Quantum speedup of classical mixing processes

Abstract

Most approximation algorithms for #P-complete problems (e.g., evaluating the permanent of a matrix or the volume of a polytope) work by reduction to the problem of approximate sampling from a distribution $\pi$ over a large set $\S$. This problem is solved using the {\em Markov chain Monte Carlo} method: a sparse, reversible Markov chain $P$ on $\S$ with stationary distribution $\pi$ is run to near equilibrium. The running time of this random walk algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} \log 1/\pi_*)$ as shown by Aldous, where $\delta$ is the spectral gap of $P$ and $\pi_*$ is the minimum value of $\pi$. A natural question is whether a speedup of this classical method to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$, the diameter of the graph underlying $P$, is possible using {\em quantum walks}. We provide evidence for this possibility using quantum walks that {\em decohere} under repeated randomized measurements. We show: (a) decoherent quantum walks always mix, just like their classical counterparts, (b) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (c) the mixing time of the decoherent quantum walk on a periodic lattice $\Z_n^d$ is $O(n d \log d)$, which is indeed $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ and is asymptotically no worse than the diameter of $\Z_n^d$ (the obvious lower bound) up to at most a logarithmic factor.Comment: 13 pages; v2 revised several part

Decoherence in quantum walks – a review

“…For odd-N cycles with no decoherence, they reported that M( ) ∼ N/ as compared to the upper bound of M( ) ∼ N log N/ 3 given in Aharonov et al (2001). Richter has recently confirmed this analytically (Richter 2007b).…”

“…In fact, the proofs in Richter (2007b) are sufficiently general that they apply equally to the continuous-time walk as to the discrete-time walk. This fills the gap between small and large decoherence rates, and proves that continuous-time walks, with O(log(1/ )) decoherence or measurement events, mix in time O(N log(1/ )) on the cycle (and the d-dimensional torus).…”

“…It is obvious that decoherence must do this, since high decoherence rates reproduce the classical random walk, which has this property. The only question is whether it does so effectively enough to be useful, and the answer is that it does for decoherence on the position (Kendon and Maloyer 2007;Richter 2007b). Figure 8 shows the mixing time M( , p) corresponding to P (x, T , p) as a function of p for cycles of size 48 to 51.…”

“…Note that these periodic mixing times decay with p and disappear altogether when p > 4k, so the quantum advantage tails off before the critical damping point is reached. Richter (2007b) offers a weak bound of O(n 3/2 log(1/ )) for the mixing time at around the critical damping point, which is slower than classical. By examining the local maxima of P [1], we can determine that the walk has approximate instantaneous hitting times to the opposite corner (1, .…”

Speedup via quantum sampling

“…The question whether a quantum speed-up to O(1/ √ δ log(1/π(x)) is possible has been examined in [9]. The author proposes a method based on quantum walks that decohere under repeated randomized measurements to attack this problem.…”