Improved QUBO Formulation of the Graph Isomorphism Problem

Abstract

In this paper, we provide a practically efficient QUBO formulation for the Graph Isomorphism Problem that is suitable for quantum annealers such as those produced by D-Wave. After proving the correctness of our new method, based on exploiting vertex degree classes, we did some experimental work on a D-Wave 2X computer. We observe that for all "hard" graphs of 6 vertices, we save around 50-95% of the number of required physical qubits over the standard QUBO formulation that was given earlier by Calude et al. (Theor Comput Sci 701:54-69, 2017). We also provide some theoretical analysis showing that, for two random graphs with the same degree sequence, our new method substantially improves in qubit savings as the number of vertices increases beyond 6.

A New QUBO Objective Function for Solving the Maximum Common Subgraph Isomorphism Problem Via Quantum Annealing

“…However, when chain length grows, the performance drops significantly. This is consistent with other experiments [16]. The D-Wave API provides a post-process function that classically optimises the quantum machine's output [10].…”

Characterization of QUBO reformulations for the maximum $k$-colorable subgraph problem

“…However, the efficacy of NISQ devices to solve this broader class of COPT problems can be highly affected by the way in which the corresponding QUBO reformulation is obtained. This is because the performance of NISQ devices is highly affected by the number of qubits and the coefficients that are required to encode a QUBO [see, e.g., 10,25,31].…”

“…where ∆ is the degree of G(V, E), and the auxiliary variable (1) to make it a homogenous quadratic). The performance of NISQ devices on solving QUBO problems is negatively affected by the use of a larger number of logical qubits and larger coefficients [see, e.g., 10,25,26,31,63]. In this context, it is natural to ask if there are improved [see, e.g., 10,25,31,61,62] QUBO reformulation for the maximum clique problem.…”

“…The performance of NISQ devices on solving QUBO problems is negatively affected by the use of a larger number of logical qubits and larger coefficients [see, e.g., 10,25,26,31,63]. In this context, it is natural to ask if there are improved [see, e.g., 10,25,31,61,62] QUBO reformulation for the maximum clique problem. For example, notice that by slightly changing the definition and number of the auxiliary variables in (1), the range of the coefficients used in (1) can be substantially reduced.…”

“…As the results in [28] highlight, this efficiency is key towards the goal of using noisy intermediate scale quantum (NISQ) devices to solve COPT problems more efficiently than with classical computers. Not surprisingly, recent articles look beyond obtaining QUBO reformulations of COPT problems such as the graph isomorphism problem as well as tree and cycle elimination problems, to look for improved QUBO reformulations of these problems for NISQ devices [see, e.g., 10,25,31,61,62]. That is, QUBO reformulations that are tailored to be more efficiently used in NISQ devices.…”