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Tailoring Many-Body Interactions to Solve Hard Combinatorial Problems.pdf

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Tailoring Many-Body Interactions to Solve Hard Combinatorial Problems

a r X i v : q u a n t - p h / 9 7 0 2 0 3 9 v 4 8 J u l 1 9 9 7 Tailoring Many-Body Interactions to Solve Hard Combinatorial Problems Haiqing Wei Department of Physics, McGill University Montreal, Quebec, Canada H3A 2T8 E-mail: dhw@physics.mcgill.ca Xin Xue Department of Natural Resource Sciences Macdonald Campus of McGill University Ste-Anne-de-Bellevue, Quebec, Canada H9X 3V9 E-mail: xkhz@musicb.mcgill.ca Abstract A quantum machine consisting of interacting linear clusters of atoms is pro- posed for the 3SAT problem. Each cluster with two relevant states of col- lective motion can be used to register a Boolean variable. Given any 3SAT Boolean formula the interactions among the clusters can be so tailored that the ground state(s) (possibly degenerate) of the whole system encodes the satisfying truth assignment(s) for it. This relates the 3SAT problem to the dynamics of the properly designed glass system. PACS numbers: 89.80.+h, 71.45.-d Typeset using REVTEX 1 Equipped with powerful algorithms, today’s electronic computers solve many problems amazingly fast. Yet there are problems hard to them in the sense that the best algorithms essentially take exponential running time. Many of the hard problems are NP even NP- complete [1]. A problem is NP provided that it can have its answer checked in polynomial time. If the problem has the further property that all problems in the NP class can be translated into it by algorithms taking polynomial time, it is NP-complete [1]. Recently attempts to search for novel machines with greater computational power are stimulated by the growing evidence that no general efficient polynomial time solution run- ning in classical computers exists for any NP-complete problem. DNA computers have been proposed to solve the directed Hamiltonian path problem [2] and the satisfiability problem [3]. These proposals took the advantage of the enormous parallelism of solution-phase chem- istry. However, the number of molecules required rises expo