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 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 O( 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.
Quantum searching for one of N marked items in an unsorted database of n items is solved in O( n/N ) steps using Grover's algorithm. Using nonlinear quantum dynamics with a Gross-Pitaevskii type quadratic nonlinearity, Childs and Young discovered an unstructured quantum search algorithm with a complexity O(min{1/g log(gn), √ n}), which can be used to find a marked item after o(log(n)) repetitions, where g is the nonlinearity strength [1]. In this work we develop a structured search on a complete graph using a time dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime O((where N ⊥ denotes the number of unmarked items and G is related to the time dependent nonlinearity. If N ⊥ ≤ N , we obtain a runtime O(1). We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 5.
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