Quantum Phase Transitions in the Transverse-Field Ising Model: A Comparative Study of Exact, Variational, and Hardware-Based Approaches

TL;DR

This study compares exact, variational, and hardware-based quantum approaches to analyze quantum phase transitions in a small transverse-field Ising model, revealing current hardware limitations and potential for modeling critical phenomena.

Contribution

It provides a comprehensive comparison of classical and quantum methods for studying quantum criticality on small systems, including experimental validation on real quantum hardware.

Findings

Shallow variational circuits reliably estimate ground-state energies

Hardware noise significantly affects correlation measurements

Quantum hardware shows limitations in capturing long-range correlations

Abstract

The quantum phase transitions provide a paradigm for studying collective quantum phenomena that are a result of competing non-commuting interactions. This paper will study the ground state properties and quantum critical dynamics of the one-dimensional transverse field Ising model through a combined perspective that includes exact diagonalisation, variational quantum eigensolver (VQE) simulations, and simulations on realistic physical quantum devices. We focus on a lattice of four spins, where we calculate the ground-state energies, magnetic order parameters and correlation functions at uniformly applied conditions, which is repeated by all systems. Precise diagonalisation provides both a benchmark, which is symmetry-conserving, and a depth-two, physics inspired variational approximation, which provides simulations accessible to hardware. The circuits that have been optimised identically are then placed on the IQM Garnet quantum processor, using a resource-efficient batched protocol. We find that the ground-state energies of shallow variational circuits are reliably captured by the circuit over the entire parameter space; the magnetic arrangement parameters and observables sensitive to correlation signal significantly more noise. The error analysis of quantitative analysis reveals a strong broadening of critical crossover on hardware, which is consistent with the noise attenuation of long-range correlations. These findings highlight the current capabilities as well as the fundamental limitations of noisy intermediate-scale quantum systems in modelling quantum critical phenomena as a benchmark to future enhancements in obtaining quantum hardware and quantum algorithms development.