Quantum Adiabatic Theorem Revisited

TL;DR

This paper presents a simplified and improved proof of the quantum adiabatic theorem, providing better bounds and a clear integral representation of the error, with applications to qubit state preparation.

Contribution

It offers a shorter, more elementary proof of the quantum adiabatic theorem with enhanced bounds and a novel integral approach, extending previous discrete methods.

Findings

New integral representation of state difference

Improved bounds on adiabatic approximation error

Application to adiabatic qubit state preparation

Abstract

In 2004 Ambainis and Regev formulated a certain form of quantum adiabatic theorem and provided an elementary proof which is especially accessible to computer scientists. Their result is achieved by discretizing the total adiabatic evolution into a sequence of unitary transformations acting on the quantum system. Here we continue this line of study by providing another elementary and shorter proof with improved bounds. Our key finding is a succinct integral representation of the difference between the target and the actual states, which yields an accurate estimation of the approximation error. Our proof can be regarded as a "continuous" version of the work by Ambainis and Regev. As applications, we show how to adiabatically prepare an arbitrary qubit state from an initial state.