Different Adiabatic Quantum Optimization Algorithms for the NP-Complete Exact Cover and 3SAT Problems

TL;DR

This paper explores various adiabatic quantum optimization algorithms for NP-complete problems, demonstrating that previous negative results are specific to certain algorithms and highlighting the need for further investigation into AQO's overall effectiveness.

Contribution

The paper introduces different AQO algorithms for Exact Cover and 3SAT, challenging prior claims of failure by showing these results are algorithm-specific.

Findings

Negative results depend on specific AQO algorithms

New algorithms for Exact Cover and 3SAT are proposed

Further research needed to assess AQO's general success

Abstract

One of the most important questions in studying quantum computation is: whether a quantum computer can solve NP-complete problems more efficiently than a classical computer? In 2000, Farhi, et al. (Science, 292(5516):472--476, 2001) proposed the adiabatic quantum optimization (AQO), a paradigm that directly attacks NP-hard optimization problems. How powerful is AQO? Early on, van Dam and Vazirani claimed that AQO failed (i.e. would take exponential time) for a family of 3SAT instances they constructed. More recently, Altshuler, et al. (Proc Natl Acad Sci USA, 107(28): 12446--12450, 2010) claimed that AQO failed also for random instances of the NP-complete Exact Cover problem. In this paper, we make clear that all these negative results are only for a specific AQO algorithm. We do so by demonstrating different AQO algorithms for the same problem for which their arguments no longer hold. Whether AQO fails or succeeds for solving the NP-complete problems (either the worst case or the average case) requires further investigation. Our AQO algorithms for Exact Cover and 3SAT are based on the polynomial reductions to the NP-complete Maximum-weight Independent Set (MIS) problem.