_n (D)\ICGn​(D) has the vertex set Zn={0,1,2,...,n−1}Z_n = \{0, 1, 2, ..., n - 1\}Zn​={0,1,2,...,n−1} and vertices aaa and bbb are adjacent if gcd⁡(a−b,n)∈D\gcd(a-b,n)\in Dgcd(a−b,n)∈D, where $D \subseteq {d : d \mid n,\ 1\leq d
The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor ΔΔΔ, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order nnn, we derive an explicit formula for every Fourier eigenvalue without assuming that n/Δn/Δn/Δ is coprime to ΔΔΔ. We prove that PST occurs only for Δ∈{3,4,8}Δ\in\{3,4,8\}Δ∈{3,4,8} and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly 111, 222, and 333. More generally, when λj=Dηjλ_j=\sqrt{D}η_jλj​=D​ηj​ with ηj∈Zη_j\in\mathbb{Z}ηj​∈Z, congruences satisfied by the integers ηjη_jηj​ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.
A continuous-time quantum walk on a graph is a matrix-valued function exp⁡(−iAt)\exp(-\mathtt{i} At)exp(−iAt) over the reals, where AAA is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices u,vu,vu,v, there is a time where the (v,u)(v,u)(v,u) entry of the matrix exponential has unit magnitude. We prove new characterizations of graphs with universal perfect state transfer. This extends results of Cameron et al. (Linear Algebra and Its Applications, 455:115-142, 2014). Also, we construct non-circulant families of graphs with universal perfect state transfer. All prior known constructions were circulants. Moreover, we prove that if a circulant, whose order is prime, prime squared, or a power of two, has universal perfect state transfer then its underlying graph must be complete. This is nearly tight since there are universal perfect state transfer circulants with non-prime-power order where some edges are missing.
We study the existence of quantum state transfer in Q\mathcal{Q}Q-graphs in this paper. The Q\mathcal{Q}Q-graph of a graph GGG, denoted by Q(G)\mathcal{Q}(G)Q(G), is the graph derived from GGG by plugging a new vertex to each edge of GGG and joining two new vertices which lie on adjacent edges of GGG by an edge. We show that, if all eigenvalues of a regular graph GGG are integers, then its Q\mathcal{Q}Q-graph Q(G)\mathcal{Q}(G)Q(G) has no perfect state transfer. In contrast, we also prove that the Q\mathcal{Q}Q-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of Q\mathcal{Q}Q-graphs having no perfect state transfer, but admitting pretty good state transfer.
Abstract. Pure states correspond to one-dimensional subspaces of [Formula: see text] represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state [Formula: see text] admits PST with another real pure state [Formula: see text], (ii) every connected graph admits PST between real pure states, and (iii) for any pair of real pure states [Formula: see text] and [Formula: see text] and for any time [Formula: see text], there exists a real symmetric matrix [Formula: see text] such that [Formula: see text] and [Formula: see text] admit PST relative to [Formula: see text] at time [Formula: see text]. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all [Formula: see text]-vertex simple unweighted graphs, the least minimum PST time between real pure states is attained by any join graph for the Laplacian case, while it is attained by the join of an empty graph and a complete graph of appropriate sizes for the adjacency case. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at none . [Formula: see text]
We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\v{s}i\'{c} and Petkovi\'{c} ({\em Applied Mathematics Letters} {\bf 22}(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\textsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where b∈{1,2}b \in \{1,2\}b∈{1,2}, nnn is a multiple of 161616 and QQQ is a subset of the odd divisors of nnn. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil (arxiv.org math/08062074).
Given a graph with Hermitian adjacency matrix HHH, perfect state transfer occurs from vertex aaa to vertex bbb if the (b,a)(b,a)(b,a)-entry of the unitary matrix exp⁡(−iHt)\exp(-iHt)exp(−iHt) has unit magnitude for some time ttt. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i\pm i±i), the oriented 333-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.