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 Ο over a large set
Β§. This problem is solved using the {\em Markov chain Monte Carlo} method: a
sparse, reversible Markov chain P on Β§ with stationary distribution Ο
is run to near equilibrium. The running time of this random walk algorithm, the
so-called {\em mixing time} of P, is O(Ξ΄β1log1/Οββ) as shown
by Aldous, where Ξ΄ is the spectral gap of P and Οββ is the minimum
value of Ο. A natural question is whether a speedup of this classical
method to O(Ξ΄β1βlog1/Οββ), 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 Zndβ is O(ndlogd), which is indeed
O(Ξ΄β1βlog1/Οββ) and is asymptotically no worse than the
diameter of Zndβ (the obvious lower bound) up to at most a logarithmic
factor.Comment: 13 pages; v2 revised several part