EPJ Quantum Technol. (2015) 2: 24
https://doi.org/10.1140/epjqt/s40507-015-0037-x

Research

Interpolation approach to Hamiltonian-varying quantum systems and the adiabatic theorem

¹ Research School of Engineering, Australian National University, Canberra, 0200, Australia
² Department of Electrical & Electronic Engineering, The University of Melbourne, Melbourne, 3010, Australia
³ Laboratoire des signaux et systèmes (L2S) Supélec, CNRS, 3 rue Joliot-Curie, Gif-Sur-Yvette, 91192, France
⁴ School of Engineering and Information Technology, University of New South Wales at ADFA, Canberra, 2600, Australia

^* e-mail: yu.pan.83.yp@gmail.com

Received: 30 June 2015
Accepted: 7 November 2015
Published online: 25 November 2015

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 resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be 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 linearly 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 in which the applicability of the adiabatic theorem is questionable.