Parallel Tensor Network Methods for Quantum Lattice Systems

: Matrix Product State Simulations on a Supercomputer

Abstract

With quantum computers still in their infancy, simulating quantum many-body systems numerically on ordinary classical computers remains an important task in both physics and chemistry. This is a difficult problem in general since quantum states are described by tensors that scale exponentially with system size. Fortunately, some physically relevant states of one- and two-dimensional systems can be efficiently simulated using tensor network methods. The prototypical example being White's density matrix renormalisation group (DMRG) algorithm, which is based on a tensor decomposition called the tensor train or matrix product state.

One way to improve the performance of classical simulations is to take advantage of modern supercomputers. However, this requires the use of efficient parallel algorithms, while most tensor network methods are inherently serial. In this thesis, I argue that the DMRG parallelisation scheme proposed by Stoudenmire and White in 2013 provides a unified framework for parallel matrix product state algorithms. To demonstrate this, I start by parallelising White and Feiguin's time-dependent DMRG algorithm. I find strong scaling up to 512 processes with a parallel efficiency greater than 70% and show that the algorithm can be used to study many-body localisation in a one-dimensional lattice comprising more than 50,000 spins (qubits). I then use the same framework to implement a parallel variant of the time-dependent variational principle algorithm, thus introducing the first parallel tensor network method capable of time evolving systems with long-range interactions.

Alongside these algorithmic developments, I explore the benefits of tensor-network parallelisation in two experimentally motivated case studies. In the first, I verify the results of a groundbreaking analogue dynamical quantum simulator in a matter of days rather than weeks. In the second, I use parallel DMRG to map out a zero-temperature phase diagram of the numerically-challenging spin-1 extended Bose-Hubbard model.

Date of Award	26 Jun 2024
Original language	English
Awarding Institution	University of Bath
Supervisor	Sergey Dolgov (Supervisor), Stephen Clark (Supervisor) & Richard James (Supervisor)