arXiv subjects

Peter C. Richter

Publications and source records attributed to Peter C. Richter.

4 recordsLinked to original sources

arXiv2008

On the hitting times of quantum versus random walks

In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.

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arXiv2007

Two remarks on the local Hamiltonian problem

In this note we present two natural restrictions of the local Hamiltonian problem which are BQP-complete under Karp reduction. Restrictions complete for QCMA, QMA_1, and MA were demonstrated previously.

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arXiv2007

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 $π$ 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.

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arXiv2006

Almost uniform sampling via quantum walks

Many classical randomized algorithms (e.g., approximation algorithms for #P-complete problems) utilize the following random walk algorithm for {\em almost uniform sampling} from a state space $S$ of cardinality $N$: run a symmetric ergodic Markov chain $P$ on $S$ for long enough to obtain a random state from within $ε$ total variation distance of the uniform distribution over $S$. The running time of this algorithm, the so-called {\em mixing time} of $P$, is $O(δ^{-1} (\log N + \log ε^{-1}))$, where $δ$ is the spectral gap of $P$. We present a natural quantum version of this algorithm based on repeated measurements of the {\em quantum walk} $U_t = e^{-iPt}$. We show that it samples almost uniformly from $S$ with logarithmic dependence on $ε^{-1}$ just as the classical walk $P$ does; previously, no such quantum walk algorithm was known. We then outline a framework for analyzing its running time and formulate two plausible conjectures which together would imply that it runs in time $O(δ^{-1/2} \log N \log ε^{-1})$ when $P$ is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.

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