Efficient optimization of perturbative gadgets

TL;DR

This paper introduces efficient methods for optimizing perturbative gadgets, reducing physical resources needed for implementing many-body interactions with two-body systems, and provides tight error bounds with significant numerical improvements.

Contribution

It develops algorithms to compute tight upper bounds on perturbation errors, enhancing gadget efficiency and accuracy, especially for commuting Hamiltonian terms.

Findings

Algorithms produce sharp error bounds for arbitrary perturbation order.

Numerical results show orders of magnitude improvement over trivial bounds.

Potential further improvements using Schrieffer-Wolff formalism.

Abstract

Perturbative gadgets are general techniques for reducing many-body spin interactions to two-body ones using perturbation theory. This allows for potential realization of effective many-body interactions using more physically viable two-body ones. In parallel with prior work (arXiv:1311.2555 [quant-ph]), here we consider minimizing the physical resource required for implementing the gadgets initially proposed by Kempe, Kitaev and Regev (arXiv:quant-ph/0406180) and later generalized by Jordan and Farhi (arXiv:0802.1874v4). The main innovation of our result is a set of methods that efficiently compute tight upper bounds to errors in the perturbation theory. We show that in cases where the terms in the target Hamiltonian commute, the bounds produced by our algorithm are sharp for arbitrary order perturbation theory. We provide numerics which show orders of magnitudes improvement over gadget constructions based on trivial upper bounds for the error term in the perturbation series. We also discuss further improvement of our result by adopting the Schrieffer-Wolff formalism of perturbation theory and supplement our observation with numerical results.