Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution

Alessandro Sinibaldi1,2, Clemens Giuliani1,2, Giuseppe Carleo1,2, and Filippo Vicentini1,2,3

1Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France

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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.

Monte Carlo variational methods are a powerful class of numerical techniques to simulate many-body quantum systems. While these algorithms have delivered state-of-the-art results for ground state search of frustrated and high-dimensional problems, variational calculations have yet to provide significant improvements over existing approaches for time evolution. Variational methods to simulate the quantum dynamics on classical or quantum hardware either integrate an explicit system of equations for the wave function parameters, the so-called time-dependent Variational Principle (TDVP), or implicitly solve a time-step optimization problem.
In this article, we first show that the classical implementation of TDVP, known as time-dependent Variational Monte Carlo (tVMC), can be affected by a statistical bias or an exponential sampling complexity when the wave function contains possibly approximate nodes. Second, we derive an extension of the implicit scheme that lowers its computational cost by several orders of magnitude and we name it projected tVMC (p-tVMC). In the final part, we apply the p-tVMC to investigate a system evolving via non-Clifford unitary dynamics with local random measurements in 2D, which is a paradigmatic model for entanglement phase transition that cannot be fully studied with other state-of-the-art techniques.

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