Adiabatic Quantum Algorithms for the NP-Complete Maximum-Weight Independent Set, Exact Cover and 3SAT Problems

TL;DR

This paper explores how tuning parameters in adiabatic quantum algorithms can avoid phase transitions and improve performance for NP-complete problems like MIS, Exact Cover, and 3SAT, challenging previous negative results.

Contribution

It introduces parameter adjustments in problem Hamiltonians to prevent phase transitions and proposes new adiabatic algorithms for NP-complete problems based on MIS reduction.

Findings

Parameter tuning can prevent first order phase transitions.

Modified algorithms show different spectral gap behaviors.

Previous negative results do not necessarily apply to new formulations.

Abstract

The problem Hamiltonian of the adiabatic quantum algorithm for the maximum-weight independent set problem (MIS) that is based on the reduction to the Ising problem (as described in [Choi08]) has flexible parameters. We show that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS on CK graphs, we can prevent the first order quantum phase transition and significantly change the minimum spectral gap. We raise the basic question about what the appropriate formulation of adiabatic running time should be. We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. We point out that the argument in Altshuler et al.(arXiv:0908.2782 [quant-ph]) that their adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover does not carry over to this new algorithm.