Finite-Size Scaling on a Digital Quantum Simulator Using Quantum Restricted Boltzmann Machine

Abstract

The critical point and the critical exponents for a phase transition can be determined using the Finite-Size Scaling (FSS) analysis. This method assumes that the phase transition occurs only in the infinite size limit. However, there has been a lot of interest recently in quantum phase transitions occurring in finite size systems such as a single two-level system interacting with a single bosonic mode e.g., in the Quantum Rabi Model (QRM). Since these phase transitions occur at a finite system size, the traditional FSS method is rendered inapplicable for these cases. For cases like this, we propose an alternative FSS method in which the truncation of the system is done in the Hilbert space instead of the physical space. This approach has previously been used to calculate the critical parameters for stability and symmetry breaking of electronic structure configurations of atomic and molecular systems. We calculate the critical point for the quantum phase transition of the QRM using this approach. We also provide a protocol to implement this method on a digital quantum simulator using the Quantum Restricted Boltzmann Machine algorithm. Our work opens up a new direction in the study of quantum phase transitions on quantum devices.

Simulations of frustrated Ising Hamiltonians using quantum approximate optimization

“…However, γ < −1 signifies a faster drop off of RMSE as compared to 1/n, which could be because of enhanced correlations between the various spins subject to the Hamiltonian. A more rigorous analysis of finite-size scaling [53,54] around each critical point could yield a careful analysis of the required lattice size for target fidelity for every phase transition. Our result based on an overall RMSE demonstrates that a 15 × 15 spin grid is already obtaining results close to the much larger 30×30 grid.…”

“…A more rigorous analysis of finite-size scaling [53,54] around each critical point could yield a careful analysis of the required lattice size for target fidelity for every phase transition. Our result based on an overall RMSE demonstrates that a 15×15 spin grid is already obtaining results close to the much larger 30×30 grid.…”

Mapping Molecular Hamiltonians into Hamiltonians of Modular cQED Processors

“…The development of quantum computing simulations for modeling chemical systems is a subject of immense interest. Recent studies have already explored the potential of quantum computing as applied to electronic structure calculations, − quantum dynamics simulations, − and simulations of molecular spectroscopy. − Currently quantum computing facilities are often called noisy intermediate-scale quantum (NISQ) computers, due to their intrinsic limitations, including architectures based on superconducting circuits, trapped ions, , and nuclear magnetic resonance. , To achieve moderate accuracy and reliability in spite of noise and decoherence, simulations of chemical systems have relied on hybrid quantum-classical algorithms, including the variational quantum eigensolver (VQE) method ,, and quantum machine learning methods , where only part of the computation is performed on the quantum computer, sometimes applied with the aid of error mitigation techniques, while the rest of the calculation is run on a conventional computer.…”