大多数近似算法用于#P-完全问题（例如，估计矩阵的积和或多面体的体积）的工作，通过简化为从大集合$§$上的分布$π$近似采样的问题。使用{ \em Markov chain Monte Carlo}方法解决该问题：具有平稳分布$π$的$§$上的稀疏可逆Markov链$P$运行到接近平衡。该随机游走算法的运行时间，即所谓的$P$的{ \em混合时间}是$O(δ^{-1} \log 1/π_*)$，如Aldous所示，其中$δ$是$P$的谱隙，而$π_*$是$π$的最小值。一个自然的问题是，使用{ \em量子行走}是否可能将这种经典方法加速到$O(\sqrt{δ^{-1}} \log 1/π_*)$，即$P$基础图的直径。我们提供了这种可能性的证据，使用量子行走，{ \em decohere}在重复的随机测量。我们展示：（a）退相干量子游走总是混合，就像它们的经典对应物一样，（b）混合时间是鲁棒的量，在退相干的任何光滑形式下基本上不变，以及（c）周期晶格$\Z_n^d$上的退相干量子游走的混合时间是$O(n d \log d)$，其实际上是$O(\sqrt{δ^{-1}} \log 1/π_*)$，并且渐近地不比$\Z_n^d$的直径（明显的下限）差，直到最多对数因子。
英文摘要：
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 $π$ over a large set $§$. This problem is solved using the {\em Markov chain Monte Carlo} method: a sparse, reversible Markov chain $P$ on $§$ with stationary distribution $π$ is run to near equilibrium. The running time of this random walk algorithm, the so-called {\em mixing time} of $P$, is $O(δ^{-1} \log 1/π_*)$ as shown by Aldous, where $δ$ is the spectral gap of $P$ and $π_*$ is the minimum value of $π$. A natural question is whether a speedup of this classical method to $O(\sqrt{δ^{-1}} \log 1/π_*)$, 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 $O(\sqrt{δ^{-1}} \log 1/π_*)$ and is asymptotically no worse than the diameter of $\Z_n^d$ (the obvious lower bound) up to at most a logarithmic factor.
Normal Mode Splitting and Antibunching in Stokes and Anti-Stokes Processes in Cavity Optomechanics: Radiation Pressure Induced Four-Wave Mixing Cavity Optomechanics
