Quantum Simulation of Quantum Phase Transitions Using the Convex Geometry of Reduced Density Matrices

TL;DR

This paper introduces a quantum computing method leveraging the geometric structure of reduced density matrices to analyze quantum phase transitions, demonstrated on a spin model with promising results despite hardware limitations.

Contribution

It presents a novel approach using the convex geometry of 2-RDMs for quantum phase transition analysis, applicable on quantum computers even for strongly correlated systems.

Findings

Quantum approach captures key features of phase transitions.

Demonstrated on IBM quantum processors with limited particles.

Results align with classical models despite hardware noise.

Abstract

Transitions of many-particle quantum systems between distinct phases at absolute-zero temperature, known as quantum phase transitions, require an exacting treatment of particle correlations. In this work, we present a general quantum-computing approach to quantum phase transitions that exploits the geometric structure of reduced density matrices. While typical approaches to quantum phase transitions examine discontinuities in the order parameters, the origin of phase transitions -- their order parameters and symmetry breaking -- can be understood geometrically in terms of the set of two-particle reduced density matrices (2-RDMs). The convex set of 2-RDMs provides a comprehensive map of the quantum system including its distinct phases as well as the transitions connecting these phases. Because 2-RDMs can potentially be computed on quantum computers at non-exponential cost, even when the quantum system is strongly correlated, they are ideally suited for a quantum-computing approach to quantum phase transitions. We compute the convex set of 2-RDMs for a Lipkin-Meshkov-Glick spin model on IBM superconducting-qubit quantum processors. Even though computations are limited to few-particle models due to device noise, comparisons with a classically solvable 1000-particle model reveal that the finite-particle quantum solutions capture the key features of the phase transitions including the strong correlation and the symmetry breaking.