Title
Church's thesis meets quantum mechanics
Author
Warren D. Smith
Abstract
``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
``intermediate'' Church's thesis (my own invention) instead says
``...with at most polynomial amplification of memory-space
requirements.'' (All the terms in quotes need to be defined.) 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 specification of some set of laws of
physics, and a precise definition 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 {\em false} in
linear quantum mechanics, and indeed even in some models of {\em open}
quantum systems. In the present paper, I show (at least, if one adopts
my assumptions and definitions -- and there are a large number of
them) that the weak Church's thesis is {\em true} for nonrelativistic
quantum mechanics with sufficiently nice interparticle potentials.
``Sufficiently nice'' includes ``Coulombic.'' I.e., {\em 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 specified) then the
simulation of quantum time evolution actually runs in quasipolynomial
time with respect to that input length.
On the other hand, QM is {\em not} algorithmic,
nor even self-consistent, in the presence of point
{\em magnetic} dipoles.
The proof strategy involves
(1) defining what ``simulation'' 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 integrands corresponding
to regularized potentials.
(4) Obtaining effectively computable
error bounds for this approximation.
(5) Finally, the quantum computer is simulated
by a conventional computer.
A different method, based on Rayleigh-Ritz approximate eigenfunctions,
also seems to yield Church's intermediate thesis. (Treated in an
appendix.) Although this method is conceptually simpler, it
apparently does not lead to efficient algorithms for a quantum
computer. But it does yield a proof that the spectral energies of
quantum bound systems form a computable real sequence.
Keywords
Church's thesis,
quantum $N$-body problem,
rigorous physics,
uncertainty principle,
quantum computers,
Feynman path integrals,
symplectic integration,
exponential splitting formulas,
Trotter product formula,
BQP,
Rayleigh Ritz variational method,
computable real numbers,
operator computability theory,
potential singularities,
Coulomb's law,
analyticity.