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arXiv:quant-ph/0104048

Solving Random Satisfiability Problems with Quantum Computers

Tad Hogg

Published Apr 9, 2001• 5 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0104048

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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.
The discrete formulation of adiabatic quantum computing is compared with other search methods, classical and quantum, for random satisfiability (SAT) problems. With the number of steps growing only as the cube of the number of variables, the adiabatic method gives solution probabilities close to 1 for problem sizes feasible to evaluate via simulation on current computers. However, for these sizes the minimum energy gaps of most instances are fairly large, so the good performance scaling seen for small problems may not reflect asymptotic behavior where costs are dominated by tiny gaps. Moreover, the resulting search costs are much higher than for other methods. Variants of the quantum algorithm that do not match the adiabatic limit give lower costs, on average, and slower growth than the conventional GSAT heuristic method.
The structure of satisfiability problems is used to improve search algorithms for quantum computers and to reduce their required coherence times by using only a single coherent evaluation of problem properties. The structure of random k-SAT allows the determination of the asymptotic average behavior of these algorithms, showing they improve on the quantum algorithms, such as amplitude amplification, that ignore detailed problem structure but remain exponential for hard problem instances. Compared to good classical methods, the algorithm performs better, on average, for weakly and highly constrained problems, but worse for hard cases. The analytic techniques introduced here apply also to other quantum algorithms, supplementing the limited evaluation possible with classical simulations, and showing how quantum computing can use the ensemble properties of NP search problems.
The deep structure of constraint satisfaction problems explains the association of hard search instances with a phase transition in problem solubility. This structure is also the basis of a quantum search algoritbm exhibiting the phase transition. In this paper, this algoritbm is modified to incorporate additional problem structure. This modification is an example of a general method for including heuristics in quantum search. The new algoritbm is evaluated empirically for random 3SAT, illustrating how quantum searches can benefit from using problem structure, on average.
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.