Scalable Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polynomial Optimization

TL;DR

This paper introduces a scalable method for certifying ground-state properties of large quantum spin systems using structured noncommutative polynomial optimization, overcoming previous scalability limitations.

Contribution

It leverages system structure to significantly improve the scalability of semidefinite programming relaxations for quantum ground-state certification.

Findings

Able to compute bounds for 16x16 quantum spin lattices.

Mitigates scalability issues in noncommutative polynomial optimization.

Provides bounds on ground state energies and observable expectations.

Abstract

A fundamental challenge in quantum physics is determining the ground-state properties of many-body systems. Whereas standard approaches, such as variational calculations, consist of writing down a wave function ansatz and minimizing over the possible states expressible by this ansatz, one can alternatively formulate the problem as a noncommutative polynomial optimization problem. This optimization problem can then be addressed using a hierarchy of semidefinite programming relaxations. In contrast to variational calculations, the semidefinite program can provide lower bounds for ground state energies and upper and lower bounds on observable expectation values. However, this approach typically suffers from severe scalability issues, limiting its applicability to small-to-medium-scale systems. In this article, we demonstrate that leveraging the inherent structures of the system can significantly mitigate these scalability challenges and thus allows us to compute meaningful bounds for quantum spin systems on up to $16\times16$ square lattices.