Church's Thesis Meets Quantum Mechanics

| \Church's thesis" is the notion that any \reasonable physical system" may be \simulated" by a Turing machine. The \strong" Church's thesis adds \...with at most polynomial slowdown." The \in-termediate" Church's thesis (my own invention) instead says \...with at most polynomial ampliication of memory-space requirements." (All the terms in quotes need to be deened.) In your favorite set of physical laws: is Church's thesis true? Church's theses are central issues for both physics and computer science. With the precise speciication of some set of laws of physics, and a precise dee-nition of \simulation" and \polynomial," Church's theses become susceptible to mathematical proof or disproof. In a previous report, I essentially settled the question for classical mechanics. Recent theoretical investigations of \quantum computers" make it look likely that the strong Church thesis is false in linear quantum mechanics, and indeed even in some models of open quantum systems. In the present paper, I show (at least, if one adopts my assumptions and deeni-tions { and there are a large number of them) that the weak Church's thesis is true for nonrelativistic quantum mechanics with suuciently nice interparti-cle potentials. \Suuciently nice" includes \Coulom-bic." I.e., quantum mechanics is simulable. We give a simulation algorithm. If the simulation is performed by a quantum computer rather than a conventional one, then the slowdown is only polynomial. In other words, even Church's strong thesis becomes true if \Turing machine" is replaced by \quantum Turing machine." With a conventional Turing machine, we then automatically get the intermediate thesis; and if the initial quantum state is represented non-sparsely, (i.e. in a format in which exponentially many complex amplitudes are speciied) then the simulation of quantum time evolution actually runs in quasipoly-nomial time with respect to that input length. The proof strategy involves (1) deening what \sim-ulation" and \reasonable physical system" should be. (2) showing that \regularizing" the potential introduces acceptably small error. (For Coulomb potentials , the most natural regularization procedure is to replace \point" charges by uniform distributions of charge within small balls centered at the point.) (3) Showing how a quantum computer can approximately evaluate Feynman path integrals with phase factor in-tegrands corresponding to regularized potentials. (4) Obtaining eeectively computable error bounds for this approximation. (5) Finally, the quantum computer is simulated by a conventional computer. A diierent method, based on Rayleigh-Ritz approximate eigenfunctions, also seems to yield Church's intermediate thesis. (Treated in an appendix .) …