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arXiv:1504.00721 · The Electronic Journal of Combinatorics

Uniform Mixing on Cayley Graphs

Chris D. Godsil, Hanmeng Zhan

Published Jul 28, 2017• 8 citations
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Abstract

We provide new examples of Cayley graphs on which the quantum walks reach uniform mixing. Our first result is a complete characterization of all $2(d+2)$-regular Cayley graphs over $\mathbb{Z}_3^d$ that admit uniform mixing at time $2\pi/9$. Our second result shows that for every integer $k\ge 3$, we can construct Cayley graphs over $\mathbb{Z}_q^d$ that admit uniform mixing at time $2\pi/q^k$, where $q=3, 4$.We also find the first family of irregular graphs, the Cartesian powers of the star $K_{1,3}$, that admit uniform mixing.

DOI
10.37236/6855

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We provide new examples of Cayley graphs on which the quantum walks reach uniform mixing. Our first result is a complete characterization of all 2(d+2)2(d+2)2(d+2)-regular Cayley graphs over Z3d\mathbb{Z}_3^dZ3d​ that admit uniform mixing at time 2π/92\pi/92π/9. Our second result shows that for every integer k≥3k\ge 3k≥3, we can construct Cayley graphs over Zqd\mathbb{Z}_q^dZqd​ that admit uniform mixing at time 2π/qk2\pi/q^k2π/qk, where q=3,4q=3, 4q=3,4.We also find the first family of irregular graphs, the Cartesian powers of the star K1,3K_{1,3}K1,3​, that admit uniform mixing.
In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. In 2003, Childs et al. found a graph in which the continuous-time quantum walk spreads exponentially faster than any classical algorithm for a certain black-box problem. Since then, there are extensive explorations on quantum walks on graphs. Among them, uniform mixing provides a uniform probability distribution of information over time which attracts a special attention. However, there are only a few known examples of graphs which admit uniform mixing. In this paper, we give a characterization of abelian Cayley graphs having uniform mixing. Some concrete constructions of such graphs are provided. Particularly, for cubelike graphs, it is shown that the Cayley graph Cay(F22k;S){\rm Cay}(\mathbb{F}_2^{2k};S)Cay(F22k​;S) having uniform mixing if and only if fff is a bent function, where fff is the characteristic function of SSS.
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 XXX with nnn vertices, then nnn is divisible by four. We also prove that if XXX is bipartite and regular, then nnn is the sum of two integer squares. Our work on bipartite graphs implies that uniform mixing does not occur on C2mC_{2m}C2m​ for m≥3m \geq 3m≥3. Using a result of Haagerup, we show that uniform mixing does not occur on CpC_pCp​ for any prime ppp such that p≥5p \geq 5p≥5. In contrast to this result, we see that ϵ\epsilonϵ-uniform mixing occurs on CpC_pCp​ for all primes ppp.
In recent years quantum algorithms have become a popular area of mathematical research. Farhi and Gutmann introduced the concept of a quantum walk in 1998. In this thesis we investigate mixing properties of continuous-time quantum walks from a mathematical perspective. We focus on the connections between mixing properties and association schemes. \n \nThere are three main goals of this thesis. Our primary goal is to develop the algebraic groundwork necessary to systematically study mixing properties of continuous-time quantum walks on regular graphs. Using these tools we achieve two additional goals: we construct new families of graphs that admit uniform mixing, and we prove that other families of graphs never admit uniform mixing. \n \nWe begin by introducing association schemes and continuous-time quantum walks. Within this framework we develop specific algebraic machinery to tackle the uniform mixing problem. Our main algebraic result shows that if a graph has an irrational eigenvalue, then its transition matrix has at least one transcendental coordinate at all nonzero times. \n \nNext we study algebraic varieties related to uniform mixing to determine information about the coordinates of the corresponding transition matrices. Combining this with our main algebraic result we prove that uniform mixing does not occur on even cycles or prime cycles. However, we show that the probability distribution of a quantum walk on a prime cycle gets arbitrarily close to uniform. \n \nFinally we consider uniform mixing on Cayley graphs of elementary abelian groups. We utilize graph quotients to connect the mixing properties of these graphs to Hamming graphs. This enables us to find new results about uniform mixing on Cayley graphs of certain elementary abelian groups.
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.