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AIP conference proceedings

Benchmarking Monte Carlo simulations for simple quantum systems in epistemically-restricted phase-space representation

Muhammad Jauhar Kholili, Siska A. Rifianti, Eny Latifah, Ahmad R. T. Nugraha

Published Jan 1, 2022• 1 citation
Publisher

Abstract

We develop an unconventional Monte Carlo simulation within the phase-space formulation of quantum mechanics taking into account a global-nonseparable random variable whose strength is epistemically restricted on the order of Planck’s constant. Compared to the ordinary variational Monte simulation, we find that energy expectation values of simple quantum systems such like harmonic oscillators can be calculated by our method with a reasonable accuracy and competitive calculation time at a large sampling N of random numbers (e.g., at N > 105 samples). We also discuss the comparison of simulations performed in different computing resources.

DOI
10.1063/5.0106325

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We develop an unconventional Monte Carlo simulation within the phase-space formulation of quantum mechanics taking into account a global-nonseparable random variable whose strength is epistemically restricted on the order of Planck’s constant. Compared to the ordinary variational Monte simulation, we find that energy expectation values of simple quantum systems such like harmonic oscillators can be calculated by our method with a reasonable accuracy and competitive calculation time at a large sampling N of random numbers (e.g., at N > 105 samples). We also discuss the comparison of simulations performed in different computing resources.
Abstract We investigate Monte Carlo simulations within the epistemically-restricted phase-space (ERPS) representation of quantum mechanics for harmonic and anharmonic oscillators. Compared to the variational Monte Carlo (VMC) simulations in the conventional representation, the calculation time in the ER phase-space representation is quite competitive. We also numerically show that the distribution function of the global random variable on the order of Planck’s constant in the ER phase-space representation does not significantly affect the energy accuracy, yet a particular choice of the distribution function may give a better calculation time. This work gives an alternative approach to the Monte Carlo simulations beyond those in the conventional representation of quantum mechanics.
We devise a classical algorithm which efficiently computes the quantum expectation values arising in a class of continuous variable quantum circuits wherein the final quantum observable-after the Heisenberg evolution associated with the circuits-is at most second order in momentum. The classical computational algorithm exploits a specific epistemic restriction in classical phase space which directly captures the quantum uncertainty relation, to transform the quantum circuits in the complex Hilbert space into classical albeit unconventional stochastic processes in the phase space. The resulting multidimensional integral is then evaluated using the Monte Carlo sampling method. The convergence rate of the classical sampling algorithm is determined by the variance of the classical physical quantity over the epistemically restricted phase space distribution. The work shows that for the specific class of computational schemes, Wigner negativity is not a sufficient resource for quantum speedup. It highlights the potential role of the epistemic restriction as an intuitive conceptual tool which may be used to study the boundary between quantum and classical computations.
We investigate the phase-space representation of quantum mechanics as a platform to perform Monte Carlo simulation of the non-interacting Jaynes-Cummings-Rabi (JCR) model involving a two-level hydrogen atom as a qubit and a quantum har-monic oscillator as a photonic cavity. Our algorithm starts by defining a trial wave function for solving the JCR model and then implement it on the “epistemically-restricted” phase-space representation, known as Budiyono-Rochrlich theory, with a Monte-Carlo sampling for the phase-space distribution, which is useful to obtain the average energy of our JCR model. We find that the numerical results are close to the analytical ones for every separate system in the JCR model, i.e., the hydrogen and the oscillator, calculated independently. Combining the two components in the non-interacting JCR model, the calculated energy values for the ground and the first excited states are then found to be —0.91 a.u. and 0.41 a.u. with corresponding standard errors of 9.12×10−8 a.u. and 1.95×10−8 a.u., respectively. We expect that the method presented in this paper can be an alternative approach to solving various qubit-photon systems.
This paper presents a novel approach to simulating quantum computing algorithms using Monte Carlo methods, specifically designed to account for the inherent probabilistic nature of quantum systems. Traditional simulation methods often rely on deterministic models, which fail to capture the fundamental uncertainty associated with quantum mechanics. Our method employs a stochastic simulation engine that accurately models the probabilistic behavior of quantum gates and measurements. The core of the approach involves generating a large number of samples, each representing a possible outcome of a quantum computation. By analyzing the statistical distribution of these samples, we can approximate the expected results of the algorithm. This technique offers a practical way to evaluate the performance of quantum algorithms, especially those with high levels of uncertainty, and provides a valuable tool for algorithm design and optimization. The method avoids the limitations of purely deterministic simulations and offers a more realistic representation of quantum computation. Key aspects include the development of a robust sampling strategy and the efficient calculation of probabilities. The approach is demonstrated through a detailed explanation and illustrative examples, focusing on the conceptual framework and the underlying mathematical principles.
We analyze the accuracy and sample complexity of variational Monte Carlo approaches to simulate the dynamics of many-body quantum systems classically. By systematically studying the relevant stochastic estimators, we are able to: (i) prove that the most used scheme, the time-dependent Variational Monte Carlo (tVMC), is affected by a systematic statistical bias or exponential sample complexity when the wave function contains some (possibly approximate) zeros, an important case for fermionic systems and quantum information protocols; (ii) show that a different scheme based on the solution of an optimization problem at each time step is free from such problems; (iii) improve the sample complexity of this latter approach by several orders of magnitude with respect to previous proofs of concept. Finally, we apply our advancements to study the high-entanglement phase in a protocol of non-Clifford unitary dynamics with local random measurements in 2D, first benchmarking on small spin lattices and then extending to large systems.
Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial wave functions, that is to variational quantum Monte Carlo. Almost all previous implementations employ samples distributed as the physical probability density of the trial wave function, and assume the central limit theorem to be valid. In this paper we provide an analysis of random error in estimation and optimization that leads naturally to new sampling strategies with improved computational and statistical properties. A rigorous lower limit to the random error is derived, and an efficient sampling strategy presented that significantly increases computational efficiency. In addition the infinite variance heavy tailed random errors of optimum parameters in conventional methods are replaced with a Normal random error, strengthening the theoretical basis of optimization. The method is applied to a number of first row systems and compared with previously published results.