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Bulletin of the American Physical Society

Quantum Monte Carlo without random numbers

Marjan Maček, Philipp T. Dumitrescu, Corentin Bertrand, Bill Triggs, Olivier Parcollet, Xavier Waintal

Published Mar 16, 2021• 0 citations
OpenAlex

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IEEE International Conference on Quantum Computing and Engineering (QCE)•Oct 1, 2021•3 citations
The chapter describes the principles of how Quantum Monte Carlo (QMC) uses random numbers to compute properties of a non-relativistic quantum mechanical system. In a many-body system, this involves computation of multidimensional integrals, which can be very effectively calculated using Monte Carlo methods. The spin of particles will be added to this framework, along with identical particles as symmetric or antisymmetric wave functions. An introduction to orbitals follows, and a discussion of the characteristics of boson and fermion ground states.
Quantum Monte Carlo (QMC) algorithms are among the most powerful classical methods for simulating quantum systems, yet their accuracy is often limited by the systematic errors in the approximations used, such as Trotterization. Here we introduce randomly compiled quantum Monte Carlo (RC-QMC) as a general framework that suppresses these systematic errors by averaging over a family of approximations rather than relying on a single fixed one. This strategy is grounded in the concept of randomized compiling from quantum computing, which suppresses errors by sampling over quantum gates, at essentially no additional computational cost. Consequently, our framework achieves a computational advantage over standard QMC methods when estimating a target state to a desired level of accuracy. We illustrate this advantage on two key Monte Carlo algorithms: (1) path integral quantum Monte Carlo for estimating thermal states, and (2) the quantum trajectories method for simulating open system dynamics. In aggregate, these results represent a cross-fertilization of quantum and classical algorithms, are readily generalizable to other QMC methods, and suggest wider applications in classical simulation.
Quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), provide quadratic speed-up compared with classical counterparts, and are therefore widely investigated, along with applications to industries such as finance. One typical feature of problems in finance is high dimensionality, or, in other words, the necessity to create numerous random numbers (RNs) to calculate the integrand. In such a situation, the original implementation of QMCI, in which states encoding the probability distributions of the RNs are generated on different quantum registers, requires a large number of qubits. In this poster, we propose an implementation of QMCI, in which pseudo-random numbers (PRNs) are sequentially generated on one register, and therefore the qubit number is tremendously reduced. Moreover, we show that, when the integrand has the form such that contributions from different RNs are separable into different terms, we can also achieve time complexity reduction with respect to the number of dimensions by combining the nested QAE and use of PRNs.