Graph comparison via nonlinear quantum search

TL;DR

This paper introduces a scalable quantum algorithm combining linear and nonlinear quantum computing to efficiently determine the maximum common subgraph size, offering a new approach to graph similarity measurement.

Contribution

It presents a novel quantum algorithm that leverages nonlinear quantum dynamics to solve graph comparison problems more efficiently than previous methods.

Findings

The linear quantum part requires $ ilde{O}(n^3)$ gates.

Nonlinear evolution scales as $ ilde{O}(rac{1}{g} n^2)$.

Demonstrates nonlinear quantum search can address NP-hard problems.

Abstract

In this paper we present an efficiently scaling quantum algorithm which finds the size of the maximum common edge subgraph for a pair of arbitrary graphs and thus provides a meaningful measure of graph similarity. The algorithm makes use of a two-part quantum dynamic process: in the first part we obtain information crucial for the comparison of two graphs through linear quantum computation. However, this information is hidden in the quantum system with vanishingly small amplitude that even quantum algorithms such as Grover's search are not fast enough to distill the information efficiently. In order to extract the information we call upon techniques in nonlinear quantum computing to provide the speed-up necessary for an efficient algorithm. The linear quantum circuit requires $\mathcal{O}(n^3 \log^3 (n) \log \log (n))$ elementary quantum gates and the nonlinear evolution under the Gross-Pitaevskii equation has a time scaling of $\mathcal{O}(\frac{1}{g} n^2 \log^3 (n) \log \log (n))$, where $n$ is the number of vertices in each graph and $g$ is the strength of the Gross-Pitaveskii non-linearity. Through this example, we demonstrate the power of nonlinear quantum search techniques to solve a subset of NP-hard problems.