2,550 research outputs found
On Approximating the Stationary Distribution of Time-reversible Markov Chains
Approximating the stationary probability of a state in a Markov chain through Markov chain Monte Carlo techniques is, in general, inefficient. Standard random walk approaches require tilde{O}(tau/pi(v)) operations to approximate the probability pi(v) of a state v in a chain with mixing time tau, and even the best available techniques still have complexity tilde{O}(tau^1.5 / pi(v)^0.5); and since these complexities depend inversely on pi(v), they can grow beyond any bound in the size of the chain or in its mixing time.
In this paper we show that, for time-reversible Markov chains, there exists a simple randomized approximation algorithm that breaks this "small-pi(v) barrier"
Explicit error bounds for lazy reversible Markov Chain Monte Carlo
We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte
Carlo methods, such as the Metropolis algorithm. The problem is to compute the
expectation (or integral) of f with respect to a measure which can be given by
a density with respect to another measure. A straight simulation of the desired
distribution by a random number generator is in general not possible. Thus it
is reasonable to use Markov chain sampling with a burn-in. We study such an
algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an
explicit error bound
Efficient Circuits for Quantum Walks
We present an efficient general method for realizing a quantum walk operator
corresponding to an arbitrary sparse classical random walk. Our approach is
based on Grover and Rudolph's method for preparing coherent versions of
efficiently integrable probability distributions. This method is intended for
use in quantum walk algorithms with polynomial speedups, whose complexity is
usually measured in terms of how many times we have to apply a step of a
quantum walk, compared to the number of necessary classical Markov chain steps.
We consider a finer notion of complexity including the number of elementary
gates it takes to implement each step of the quantum walk with some desired
accuracy. The difference in complexity for various implementation approaches is
that our method scales linearly in the sparsity parameter and
poly-logarithmically with the inverse of the desired precision. The best
previously known general methods either scale quadratically in the sparsity
parameter, or polynomially in the inverse precision. Our approach is especially
relevant for implementing quantum walks corresponding to classical random walks
like those used in the classical algorithms for approximating permanents and
sampling from binary contingency tables. In those algorithms, the sparsity
parameter grows with the problem size, while maintaining high precision is
required.Comment: Modified abstract, clarified conclusion, added application section in
appendix and updated reference
Quantum speedup of classical mixing processes
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 over a large set
. This problem is solved using the {\em Markov chain Monte Carlo} method: a
sparse, reversible Markov chain on with stationary distribution
is run to near equilibrium. The running time of this random walk algorithm, the
so-called {\em mixing time} of , is as shown
by Aldous, where is the spectral gap of and is the minimum
value of . A natural question is whether a speedup of this classical
method to , the diameter of the graph
underlying , 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 is , which is indeed
and is asymptotically no worse than the
diameter of (the obvious lower bound) up to at most a logarithmic
factor.Comment: 13 pages; v2 revised several part
Estimating the spectral gap of a trace-class Markov operator
The utility of a Markov chain Monte Carlo algorithm is, in large part,
determined by the size of the spectral gap of the corresponding Markov
operator. However, calculating (and even approximating) the spectral gaps of
practical Monte Carlo Markov chains in statistics has proven to be an extremely
difficult and often insurmountable task, especially when these chains move on
continuous state spaces. In this paper, a method for accurate estimation of the
spectral gap is developed for general state space Markov chains whose operators
are non-negative and trace-class. The method is based on the fact that the
second largest eigenvalue (and hence the spectral gap) of such operators can be
bounded above and below by simple functions of the power sums of the
eigenvalues. These power sums often have nice integral representations. A
classical Monte Carlo method is proposed to estimate these integrals, and a
simple sufficient condition for finite variance is provided. This leads to
asymptotically valid confidence intervals for the second largest eigenvalue
(and the spectral gap) of the Markov operator. In contrast with previously
existing techniques, our method is not based on a near-stationary version of
the Markov chain, which, paradoxically, cannot be obtained in a principled
manner without bounds on the spectral gap. On the other hand, it can be quite
expensive from a computational standpoint. The efficiency of the method is
studied both theoretically and empirically
Hitting Time of Quantum Walks with Perturbation
The hitting time is the required minimum time for a Markov chain-based walk
(classical or quantum) to reach a target state in the state space. We
investigate the effect of the perturbation on the hitting time of a quantum
walk. We obtain an upper bound for the perturbed quantum walk hitting time by
applying Szegedy's work and the perturbation bounds with Weyl's perturbation
theorem on classical matrix. Based on the definition of quantum hitting time
given in MNRS algorithm, we further compute the delayed perturbed hitting time
(DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the
upper bound for DPQHT is actually greater than the difference between the
square root of the upper bound for a perturbed random walk and the square root
of the lower bound for a random walk.Comment: 9 page
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