How Fast Do Quantum Walks Mix?

Abstract

The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications, is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the quantum mixing time. However this quantity is only known for a handful of specific networks. In this letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e. the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example for dense random networks, where p is a constant, we show that the quantum mixing time is O(n 3/2+o(1) ). Besides opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.The quantum dynamics of any discrete system can be captured by a quantum walk on a network, which is a universal model for quantum computation [1]. Besides being a useful primitive to design quantum algorithms [2][3][4][5][6], quantum walks are a powerful tool to model transport in quantum systems such as the transfer of excitations in light-harvesting systems [7][8][9]. Studying the long-time dynamics of quantum walks on networks is crucial to the understanding of these diverse problems. As quantum evolutions are unitary and hence distancepreserving, quantum walks never converge to a limiting distribution, unlike their classical counterpart. However, given a network of n nodes, one can define the limiting distribution of quantum walk on the network as the long-time average probability distribution of finding the walker in each node [10]. Of particular interest is the quantum mixing time: starting from some initial state, the minimum time after which the underlying quantum walk remains close to its limiting distribution.The importance of the problem of mixing for quantum walks cannot be overstated: this is at the heart of quantum speedups for a number of quantum algorithms [2,11] and is also key to demonstrating the equivalence between the standard (circuit) and Hamiltonian-based models of quantum computation [12,13]. Unfortunately, no general result exists for quantum mixing time on networks: it has been estimated for a handful of specific graphs (graphs and networks are ...

Analog quantum algorithms for the mixing of Markov chains

“…We provide a purely analog quantum algorithm to solve the QSSamp problem while, for the QLSamp problem, we expand and extend upon the results of Ref. [6], where we prove an upper bound for the quantum mixing time for almost all graphs.…”

“…with probability 1 − o (1), which implies that Ā G(n,p) ≈ 1 [6,55]. It can also be shown that, for the same range of p, the second highest eigenvalue λ n−1 can be upper bounded as…”

“…In this section, we elaborate on the results of Ref. [6] and provide numerical evidence to back up our analytical bounds. In particular, we focus on the random matrix theory aspects of our proof, elaborating on the underlying concepts.…”

“…Recently, we proved an upper bound for the mixing time of quantum walks for almost all graphs [6]. This implies that the fraction of graphs of n nodes, for which our upper bound holds, goes to 1 as n goes to infinity or, equivalently, if a graph is picked uniformly at random from the set of all graphs, our result provides an upper bound on the quantum mixing time almost surely, i.e., with probability 1 − o (1).…”

“…We proved this by obtaining the mixing time for quantum walks on Erdős-Rényi random graphs: graphs of n nodes such that the probability of an edge existing between any two nodes is p, typically denoted as G(n, p). Here, we expand upon the results of [6]. In particular, our goal is to offer an intuitive explanation of our proof techniques with an emphasis on the several recently developed random matrix theory tools that were used to derive the aforementioned results.…”