Atom Cavity Encoding for NP-Complete Problems

TL;DR

This paper develops encoding schemes for NP-complete problems using an atom-cavity system, enabling quantum solutions with linear or quadratic resource costs, and guides future quantum advantage research.

Contribution

It introduces efficient encoding schemes for many NP-complete problems in atom-cavity systems, expanding quantum problem-solving capabilities.

Findings

Many NP-complete problems can be encoded at linear atom cost.

Some problems require quadratic or quartic atom costs for encoding.

The schemes may be applicable to other optical systems with similar Hamiltonians.

Abstract

We consider an atom-cavity system having long-range atomic interactions mediated by cavity modes. It has been shown that quantum simulations of spin models with this system can naturally be used to solve number partition problems. Here, we present encoding schemes for numerous NP-complete problems, encompassing the majority of Karp's 21 NP-complete problems. We find a number of such computation problems can be encoded by the atom-cavity system at a linear cost of atom number. There are still certain problems that cannot be encoded by the atom-cavity as efficiently, such as quadratic unconstrained binary optimization (QUBO), and the Hamiltonian cycle. For these problems, we provide encoding schemes with a quadratic or quartic cost in the atom number. We expect this work to provide important guidance to search for the practical quantum advantage of the atom-cavity system in solving NP-complete problems. Moreover, the encoding schemes we develop here may also be adopted in other optical systems for solving NP-complete problems, where a similar form of Mattis-type spin glass Hamiltonian as in the atom-cavity system can be implemented.