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arXiv:2206.10283 · Physical Review A

Dynamical triplet unraveling : A quantum Monte Carlo algorithm for reversible dynamics

Romain Chessex, Massimo Borrelli, Hans Christian Öttinger

Published Aug 31, 2022• 2 citations
arXivHTMLPDF

Abstract

We introduce a quantum Monte Carlo method to simulate the reversible dynamics of correlated many-body systems. Our method is based on the Laplace transform of the time-evolution operator, which, as opposed to most quantum Monte Carlo methods, makes it possible to access the dynamics at longer times. The Monte Carlo trajectories are realized through a piecewise stochastic-deterministic reversible evolution where free dynamics is interspersed with two-process quantum jumps. The dynamical sign problem is bypassed via the so-called deadweight approximation, which stabilizes the many-body phases at longer times. We benchmark our method by simulating spin excitation propagation in the one-dimensional (1D) XXZ spin chain, dynamical confinement in the 1D quantum Ising model and quenched magnetization dynamics in the two-dimensional square Ising model. Finally, we also show how to extract dynamical information directly from the Laplace representation.

DOI
10.1103/physreva.106.022222

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We introduce a quantum Monte Carlo method to simulate the reversible dynamics of correlated many-body systems. Our method is based on the Laplace transform of the time-evolution operator, which, as opposed to most quantum Monte Carlo methods, makes it possible to access the dynamics at longer times. The Monte Carlo trajectories are realized through a piecewise stochastic-deterministic reversible evolution where free dynamics is interspersed with two-process quantum jumps. The dynamical sign problem is bypassed via the so-called deadweight approximation, which stabilizes the many-body phases at longer times. We benchmark our method by simulating spin excitation propagation in the one-dimensional (1D) XXZ spin chain, dynamical confinement in the 1D quantum Ising model and quenched magnetization dynamics in the two-dimensional square Ising model. Finally, we also show how to extract dynamical information directly from the Laplace representation.
We present an alternative approach to the study of equilibrium properties in many-body quantum physics. Our method takes inspiration from density-matrix quantum Monte Carlo in dealing with the sign problem and incorporates additional features. First of all, the dynamics is transferred to the Laplace representation where an exact equation can be derived and solved using an evolution step that, unlike most Monte Carlo methods, is not a priori physically bounded. Moreover, the spawning events are formulated in terms of two-process stochastic unravellings of quantum master equations, a formalism that is particularly useful when working with density matrices. Last, our approach is perturbative in nature, with the free part being integrated exactly and, as such, the convergence is greatly accelerated when the interaction parameter is small. We benchmark our method by applying it to two case studies in condensed-matter physics, perform some ground-state calculations, show its accuracy, and further discuss its efficiency.
We present a new approach to the study of equilibrium properties in many-body quantum physics. Our method takes inspiration from Density Matrix Quantum Monte Carlo and incorporates new crucial features. First of all, the dynamics is transferred to the Laplace representation where an exact equation can be derived and solved using a simulation-step that, unlike most Monte Carlo methods, is not a priori physically bounded. Moreover, the spawning events are formulated in terms of two-process stochastic unravellings of quantum master equations, a formalism that is particularly useful when working with density matrices. And last, this is equivalent to an interaction picture, where the free part is integrated exactly and the convergence rate can be greatly increased if the interaction parameter is small. We benchmark our method by applying it to two case-studies in condensed matter physics, show its accuracy and further discuss its efficiency.
We introduce a method to simulate open quantum many-body dynamics by combining time-dependent variational Monte Carlo (tVMC) with quantum trajectory techniques. Our approach unravels the Lindblad master equation into an ensemble of stochastic Schrödinger equations for a variational ansatz, avoiding the exponential cost of density matrix evolution. The method is compatible with generic ansatzë, including expressive neural-network wavefunctions. We derive the nonlinear stochastic equations of motion for the variational parameters and employ suitable Stratonovich numerical solvers. To validate our approach, we simulate quenches in the locally dissipative long-range Ising model in a transverse field, accurately capturing non-equilibrium magnetization and spin squeezing dynamics relevant to trapped-ion and Rydberg atom experiments. The framework is computationally efficient, scalable on high-performance computing platforms, and can be readily integrated into existing tVMC implementations. This work enables large-scale simulations of complex, dissipative quantum systems in higher dimensions, with broad implications for quantum technology and fundamental science.
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Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.