Abstract

We consider continuous-time quantum walks on distance-regular graphs. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing.First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph $X$ with $n$ vertices, then $n$ is divisible by four. We also prove that if $X$ is bipartite and regular, then $n$ is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on $C_{2m}$ for $m \geq 3$. Using a result of Haagerup, we show that uniform mixing does not occur on $C_p$ for any prime $p$ such that $p \geq 5$. In contrast to this result, we see that $\epsilon$-uniform mixing occurs on $C_p$ for all primes $p$.

Highlights

  • Quantum walks are a quantum analogue of a random walk on a graph, and have recently been the subject of much investigation

  • A quantum walk can behave quite differently from a classical walk: for example Childs et al [9] found a graph in which the time to propagate from one node to another was sped up exponentially compared to any classical walk

  • We describe the physical motivation for continuous-time quantum walks

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Summary

Introduction

Quantum walks are a quantum analogue of a random walk on a graph, and have recently been the subject of much investigation. | k|ψ(t) |2 = k|eiLt|j 2 = (eiLt)kj 2 If this probability distribution is uniform over all vertices k, and this is true for all initial states |j , the graph is said to have uniform mixing. In this paper we focus on the question of which graphs have uniform mixing: can a quantum walk beginning at a single vertex result in a state whose probability distribution is uniform over all vertices? (c) The Cartesian product of any two graphs that admit uniform mixing at the same time. A number of Cayley graphs for Zd2 admit uniform mixing, but no classification is yet known. Some work in this direction appears in [3]. Since uniform mixing seems to be rare, we consider a relaxed version, called ǫ-uniform mixing, and we demonstrate that this does take place on all cycles of prime length

Uniform mixing
Type-II Matrices
Complex Hadamards and SRGs
Uniform mixing on SRGs
Regular symmetric Hadamard matrices
Symplectic-type graphs
Regular conference matrix type graphs
Conference graphs
Bipartite graphs
Cycles
Cyclotomic number theory
10 Future work
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