How Fast Do Quantum Walks Mix?

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 $\mathcal{O}(n^{3/2+o(1)})$. In addition to 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.

Figure 1

Summary of the quantum mixing time (time after which the instantaneous distribution of a quantum walk is $\epsilon$ close to its limiting distribution) for $G(n,p)$ for different regimes of $p$: For dense random networks (when $p$ is some constant $c$), the quantum mixing time is in $\tilde{\mathcal{O}}(n^{3/2}/\epsilon )$. For sparse random networks, for $p\ge n^{-1+\zeta }$, where $\zeta \ge 2/3$, the quantum mixing time is in $\mathcal{O}(n^{2-\zeta /2}/\epsilon )$. Finally, for sparser random networks, i.e., when ${\log }^8(n)/n\le p<n^{-1/3}$, the quantum mixing time is in $\tilde{\mathcal{O}}(n^{5/2}\sqrt{p}/\epsilon )$.