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arXiv:cs/9603101

Quantum Computing and Phase Transitions in Combinatorial Search

Tad Hogg

Published Mar 1, 1996• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cs/9603101

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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.
Phase transitions have long been studied empirically in various combinatorial searches and theoretically in simplified models [91, 264, 301, 490]. The analogy with statistical physics [397], explored throughout this volume, shows how the many local choices made during search relate to global properties such as the resulting search cost. These studies have led to a better understanding of typical search behaviors [514] and improved search methods [195, 247, 261, 432, 433]. Among the current research questions in this field are the range of algorithms exhibiting the transition behavior and the algorithm-independent problem properties associated with the difficult instances concentrated near the transition. Towards this end, the present chapter examines quantum computer [123, 126, 158, 486] algorithms for nondeterministic polynomial (NP) combinatorial search problems [191]. As with many conventional methods, they exhibit the easy-hard-easy pattern of computational cost as the degree of constraint in the problems varies. We describe how properties of the search space affect the algorithms and identify an additional structural property, the energy gap, motivated by one quantum algorithm but applicable to a variety of techniques, both quantum and classical. Thus, the study of quantum search algorithms not only extends the range of algorithms exhibiting phase transitions, but also helps identify underlying structural properties. Specifically, the next two sections describe a class of hard search problems and the form of quantum search algorithms proposed to date. The remainder of the chapter presents algorithm behaviors, relevant problem structure, arid an approximate asymptotic analysis of their cost scaling. The final section discusses various open issues in designing and evaluating quantum algorithms, and relating their behavior to problem structure. The k-satisfiability (k -SAT) problem, as discussed earlier in this volume, consists of n Boolean variables and m clauses. A clause is a logical OR of k variables, each of which may be negated. A solution is an assignment, that is, a value for each variable, TRUE or FALSE, satisfying all the clauses. An assignment is said to conflict with any clause it does not satisfy.
A quantum algorithm for combinatorial search is presented that provides a simple framework for utilizing search heuristics. The algorithm is evaluated in a new case that is an unstructured version of the graph coloring problem. It performs significantly better than the direct use of quantum parallelism, on average, in cases corresponding to previously identified phase transitions in search difficulty. The conditions underlying this improvement are described. Much of the algorithm is independent of particular problem instances, making it suitable for implementation as a special purpose device.
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.