Interpolation Approach to Hamiltonian-varying Quantum Systems and the Adiabatic Theorem

TL;DR

This paper develops an interpolation-based method to estimate the error in adiabatic quantum evolution, providing guidelines for more accurate adiabatic control by analyzing Hamiltonian variations.

Contribution

It introduces a precise error estimation technique for adiabatic approximation based on interpolation between Hamiltonians, challenging traditional assumptions about error dependence.

Findings

Error can be precisely estimated for arbitrary interpolating functions

Adiabatic error is not always proportional to the variation speed or inverse energy gaps

Application to a questionable case of adiabatic theorem validity

Abstract

Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure could be taken as the error of adiabatic approximation. We prove under certain conditions, this error can be precisely estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example on which the applicability of the adiabatic theorem is questionable.