Out-of-equilibrium many-body quantum systems with respect to adiabatic quantum computing through the lens of the Pechukas–Yukawa framework

We investigate the theoretical description of adiabatic quantum computing (AQC) algorithms using the evolution of the Hamiltonian eigenvalues in the framework of the Pechukas–Yukawa formalism, exactly mapping the eigenvalues to the dynamics of a fictitious one-dimensional classical gas with cubic repulsion. We exploit the properties of the Pechukas-Yukawa model to describe the behaviour of quantum algorithms used in AQC. Specifically, we derive the non-equilibrium nonstationary statistical mechanics of the Pechukas–Yukawa gas based on the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) chain of equations with the goal of increasing the efficiency of direct numerical simulation. We extended our research to consider the impacts of level crossings and avoided crossings to evaluate the compatibility of the Pechukas–Yukawa formalism and the Landau–Zener description of these occurrences. This is valuable to the investigation of decoherence in a quantum system and carries scope for research on the description of state dynamics through the energy level dynamics. We relate the evolution of a quantum state of a system under external perturbation to that of its energy levels. Using this relationship, we produced a cumulant expansion with improved efficiency compared to traditional methods of approximate quantum state evolution description. It is especially significant for the investigation of decoherence in an evolving quantum system.