Solving Set Cover with Pairs Problem Using Quantum Annealing

TL;DR

This paper explores using quantum annealing to solve the NP-hard Set Cover with Pairs problem by constructing specific Ising Hamiltonians and testing their performance through simulations and embeddings on D-Wave hardware.

Contribution

It presents an explicit Hamiltonian construction for SCP, simulation results comparing quantum and simulated annealing, and embedding strategies for D-Wave Chimera graphs.

Findings

Quantum annealing can encode SCP solutions in Ising Hamiltonians.

Simulations compare quantum annealing performance with classical methods.

Embedding strategies preserve problem structure on D-Wave hardware.

Abstract

Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that play an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schr\"{o}dinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with.