Variational Monte Carlo with the Multi-Scale Entanglement Renormalization Ansatz

TL;DR

This paper introduces a variational Monte Carlo method for optimizing the multi-scale entanglement renormalization ansatz (MERA), reducing computational complexity by sampling over an effective lattice and ensuring unitary constraints during optimization.

Contribution

It presents a novel variational Monte Carlo approach tailored for MERA, addressing unitary tensor optimization and efficient sampling over a simplified effective lattice.

Findings

Effective sampling over a log(L)-sized lattice reduces computational cost.

Modified steepest descent method ensures robust optimization of unitary tensors.

Demonstrated success on a critical quantum spin chain.

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.