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

TL;DR

This paper introduces a novel finite-size scaling method tailored for finite quantum systems, utilizing a Hilbert space truncation approach and implementing it on a digital quantum simulator with a Quantum Restricted Boltzmann Machine.

Contribution

It proposes an alternative FSS method for finite quantum systems and demonstrates its application to the Quantum Rabi Model on a quantum simulator.

Findings

Critical point for QRM quantum phase transition calculated

Method implemented on digital quantum simulator

New approach for studying finite-size quantum phase transitions

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 occuring 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.