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

Rudraksh Sharma

TL;DR

This study benchmarks the quantum phase transition of the one-dimensional transverse-field Ising model on a four-spin system using exact diagonalization, a fixed-depth two-layer VQE, and experiments on the IQM Garnet quantum processor. Energies are reproduced with high fidelity by the VQE, while magnetic order and correlation observables suffer from hardware noise, especially near the quantum critical region, leading to a broadened crossover on real devices. The work emphasizes the relative robustness of energy measurements to current NISQ limitations and highlights the challenges in accurately capturing critical behavior and long-range correlations on noisy hardware. It provides a controlled framework for cross-method benchmarking that can guide future hardware improvements and variational ansatz design for quantum simulations of many-body phenomena.

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.

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

TL;DR

This study benchmarks the quantum phase transition of the one-dimensional transverse-field Ising model on a four-spin system using exact diagonalization, a fixed-depth two-layer VQE, and experiments on the IQM Garnet quantum processor. Energies are reproduced with high fidelity by the VQE, while magnetic order and correlation observables suffer from hardware noise, especially near the quantum critical region, leading to a broadened crossover on real devices. The work emphasizes the relative robustness of energy measurements to current NISQ limitations and highlights the challenges in accurately capturing critical behavior and long-range correlations on noisy hardware. It provides a controlled framework for cross-method benchmarking that can guide future hardware improvements and variational ansatz design for quantum simulations of many-body phenomena.

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.
Paper Structure (26 sections, 6 equations, 5 figures, 2 tables)

This paper contains 26 sections, 6 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Methodology
  4. Transverse-Field Ising Model
  5. Exact Diagonalization
  6. Variational Quantum Eigensolver
  7. Ansatz Design
  8. Classical Optimization
  9. Observable Evaluation
  10. Quantum Hardware Implementation
  11. Batched Execution
  12. Measurement Strategy
  13. Statistical Analysis and Error Estimation
  14. Results
  15. Ground-State Energy
  16. ...and 11 more sections

Figures (5)

  • Figure 1: Ground-state energy of the four-spin transverse-field Ising model as a function of the transverse field
  • Figure 2: Magnetic order parameter as a function of the transverse field
  • Figure 3: Absolute longitudinal magnetization measured on IQM quantum hardware as a function of the transverse field.
  • Figure 4: Ground-state energy reconstructed from IQM hardware measurements for different transverse field values.
  • Figure 5: Nearest-neighbor spin correlations $\langle Z_i Z_{i+1}\rangle$ measured on IQM quantum hardware as a function of the transverse field