Tunneling in projective quantum Monte Carlo simulations with guiding wave functions

Abstract

Quantum tunneling is a valuable resource exploited by quantum annealers to solve complex optimization problems. Tunneling events also occur during projective quantum Monte Carlo (PQMC) simulations, and in a class of problems characterized by a double-well energy landscape their rate was found to scale linearly with the first energy gap, i.e., even more favorably than in physical quantum annealers, where the rate scales with the gap squared. Here we investigate how a guiding wave function -which is essential to make many-body PQMC simulations computationally feasible -affects the tunneling rate. The chosen testbeds are a continuous-space double-well problem, the ferromagnetic quantum Ising chain, and the recently introduced shamrock model. As guiding wave function, we consider an approximate Boltzmann-type ansatz, the numerically-exact ground state of the double-well model, and a neural-network wave function based on a Boltzmann machine. Remarkably, for each ansatz we find the same asymptotic linear scaling of the tunneling rate that was previously found in the PQMC simulations performed without a guiding wave function. We also provide a semiclassical theory for the double-well with exact guiding wave function that explains the observed linear scaling. These findings suggest that PQMC simulations guided by an accurate ansatz represent a valuable benchmark for physical quantum annealers and a potentially competitive quantum-inspired optimization technique.

Neural networks in quantum many-body physics: a hands-on tutorial

“…Such progress has led to a burst of research where neural networks have been repurposed to tackle fundamental questions in condensed matter physics, quantum computing, statistical physics, and atomic, molecular and optical physics. Machine learning, and in particular deep neural networks, have been used to identify phases of matter in numerical simulations and experiments [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], to increase the performance of Monte Carlo simulations [22][23][24][25][26][27][28][29], to accurately describe the state of classical [30] and quantum systems [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45], to develop novel quantum control strategies [46][47][48][49][50]…”

Simulating disordered quantum Ising chains via dense and sparse restricted Boltzmann machines

“…As a future perspective, sparse RBMs could find use in studies of combinatorial optimization problems. In that context, projective quantum Monte Carlo algorithms have emerged as a stringent benchmark for physical quantum annealers [47,[77][78][79]. However, the lack of guiding functions appropriate for the typical instances of complex optimization problems, which can be mapped to spin-glass models, has limited their success [80].…”

Simulating disordered quantum systems via dense and sparse restricted Boltzmann machines

“…As a future perspective, sparse RBMs could find use in studies of combinatorial optimization problems. In that context, projective quantum Monte Carlo algorithms have emerged as a stringent benchmark for physical quantum annealers [49,[77][78][79]. However, the lack of guiding functions appropriate for the typical instances of complex optimization problems, which can be mapped to spin-glass models, has limited their success [80].…”