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

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 Schrödinger 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.