Adiabatic approximation with exponential accuracy for many-body systems and quantum computation

TL;DR

This paper presents an adiabatic theorem tailored for quantum computation, demonstrating that the error in approximating the final state can be exponentially small with respect to the evolution time, under certain smoothness and gap conditions.

Contribution

It introduces an adiabatic theorem with exponential accuracy applicable to many-body quantum systems, emphasizing controllable interpolation and spectral gap assumptions.

Findings

Error between states is exponentially small in evolution time.

Error scales with the square of the Hamiltonian's time-derivative norm.

Evolution time scales with the inverse cube of the minimal spectral gap.

Abstract

We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real time axis, that some number of its time-derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is non-degenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time-derivative of the Hamiltonian, divided by the cube of the minimal gap.