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Computational study demonstrates quadratic run time for finding maximum cliques in random graphs up to 18 vertices, suggesting quantum speedup over classical algorithms.
Quantum adiabatic evolution provides a general technique for the solution of combinatorial search problems on quantum computers. We present the results of a numerical study of a particular application of quantum adiabatic evolution, the problem of finding the largest clique in a random graph. An n-vertex random graph has each edge included with probability 1/2, and a clique is a completely connected subgraph. There is no known classical algorithm that finds the largest clique in a random graph with high probability and runs in a time polynomial in n. For the small graphs we are able to investigate (n ≤ 18), the quantum algorithm appears to require only a quadratic run time.
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Childs et al. (2002) studied this question.
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