Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution
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
| Published: | 2023-10-10, volume 7, page 1131 |
| Eprint: | arXiv:2305.14294v3 |
| Doi: | https://doi.org/10.22331/q-2023-10-10-1131 |
| Citation: | Quantum 7, 1131 (2023). |
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Abstract
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.
Featured image: Sketch of the failure of the tVMC when the state features zeros (red) and of the dynamics generated by the p-tVMC algorithm (blue). In p-tVMC, the optimization problem to solve consists in minimizing a distance in the Hilbert space, which is the infidelity $\mathcal{I}$ (as shown in the right panel).
Popular summary
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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