Graph isomorphism and adiabatic quantum computing

Abstract

In the Graph Isomorphism (GI) problem two N -vertex graphs G and G are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G → G . If yes, then G and G are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and which also provides a novel approach to determining all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithm's quantum dynamics and show that it correctly: (i) distinguishes non-isomorphic graphs; (ii) recognizes isomorphic graphs and determines the permutation(s) that connect them; and (iii) finds the automorphism group of a given graph G. We then discuss the GI quantum algorithm's experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-Complete Sub-Graph Isomorphism problem. The computational complexity of an adiabatic quantum algorithm is largely determined by the minimum energy gap ∆(N ) separating the ground-and first-excited states in the limit of large problem size N 1. Calculating ∆(N ) in this limit is a fundamental open problem in adiabatic quantum computing, and so it is not possible to determine the computational complexity of adiabatic quantum algorithms in general, nor consequently, of the specific adiabatic quantum algorithms presented here. Adiabatic quantum computing has been shown to be equivalent to the circuit-model of quantum computing, and so development of adiabatic quantum algorithms continues to be of great interest.

A Comparison of Approaches for Solving Hard Graph-Theoretic Problems

“…Section 2 describes an approach for solving the miminum identifying code problem using adiabatic quantum optimization [9], which for small enough problem instances can be implemented on a D-Wave quantum annealing processor [12]. While to our knowledge this is the first time adiabatic quantum optimization has been applied to the identifying code problem, it has been studied for other graph-theoretic problems including graph coloring [23] and the graph isomorphism problem [10], [25]; in addition, [17] showed how to formulate a number of NP-complete problems as Ising models. Other approaches are considered as follows: Subsection 3.1 explores a parallel computing algorithm and Subsection 3.2 illustrates the method of using Satisfiability Modulo Theory (SMT) solvers.…”

Recall Performance for Content-Addressable Memory Using Adiabatic Quantum Optimization

“…This finite-temperature, open system model is expected to more accurately describe the dynamics underlying the QPU [8]. Nevertheless, several experimental tests of the D-Wave QPU have been carried out including applications of machine learning, binary classification, protein folding, graph analysis, and network analysis [9][10][11][12][13][14][15][16][17][18]. Demonstrations of enhanced performance using the D-Wave QPU have been found only for a few selected and highly contrived problem instances [19][20][21].…”

An integrated programming and development environment for adiabatic quantum optimization

“…Examples include problems in classification [15,16], machine learning [17], graph theory [18,19,20], artificial neural networks [21], and protein folding [22] among others [23].…”