  This paper proves that the computational power of quantum interactive proof
systems, with a double-exponentially small gap in acceptance probability
between the completeness and soundness cases, is precisely characterized by
EXP, the class of problems solvable in exponential time by deterministic Turing
machines. This fact, and our proof of it, has implications concerning quantum
and classical interactive proof systems in the setting of unbounded error that
include the following:
  * Quantum interactive proof systems are strictly more powerful than their
classical counterparts in the unbounded-error setting unless PSPACE=EXP, as
even unbounded error classical interactive proof systems can be simulated in
PSPACE.
  * The recent proof of Jain, Ji, Upadhyay, and Watrous (STOC 2010)
establishing QIP=PSPACE relies heavily on the fact that the quantum interactive
proof systems defining the class QIP have bounded error. Our result implies
that some nontrivial assumption on the error bounds for quantum interactive
proofs is unavoidable to establish this result (unless PSPACE=EXP).
  * To prove our result, we give a quantum interactive proof system for EXP
with perfect completeness and soundness error 1-2^{-2^poly}, for which the
soundness error bound is provably tight. This establishes another respect in
which quantum and classical interactive proof systems differ, because such a
bound cannot hold for any classical interactive proof system: distinct
acceptance probabilities for classical interactive proof systems must be
separated by a gap that is at least (single-)exponentially small.
  We also study the computational power of a few other related unbounded-error
complexity classes.
