This paper investigates the power of polynomial-time quantum computation in
which only a very limited number of qubits are initially clean in the |0>
state, and all the remaining qubits are initially in the totally mixed state.
No initializations of qubits are allowed during the computation, nor
intermediate measurements. The main results of this paper are unexpectedly
strong error-reducible properties of such quantum computations. It is proved
that any problem solvable by a polynomial-time quantum computation with
one-sided bounded error that uses logarithmically many clean qubits can also be
solvable with exponentially small one-sided error using just two clean qubits,
and with polynomially small one-sided error using just one clean qubit. It is
further proved in the case of two-sided bounded error that any problem solvable
by such a computation with a constant gap between completeness and soundness
using logarithmically many clean qubits can also be solvable with exponentially
small two-sided error using just two clean qubits. If only one clean qubit is
available, the problem is again still solvable with exponentially small error
in one of the completeness and soundness and polynomially small error in the
other. As an immediate consequence of the above result for the two-sided-error
case, it follows that the TRACE ESTIMATION problem defined with fixed constant
threshold parameters is complete for the classes of problems solvable by
polynomial-time quantum computations with completeness 2/3 and soundness 1/3
using logarithmically many clean qubits and just one clean qubit. The
techniques used for proving the error-reduction results may be of independent
interest in themselves, and one of the technical tools can also be used to show
the hardness of weak classical simulations of one-clean-qubit computations
(i.e., DQC1 computations).Comment: 44 pages + cover page; the results in Section 8 are overlapping with
the main results in arXiv:1409.677
