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Physical Review A

Quantum search heuristics

Tad Hogg

Published Apr 14, 2000• 70 citations
Publisher

Abstract

An alternative quantum algorithm for combinatorial search, adjusting amplitudes based on number of conflicts in search states, performs well, on average, for hard random satisfiability problems near a phase transition in search difficulty. The algorithm exploits correlations among problem properties more effectively than some current heuristics, and improves on prior quantum algorithms that ignore these correlations.

DOI
10.1103/physreva.61.052311

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An alternative quantum algorithm for combinatorial search, adjusting amplitudes based on number of conflicts in search states, performs well, on average, for hard random satisfiability problems near a phase transition in search difficulty. The algorithm exploits correlations among problem properties more effectively than some current heuristics, and improves on prior quantum algorithms that ignore these correlations.
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Quantum computer algorithms can exploit the structure of random satisfiability problems. This paper extends a previous empirical evaluation of such an algorithm and gives an approximate asymptotic analysis accounting for both the average and variation of amplitudes among search states with the same costs. The analysis predicts good performance, on average, for a variety of problems including those near a phase transition associated with a high concentration of hard cases. Based on empirical evaluation for small problems, modifying the algorithm in light of this analysis improves its performance. The algorithm improves on both GSAT, a commonly used conventional heuristic, and quantum algorithms ignoring problem structure.
We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems.
We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems.
We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems.
We present a quantum computer heuristic search algorithm for graph coloring. This algorithm uses a new quantum operator, appropriate for nonbinary-valued constraint satis-faction problems, and information available in partial col-orings. We evaluate the algorithm empirically with small graphs near a phase transition in search performance. It im-proves on two prior quantum algorithms: unstructured search and a heuristic applied to the satisfiability (SAT) encoding of graph coloring. An approximate asymptotic analysis sug-gests polynomial-time cost for hard graph coloring problems, on average.