Error Reduction in Quantum Annealing using Boundary Cancellation: Only the End Matters

TL;DR

This paper extends boundary cancellation techniques to open quantum systems, showing that error reduction depends only on the end of the evolution and can be achieved through system control, with proven stability and applicability.

Contribution

It generalizes boundary cancellation to open systems, demonstrating error reduction depends only on the end, and shows control strategies can satisfy necessary conditions.

Findings

Error bounds improve with vanishing derivatives at the end of evolution.

Control of the system alone suffices to achieve boundary cancellation.

Results are stable under imperfections and applicable to Hamiltonian models.

Abstract

The adiabatic theorem of quantum mechanics states that the error between an instantaneous eigenstate of a time-dependent Hamiltonian and the state given by quantum evolution of duration $\tau$ is upper bounded by $C/\tau$ for some positive constant $C$. It has been known for decades that this error can be reduced to $C_{k}/\tau^{k+1}$ if the Hamiltonian has vanishing derivatives up to order $k$ at the beginning and end of the evolution. Here we extend this result to open systems described by a time-dependent Liouvillian superoperator. We find that the same results holds provided the Liouvillian has vanishing derivatives up to order $k$ only at the end of the evolution. This asymmetry is ascribable to the arrow of time inherent in open system evolution. We further investigate whether it is possible to satisfy the required assumptions by controlling only the system, as required for realistic implementations. Surprisingly, we find the answer to be affirmative. We establish this rigorously in the setting of the Davies-Lindblad adiabatic master equation, and numerically in the setting of two different time-dependent Redfield-type master equations we derive. The results are shown to be stable with respect to imperfections in the preparation. Finally, we prove that the results hold also in a fully Hamiltonian model.