Quantum tunneling and quantum walks as algorithmic resources to solve hard K-SAT instances

Abstract

Abstract We present a new quantum heuristic algorithm aimed at finding satisfying assignments for hard K-SAT instances using a continuous time quantum walk that explicitly exploits the properties of quantum tunneling. Our algorithm uses a Hamiltonian $$H_A(F)$$ H A ( F ) which is specifically constructed to solve a K-SAT instance F. The heuristic algorithm aims at iteratively reducing the Hamming distance between an evolving state $${|{\psi _j}\rangle }$$ | ψ j ⟩ and a state that represents a satisfying assignment for F. Each iteration consists on the evolution of $${|{\psi _j}\rangle }$$ | ψ j ⟩ (where j is the iteration number) under $$e^{-iH_At}$$ e - i H A t , a measurement that collapses the superposition, a check to see if the post-measurement state satisfies F and in the case it does not, an update to $$H_A$$ H A for the next iteration. Operator $$H_A$$ H A describes a continuous time quantum walk over a hypercube graph with potential barriers that makes an evolving state to scatter and mostly follow the shortest tunneling paths with the smaller potentials that lead to a state $${|{s}\rangle }$$ | s ⟩ that represents a satisfying assignment for F. The potential barriers in the Hamiltonian $$H_A$$ H A are constructed through a process that does not require any previous knowledge on the satisfying assignments for the instance F. Due to the topology of $$H_A$$ H A each iteration is expected to reduce the Hamming distance between each post measurement state and a state $${|{s}\rangle }$$ | s ⟩ . If the state $${|{s}\rangle }$$ | s ⟩ is not measured after n iterations (the number n of logical variables in the instance F being solved), the algorithm is restarted. Periodic measurements and quantum tunneling also give the possibility of getting out of local minima. Our numerical simulations show a success rate of 0.66 on measuring $${|{s}\rangle }$$ | s ⟩ on the first run of the algorithm (i.e., without restarting after n iterations) on thousands of 3-SAT instances of 4, 6, and 10 variables with unique satisfying assignments.