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P/NP, and the quantum field computer. (English) Zbl 0895.68053
Summary: The central problem in computer science is the conjecture that two complexity classes, P (polynomial time) and NP (nondeterministic polynomial time-roughly those decision problems for which a proposed solution can be checked in polynomial time), are distinct in the standard Turing model of computation: P\(\neq \)NP. As a generality, we propose that each physical theory supports computational models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of implementation of the Turing machine. Non-Abelian topological quantum field theories exhibit the mathematical features necessary to support a model capable of solving all \(\#\)P problems, a computationally intractable class, in polynomial time. Specifically, E. Witten [Commun. Math. Phys. 121, No. 3, 351-399 (1989; Zbl 0667.57005)] has identified expectation values in a certain \(SU(2)\)-field theory with values of the Jones polynomial [V. Jones, Bull. Am. Math. Soc., New Ser. 12, 103-111 (1985; Zbl 0564.57006)] that are #P-hard [F. Jaeger, D. Vertigen and D. Welsh, Math. Proc. Camb. Philos. Soc. 108, No. 1, 35-53 (1990; Zbl 0747.57006)]. This suggests that some physical system whose effective Lagrangian contains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving NP or even #P-hard problems in polynomial time. Defining such a system and addressing the accuracy issues inherent in preparation and measurement is a major unsolved problem.

68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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