Power of Quantum Computation with Few Clean Qubits

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

Space-Efficient Error Reduction for Unitary Quantum Computations

This paper presents a general space-efficient method for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness c and soundness s, either with or without a witness (corresponding to QMA and BQP, respectively). To convert this computation into a new computation with error at most 2^{-p}, the most space-efficient method known requires extra workspace of O(p*log(1/(c-s))) qubits. This space requirement is too large for scenarios like logarithmic-space quantum computations. This paper shows an errorreduction method for unitary quantum computations (i.e., computations without intermediate measurements) that requires extra workspace of just O(log(p/(c-s))) qubits. This in particular gives the first method of strong amplification for logarithmic-space unitary quantum computations with two-sided bounded error. This also leads to a number of consequences in complexity theory, such as the uselessness of quantum witnesses in bounded-error logarithmic-space unitary quantum computations, the PSPACE upper bound for QMA with exponentially-small completeness-soundness gap, and strong amplification for matchgate computations

Completeness Results for Parameterized Space Classes

The parameterized complexity of a problem is considered "settled" once it has been shown to lie in FPT or to be complete for a class in the W-hierarchy or a similar parameterized hierarchy. Several natural parameterized problems have, however, resisted such a classification. At least in some cases, the reason is that upper and lower bounds for their parameterized space complexity have recently been obtained that rule out completeness results for parameterized time classes. In this paper, we make progress in this direction by proving that the associative generability problem and the longest common subsequence problem are complete for parameterized space classes. These classes are defined in terms of different forms of bounded nondeterminism and in terms of simultaneous time--space bounds. As a technical tool we introduce a "union operation" that translates between problems complete for classical complexity classes and for W-classes.Comment: IPEC 201