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

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.

💡 Research Summary

The paper addresses a central question in quantum computation: can an adiabatic quantum optimizer (AQO) solve NP‑complete problems more efficiently than classical algorithms? After recalling Farhi et al.’s original proposal of AQO as a direct approach to hard optimization problems, the authors focus on two well‑studied negative results. The first, by van Dam and Vazirani, claims exponential runtime for a specially constructed family of 3‑SAT instances when a “standard” AQO Hamiltonian is used. The second, by Altshuler et al., argues that random instances of the Exact Cover problem also cause AQO to fail because the minimum spectral gap shrinks exponentially. Both arguments rely on a particular choice of problem Hamiltonian (H_P) (typically a clause‑cloud or literal‑cloud construction) and a simple transverse‑field driver (H_B).

The central contribution of the present work is to demonstrate that these failures are not inherent to AQO itself but are artifacts of the specific Hamiltonian design. The authors achieve this by reducing Exact Cover and 3‑SAT to the Maximum‑Weight Independent Set (MIS) problem, which is known to be NP‑complete. In the MIS formulation, each variable or clause becomes a vertex in a graph, edges encode mutual exclusion (e.g., overlapping sets in Exact Cover or contradictory literals in 3‑SAT), and vertex weights encode the contribution of a satisfied clause or selected set. The reduction is polynomial‑time and preserves the solution structure: a maximum‑weight independent set corresponds directly to a satisfying assignment of the original instance.

With the MIS representation in hand, the authors construct a new problem Hamiltonian \