Quantum Monte Carlo guided by quantum computing

Stefan Heinrich

Estelle M. Inack

In this thesis, we implement projective quantum Monte Carlo (PQMC) methods to simulate quantum annealing on classical computers. A detailed derivation of the algorithm is presented. We show that in the regime where the systematic errors are well controlled, PQMC algorithms are indeed capable of simulating the imaginary-time dynamics of the Schroedinger equation both on continuous space models and discrete basis systems. We also demonstrate that the tunneling time of the PQMC method is quadratically faster than the one of incoherent quantum tunneling. It also shows remarkable stability when applied to frustrated systems compared to the path integral Monte Carlo algorithm, the algorithm mostly chosen to do comparisons with quantum annealers. However, a major drawback of the method comes from the finite number of random walkers needed to implement the simulations. It grows exponentially with the system size when no or poor guiding wave-functions are utilized. Nevertheless, we demonstrated that when good enough guiding wave-functions are used -- in our case we choose artificial neural networks -- the computational complexity seems to go from exponential to polynomial in the system size. We advocate for a search of more efficient guiding wave functions since they determine when and how feasible the PQMC simulations are on classical computers, a question closely related to a provable need or speed-up of a quantum computer.

Sebastiano Pilati, Estelle M. Inack, P. Pieri

The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.

Lavoisier Wah, Remmy Zen, Flore K. Kunst

Accurately determining the ground-state properties of quantum many-body systems remains a central challenge. In this work, we introduce an autoregressive projective quantum Monte Carlo (PQMC) framework that leverages recurrent neural networks (RNNs) to guide the stochastic dynamics. By incorporating autoregressive sampling into PQMC, we demonstrate substantial improvements in accuracy compared to standard unguided PQMC, while retaining polynomial computational cost. We benchmark our approach against conventional variational RNN ansätze and find that the autoregressive PQMC consistently achieves lower energies and higher fidelity, regardless of system size or whether the Hamiltonian is Hermitian or non-Hermitian. Our results highlight the versatility and power of neural-guided PQMC methods, paving the way for promising scalable simulations of low-energy states in complex quantum many-body systems.

Michael Dzierzawa, X. Zotos

Stephen Fahy