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arXiv:quant-ph/0112152

From Monte Carlo to Quantum Computation

Stefan Heinrich

Published Dec 23, 2001• 5 citations
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Abstract

Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this type of problem. In this paper we give a short introduction to the basic ideas of quantum computing and survey recent results on high dimensional integration. We discuss connections to the Monte Carlo methology and compare the optimal error rates of quantum algorithms to those of classical deterministic and randomized algorithms.

DOI
10.48550/arxiv.quant-ph/0112152

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The Monte Carlo method is a powerful scheme to solve many mathematical equations and is in fact, made practical only through high speed computers, particularly for those physical problems that are of high dimensionality. Alternative numerical schemes of spanning high dimensional spaces uniformly with a grid, increase in computational time exponentially with the dimensionality and thus became rapidly impractical as the dimensionality increases. The key to success of a Monte Carlo scheme is that it samples the space selectively, not with some uniform random distribution but where it matters most, by a probability density called an importance function. Furthermore, the Monte Carlo scheme is well adapted to computers since it consists of a simple, highly repetitive algorithm that takes advantage of the fast arithmetic capabilities of the machines. Consideration of how to arrange such calculations efficiently on various machines of different architecture is discussed.
This Perspective focuses on the several overlaps between quantum algorithms and Monte Carlo methods in the domains of physics and chemistry. We will analyze the challenges and possibilities of integrating established quantum Monte Carlo solutions into quantum algorithms. These include refined energy estimators, parameter optimization, real and imaginary-time dynamics, and variational circuits. Conversely, we will review new ideas for utilizing quantum hardware to accelerate the sampling in statistical classical models, with applications in physics, chemistry, optimization, and machine learning. This review aims to be accessible to both communities and intends to foster further algorithmic developments at the intersection of quantum computing and Monte Carlo methods. Most of the works discussed in this Perspective have emerged within the last two years, indicating a rapidly growing interest in this promising area of research.