Estimating the Number of States via the Rodeo Algorithm for Quantum Computation

TL;DR

This paper introduces a quantum algorithm called the rodeo algorithm to estimate the number of quantum states at various energy levels without prior eigenstate knowledge, demonstrated on the 1D transverse-field Ising model.

Contribution

The work applies the rodeo algorithm to quantum systems, enabling state counting without eigenstate solutions, advancing quantum thermodynamics analysis.

Findings

Successfully estimated the number of states for the 1D transverse-field Ising model

Computed the specific heat from the estimated state counts

Validated the reliability of the rodeo algorithm in quantum thermodynamics

Abstract

In the realm of statistical physics, the number of states in which a system can be realized with a given energy is a key concept that bridges the microscopic and macroscopic descriptions of physical systems. For quantum systems, many approaches rely on the solution of the Schr\"odinger equation. In this work, we demonstrate how the recently developed rodeo algorithm can be utilized to determine the number of states associated with all energy levels without any prior knowledge of the eigenstates. Quantum computers, with their innate ability to address the intricacies of quantum systems, make this approach particularly promising for the study of the thermodynamics of those systems. To illustrate the procedure's effectiveness, we apply it to compute the number of states of the 1D transverse-field Ising model and, consequently, its specific heat, proving the reliability of the method presented here.