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Time Evolution of Quantum Many-Body Systems with Real-Time Quantum Monte Carlo

Jamie Taylor

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

We develop a new implementation of the real-time full configuration interaction quantum Monte Carlo method, and apply it to the dynamics of the Heisenberg spin chain and the spectrum of the one-dimensional Fröhlich polaron. This is a general method for the numerical integration of the Schrödinger equation to evolve quantum systems in real time, introduced by Guther et al. as an extension of the imaginary-time version of the algorithm that finds ground-state properties. We introduce a new integrator for improved efficiency, and a scaling procedure for better control of simulations. We study the accuracy of the method, including the effect of discretising time and of the stochastic errors introduced by the Monte Carlo elements of the algorithm. Comparing the method to a hardware quantum simulator for the quench dynamics of a Heisenberg spin chain, we find that for this model the discretisation and stochastic errors grow quickly, limiting the effectiveness of the algorithm for finding accurate evolution over long times. We find that spectral information can be generated successfully using an energy-dependent damping, with either a tilted complex contour time evolution or a strong second-order damping. Specifically, we reproduce the low-energy spectrum of the Fr ̈ohlich polaron model and confirm the existence of a bound excited state.

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We develop a new implementation of the real-time full configuration interaction quantum Monte Carlo method, and apply it to the dynamics of the Heisenberg spin chain and the spectrum of the one-dimensional Fröhlich polaron. This is a general method for the numerical integration of the Schrödinger equation to evolve quantum systems in real time, introduced by Guther et al. as an extension of the imaginary-time version of the algorithm that finds ground-state properties. We introduce a new integrator for improved efficiency, and a scaling procedure for better control of simulations. We study the accuracy of the method, including the effect of discretising time and of the stochastic errors introduced by the Monte Carlo elements of the algorithm. Comparing the method to a hardware quantum simulator for the quench dynamics of a Heisenberg spin chain, we find that for this model the discretisation and stochastic errors grow quickly, limiting the effectiveness of the algorithm for finding accurate evolution over long times. We find that spectral information can be generated successfully using an energy-dependent damping, with either a tilted complex contour time evolution or a strong second-order damping. Specifically, we reproduce the low-energy spectrum of the Fr ̈ohlich polaron model and confirm the existence of a bound excited state.
We present a stochastic method for solving the time-dependent Schrödinger equation, generalizing a ground state full configuration interaction quantum Monte Carlo method. By performing the time integration in the complex plane close to the real-time axis, the numerical effort is kept manageable and the analytic continuation to real frequencies is efficient. This allows us to perform ab initio calculation of electron spectra for strongly correlated systems. The method can be used as a cluster solver for embedding schemes.
Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. The conventional algorithm quantum phase estimation uses deep circuits and requires fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit shallow circuits. In this work, we combine quantum Monte Carlo with quantum computing and propose an algorithm for simulating the imaginary-time evolution and solving the ground-state problem. By sampling the real-time evolution operator with a random evolution time according to a modified Cauchy-Lorentz distribution, we can compute the expected value of an observable in imaginary-time evolution. Our algorithm approaches the exact solution given a circuit depth increasing polylogarithmically with the desired accuracy. Compared with quantum phase estimation, the Trotter step number, i.e. the circuit depth, can be thousands of times smaller to achieve the same accuracy in the ground-state energy. We verify the resilience to Trotterisation errors caused by the finite circuit depth in the numerical simulation of various models. The results show that Monte Carlo quantum simulation is promising even without a fully fault-tolerant quantum computer.
A method is presented for discrete real-time learning (DRTL) of the evolution of a many-body quantum state in the presence of time-dependent control fields. The DRTL method is based on the recent adoption of the artificial neural network (ANN) ansatz within the framework of the time-dependent variational Monte Carlo method in learning large-scale quantum many-body dynamics. The method leverages accurate short-time quantum-state propagation with the Schr\"odinger equation and the efficient stochastic quantum natural gradient descent algorithm in machine learning. Importantly, the DRTL method is devoid of high numerical costs resulting from the exponentially large many-body Hilbert space by replacing the full state propagation with short-time propagations of a sequence of partial stochastic samples of the ANN state and may be considered as a synergism between the Schr\"odinger equation and the ANN that streamlines the computation of the many-body controlled quantum dynamics. Specifically, this method minimizes the normed distances between the time-dependent variational ANN ansatz and a set of partial quantum states over a sequence of stochastically sampled subspaces that are connected to each other in small discrete time steps. We demonstrate the high accuracy and stability of the DRTL method for the spin excitation dynamics of $4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6, 8\ifmmode\times\else\texttimes\fi{}8$, and $10\ifmmode\times\else\texttimes\fi{}10$ two-dimensional (2D) Heisenberg spin-1/2 lattices driven by time-dependent control fields applied to a corner spin along with strong exchange coupling between the nearest-neighbored spins. It is found that the number of ANN parameters for properly following the dynamically occupied subspaces (i.e., starting from the pure tensor-product spin configuration with all spins down) approximately increases quadratically with the number of spins in the system. Importantly, we show that the DRTL method can accurately and efficiently follow the small, yet importantly, occupied parts of the Hilbert space of the 2D spin square lattices during the interaction with the strong time-dependent control fields over a physically significant time period. This finding provides a basis for future work to perform quantum optimal control simulations of 2D spin lattices and other multiparticle systems.
An algorithm is developed for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast into the form of a many-dimensional integral through the use of the Hubbard-Stratonovitch representation of the imaginary time many- body evolution operator. The resulting functional integral is then evaluated stochastically. The algorithm is tested for an exactly soluble boson system and is then extended to include fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Improved computational efficiency is attained by performing the calculations in momentum space.
On the basis of a Feynman–Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.
We develop a real-time full configuration-interaction quantum Monte Carlo approach to model driven-dissipative open quantum systems with Markovian system-bath coupling. The method enables stochastic sampling of the Liouville--von Neumann time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the nonequilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce the statistical error. Finally, we demonstrate the efficiency of our approach by applying it to the driven-dissipative two-dimensional XYZXYZXYZ spin-12\frac{1}{2}21​ model on a lattice.