Variational Monte Carlo with the Multi-Scale Entanglement Renormalization Ansatz

Abstract

Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here we put forward a variational Monte Carlo approach for the multi-scale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with non-unitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just log(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two dimensional systems.

11 pages, 12 figures, a variety of minor clarifications and corrections

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Cited by in corpus (16)

• Quantum Entanglement in Deep Learning Architectures
• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
• Perfect Sampling with Unitary Tensor Networks
• Tree Tensor Networks for Generative Modeling
• Many-body magic via Pauli-Markov chains -- from criticality to gauge theories
• Higher order tensor renormalization group for relativistic fermion systems
• Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model
• Generalized Transfer Matrix States from Artificial Neural Networks
• Efficient Simulation of Dynamics in Two-Dimensional Quantum Spin Systems with Isometric Tensor Networks
• The area law and real-space renormalization
• Projector quantum Monte Carlo with matrix product states
• Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems
• Generalized mean field description of entanglement in dimerized spin systems
• All-mode Renormalization for Tensor Network with Stochastic Noise
• Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo
• Analog simulation of noisy quantum circuits