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arXiv:1701.04145

Universality in perfect state transfer

Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon

Published Jan 16, 2017• 0 citations
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Abstract

A continuous-time quantum walk on a graph is a matrix-valued function $\exp(-\mathtt{i} At)$ over the reals, where $A$ is the adjacency matrix of the graph. Such a quantum walk has universal perfect state transfer if for all vertices $u,v$, there is a time where the $(v,u)$ 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.

DOI
10.48550/arxiv.1701.04145

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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.
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.
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 $\ICG_n (D)$ 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 represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.
A continuous quantum walk on a graph XXX with adjacency matrix AAA is specified by the 1-parameter family of unitary matrices U(t)=exp⁡(itA)U(t)=\exp(itA)U(t)=exp(itA). These matrices act on the state space of a quantum system, the states of which we may represent by density matrices, positive semidefinite matrices with rows and columns indexed by V(X)V(X)V(X) and with trace 111. The square of the absolute values of the entries of a column of U(t)U(t)U(t) define a probability density on V(X)V(X)V(X), and it is precisely these densities that predict the outcomes of measurements. There are two special cases of physical interest: when the column density is supported on a vertex, and when it is uniform. In the first case we have perfect state transfer; in the second, uniform mixing. There are many results concerning state transfer and uniform mixing. In this paper we show that these results on perfect state transfer hold largely because at the time it occurs, the density matrix is real. We also show that the results on uniform mixing obtained so far hold because the entries of the density matrix are algebraic numbers. As a consequence of these we derive strong restrictions on the occurence of uniform mixing on bipartite graphs and on oriented graphs.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.