Abstract
Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays 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 Schrödinger 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.
Highlights
We consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry
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
In general w(S) increases as S increases, leading to a decrease in R. We numerically investigate this with an Ising system of N = 17 spins generated from an SCP instance via the construction in Theorem 1
Summary
Yudong Cao[1], Shuxian Jiang[1], Debbie Perouli2 & Sabre Kais[3,4] received: 12 June 2016 accepted: 02 September 2016 Published: 27 September 2016. If the ground state of HP is NP-complete to find (for instance consider the case for Ising spin glass49), the adiabatic evolution H(s) could be used as a heuristic for solving the problem. We construct Ising Hamiltonians whose ground state encodes the solution to an arbitrary instance of the SCP problem. Given an instance of the Set Cover with Pairs Problem SCP(G, U, S) as in Definition 1, there exists an efficient (classical) algorithm that computes an instance of the Ising Hamiltonian ground state problem ISING(h, J) with h ∈ M and J ∈ M×M where the number of qubits involved in the Hamiltonian is M =O(nm2) with n =|U| and m =|S|.
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