# Pechukas-Yukawa approach to the evolution of the quantum state of a parametrically perturbed system

## TL;DR

This paper explores the evolution of quantum states under parametric perturbations using the Pechukas-Yukawa approach, comparing approximation methods and applying findings to a quantum computational problem.

## Contribution

It introduces the use of the Magnus series expansion for better non-adiabatic quantum evolution description within the Pechukas-Yukawa framework.

## Key findings

- Magnus series effectively describes non-adiabatic evolution.
- Comparison shows advantages of the Magnus expansion over other methods.
- Application to 3-satisfiability problem reveals state occupation dynamics.

## Abstract

We consider the evolution of a quantum state of a Hamiltonian which is parametrically perturbed via a term proportional to the adiabatic parameter \lambda (t). Starting with the Pechukas-Yukawa mapping of the energy eigenvalues evolution on a generalised Calogero-Sutherland model of 1D classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of d\lambda/dt and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of non-adiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfability problem, we obtain the occupation dynamics which provides insight on the population of states.