Quantum-Inspired Tempering for Ground State Approximation using Artificial Neural Networks

Abstract

A large body of work has demonstrated that parameterized artificial neural networks (ANNs) can efficiently describe ground states of numerous interesting quantum many-body Hamiltonians. However, the standard variational algorithms used to update or train the ANN parameters can get trapped in local minima, especially for frustrated systems and even if the representation is sufficiently expressive. We propose a parallel tempering method that facilitates escape from such local minima. This methods involves training multiple ANNs independently, with each simulation governed by a Hamiltonian with a different "driver" strength, in analogy to quantum parallel tempering, and it incorporates an update step into the training that allows for the exchange of neighboring ANN configurations. We study instances from two classes of Hamiltonians to demonstrate the utility of our approach using Restricted Boltzmann Machines as our parameterized ANN. The first instance is based on a permutation-invariant Hamiltonian whose landscape stymies the standard training algorithm by drawing it increasingly to a false local minimum. The second instance is four hydrogen atoms arranged in a rectangle, which is an instance of the second quantized electronic structure Hamiltonian discretized using Gaussian basis functions. We study this problem in a minimal basis set, which exhibits false minima that can trap the standard variational algorithm despite the problem's small size. We show that augmenting the training with quantum parallel tempering becomes useful to finding good approximations to the ground states of these problem instances.

arXiv comment13 pages, 6 figures. v2: updated Figs. 3, 4, and 5. v3. included two new QPT schemes. Updated format to SciPost format. 30 pages, 11 figures. v4. Fixed typo in Eqt. 10 and minor correction in Conclusion. 30 pages, 11 figures. Submission to SciPost

References in corpus (29)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Exchange Monte Carlo Method and Application to Spin Glass Simulations
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Defining and detecting quantum speedup
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks
• Generalized Lanczos Algorithm for Variational Quantum Monte Carlo
• Green Function Monte Carlo with Stochastic Reconfiguration
• Feedback-optimized parallel tempering Monte Carlo
• Fermionic neural-network states for ab-initio electronic structure
• Study of the Two-Dimensional Frustrated J1-J2 Model with Neural Network Quantum States
• Quantum many-body dynamics in two dimensions with artificial neural networks
• A Constrained Path Quantum Monte Carlo Method for Fermion Ground States
• Correlated geminal wave function for molecules: an efficient resonating valence bond approach
• How Powerful is Adiabatic Quantum Computation?
• The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs
• Tunneling and speedup in quantum optimization for permutation-symmetric problems
• Benchmarking a quantum annealing processor with the time-to-target metric
• Nature of the spin-glass phase at experimental length scales
• Non-Reversible Parallel Tempering: a Scalable Highly Parallel MCMC Scheme
• Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape
• Geometry of learning neural quantum states
• ALGORITHM 937: MINRES-QLP for Singular Symmetric and Hermitian Linear Equations and Least-Squares Problems
• Investigating ultrafast quantum magnetism with machine learning
• Off-Diagonal Expansion Quantum Monte Carlo
• Iterative Retraining of Quantum Spin Models Using Recurrent Neural Networks
• Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy

Cited by in corpus (1)

• Optimizing Temperature Distributions for Training Neural Quantum States using Parallel Tempering