Journal of Artificial Intelligence Research 10 (1999), pp. 39-66. Submitted 9/98; published 2/99.
© 1999 AI Access Foundation and Morgan Kaufmann Publishers. All rights reserved.

Solving Highly Constrained Search Problems with Quantum Computers

Tad Hogg
Xerox Palo Alto Research Center
3333 Coyote Hill Road, Palo Alto, CA 94304
hogg@parc.xerox.com

Abstract:

A previously developed quantum search algorithm for solving 1-SAT problems in a single step is generalized to apply to a range of highly constrained k-SAT problems. We identify a bound on the number of clauses in satisfiability problems for which the generalized algorithm can find a solution in a constant number of steps as the number of variables increases. This performance contrasts with the linear growth in the number of steps required by the best classical algorithms, and the exponential number required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems in fact have no solutions, unlike previously proposed quantum search algorithms.

• 1 Introduction
• 2 Quantum Computers
• 3 The Satisfiability Problem
  • 3.1 Highly Constrained SAT Problems
  • 3.2 Random SAT Problems
    • 3.2.1 Prespecified Solution
    • 3.2.2 Balanced Clauses
• 4 Solving 1-SAT
  • 4.1 Motivation
  • 4.2 The Algorithm for Maximally Constrained 1-SAT
  • 4.3 Examples
  • 4.4 Performance of the Algorithm
  • 4.5 Implementation
    • 4.5.1 Forming the Initial Superposition
    • 4.5.2 Adjusting Phases
    • 4.5.3 The Mixing Matrix
    • 4.5.4 Required Search Time
• 5 Applying the Algorithm to k-SAT
• 6 Solving Maximally Constrained k-SAT
• 7 Solving Highly Constrained Random k-SAT
  • 7.1 Random k-SAT
  • 7.2 Balanced Clauses
• 8 Discussion
• References
• About this document ...