Phase Transitions for Quantum Search Algorithms

Abstract

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.