Adiabatic Quantum Computation with a 1D projector Hamiltonian

TL;DR

This paper analyzes a specific class of adiabatic quantum evolutions involving one-dimensional projector Hamiltonians, revealing how the minimum energy gap relates to ground state overlaps and proposing methods for faster evolution.

Contribution

It introduces a detailed analysis of adiabatic evolutions with projector Hamiltonians, linking energy gaps to ground state overlaps and suggesting optimized evolution strategies.

Findings

Minimum energy gap proportional to ground state overlap

Rapid crossover near the transition point

Partial adiabatic evolution improves speed

Abstract

Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyse a particular class of qauntum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum energy gap which governs the time required for a successful evolution is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.