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arXiv:1911.07495

Uniform mixing on abelian Cayley graphs

Xiwang Cao, Keqin Feng

Published Nov 18, 2019• 0 citations
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Abstract

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 ${\rm Cay}(\mathbb{F}_2^{2k};S)$ having uniform mixing if and only if $f$ is a bent function, where $f$ is the characteristic function of $S$.

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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 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 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.
Abstract Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks. Quantum walks show promise for faster mixing times than classical methods but lack universal proof, especially in finite group settings. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group D 2n for odd n, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order Ω ( n 2 log ( 1 / 2 ϵ ) ) , the continuous-time quantum walk mixing time on D 2n is of order O ( n ( log n ) 5 log ( 1 / ϵ ) ) , achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on D 2n . This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.