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arXiv:1207.1712 · Physical Review A

Solving the graph-isomorphism problem with a quantum annealer

Itay Hen, A. P. Young

Published Oct 10, 2012• 33 citations
arXivHTMLPDF

Abstract

We propose a method using a quantum annealer---an analog quantum computer based on the principles of quantum adiabatic evolution---to solve the graph isomorphism problem, in which one has to determine whether two graphs are isomorphic (i.e., can be transformed into each other simply by a relabeling of the vertices). We demonstrate the capabilities of the method by analyzing several types of graph families, focusing on graphs with particularly high symmetry called strongly regular graphs. We also show that our method is applicable, within certain limitations, to currently available quantum hardware such as D-Wave One.

DOI
10.1103/physreva.86.042310

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We propose a method using a quantum annealer---an analog quantum computer based on the principles of quantum adiabatic evolution---to solve the graph isomorphism problem, in which one has to determine whether two graphs are isomorphic (i.e., can be transformed into each other simply by a relabeling of the vertices). We demonstrate the capabilities of the method by analyzing several types of graph families, focusing on graphs with particularly high symmetry called strongly regular graphs. We also show that our method is applicable, within certain limitations, to currently available quantum hardware such as D-Wave One.
In the graph isomorphism (GI) problem two NNN-vertex graphs GGG and ${G}^{\ensuremath{'}}$ are given and the task is to determine whether there exists a permutation of the vertices of GGG that preserves adjacency and transforms $G\ensuremath{\rightarrow}{G}^{\ensuremath{'}}$. If yes, then GGG and ${G}^{\ensuremath{'}}$ are said to be isomorphic; otherwise they are nonisomorphic. 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 an 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 nonisomorphic graphs; (ii) recognizes isomorphic graphs and determines the permutation(s) that connect them; and (iii) finds the automorphism group of a given graph GGG. 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 subgraph isomorphism problem. The computational complexity of an adiabatic quantum algorithm is largely determined by the minimum energy gap $\ensuremath{\Delta}(N)$ separating the ground and first-excited states in the limit of large problem size $N\ensuremath{\gg}1$. Calculating $\ensuremath{\Delta}(N)$ 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.
We show that the quantum dynamics of interacting and noninteracting quantum particles are fundamentally different in the context of solving a particular computational problem. Specifically, we consider the graph isomorphism problem, in which one wishes to determine whether two graphs are isomorphic (related to each other by a relabeling of the graph vertices), and focus on a class of graphs with particularly high symmetry called strongly regular graphs (SRGs). We study the Green's functions that characterize the dynamical evolution single-particle and two-particle quantum walks on pairs of nonisomorphic SRGs and show that interacting particles can distinguish nonisomorphic graphs that noninteracting particles cannot. We obtain the following specific results. (1) We prove that quantum walks of two noninteracting particles, fermions or bosons, cannot distinguish certain pairs of nonisomorphic SRGs. (2) We demonstrate numerically that two interacting bosons are more powerful than single particles and two noninteracting particles, in that quantum walks of interacting bosons distinguish all nonisomorphic pairs of SRGs that we examined. By utilizing high-throughput computing to perform over 500 million direct comparisons between evolution operators, we checked all tabulated pairs of nonisomorphic SRGs, including graphs with up to 64 vertices. (3) By performing a short-time expansion of the evolution operator, we derive distinguishing operators that provide analytic insight into the power of the interacting two-particle quantum walk.
The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well \cite{kobler93,fortin96}. No polynomial-time algorithm for solving GI is known. We investigate classical and quantum physics-based polynomial-time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. We show that a classical dynamical algorithm proposed by Gudkov and Nussinov \cite{gudkov02} as well as its simplest quantum generalization fail to distinguish pairs of non-isomorphic strongly regular graphs. However, by combining the algorithm of Gudkov and Nussinov with a construction proposed by Rudolph \cite{rudolph02} in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested with up to 29 vertices.
We investigate the quantum networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information one part of a network shares with the other part of the system. The networks which we studied in this paper, are called strongly regular graphs (SRG). These kinds of graphs have some special properties like they have three strata in the stratification basis. The Schur complement method is used to calculate the Schmidt numbers and entanglement entropy between two parts of graph. We could obtain analytically, all blocks of adjacency matrix in several scalable sets of strongly regular graphs. Also the entanglement entropy in the large coupling limit is considered in these graphs and the relationship between entanglement entropy and the ratio of size of boundary to size of the system is found. After that, area-law is studied to show that there are no entanglement entropy for the highest size of system. Then, the graph isomorphism problem is considered in SRGs by using the entanglement entropies in some partitioning of graph. Two SRGs with the same parameters: are isomorphic if they can be made identical by relabeling their vertices. So the adjacency matrices of two isomorphic SRGs become identical by replacing of rows and columns. The non-isomorph SRGs could be distinguished by using the elements of blocks of adjacency matrices in the stratification basis, numerically.
We investigate classical and quantum physics-based algorithms for solving the graph isomorphism problem. Our work integrates and extends previous work by Gudkov et al. (cond-mat/0209112) and by Rudolph (quant-ph/0206068). Gudkov et al. propose an algorithm intended to solve the graph isomorphism problem in polynomial time by mimicking a classical dynamical many-particle process. We show that this algorithm fails to distinguish pairs of non-isomorphic strongly regular graphs, thus providing an infinite class of counterexamples. We also show that the simplest quantum generalization of the algorithm also fails. However, by combining Gudkov et al.'s algorithm with a construction proposed by Rudoph in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested (with up to 29 vertices).
We present a quantum algorithm for solving graph isomorphism problems that is based on an adiabatic protocol. We use a collection of continuous time quantum walks, each one generated by an XY Hamiltonian, to visit the configuration space. In this way we avoid a diffusion over all the possible configurations and significantly reduce the dimensionality of the accessible Hilbert space. Within this restricted space, the graph isomorphism problem can be translated into searching for a satisfying assignment to a 2-SAT (satisfiable) formula and mapped onto a 2-local Hamiltonian without resorting to perturbation gadgets or projective techniques. We present an analysis of the time for execution of the algorithm on small graph isomorphism problem instances and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware.