6,365 research outputs found

    Unbounded-error quantum computation with small space bounds

    Full text link
    We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound s s satisfying s(n)=o(loglogn) s(n)=o(\log \log n) . For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for s(n)=o(logn)s(n)=o(\log n) . We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.Comment: A preliminary version of this paper appeared in the Proceedings of the Fourth International Computer Science Symposium in Russia, pages 356--367, 200

    Unbounded-Error Classical and Quantum Communication Complexity

    Full text link
    Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, \cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum} communication complexity in the {\em one-way communication} model can also be investigated using the arrangement, and showed that it is exactly (without a difference of even one qubit) half of the classical one-way communication complexity. In this paper, we extend the arrangement argument to the {\em two-way} and {\em simultaneous message passing} (SMP) models. As a result, we show similarly tight bounds of the unbounded-error two-way/one-way/SMP quantum/classical communication complexities for {\em any} partial/total Boolean function, implying that all of them are equivalent up to a multiplicative constant of four. Moreover, the arrangement argument is also used to show that the gap between {\em weakly} unbounded-error quantum and classical communication complexities is at most a factor of three.Comment: 11 pages. To appear at Proc. ISAAC 200

    Small Depth Quantum Circuits

    Full text link
    Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.National Security Agency; Advanced Research and Development Agency under Army Research Office (DAAD 19-02-1-0058

    Quantum computation with devices whose contents are never read

    Full text link
    In classical computation, a "write-only memory" (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented with WOM can solve problems that neither a classical computer with WOM nor a quantum computer without WOM can solve, when all other resource bounds are equal. We focus on realtime quantum finite automata, and examine the increase in their power effected by the addition of WOMs with different access modes and capacities. Some problems that are unsolvable by two-way probabilistic Turing machines using sublogarithmic amounts of read/write memory are shown to be solvable by these enhanced automata.Comment: 32 pages, a preliminary version of this work was presented in the 9th International Conference on Unconventional Computation (UC2010

    Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity

    Get PDF
    In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between quantum branching programs and nonuniform quantum Turing machines are presented which allow to transfer lower and upper bound results between the two models. In the second part of the paper, different variants of quantum OBDDs are compared with their deterministic and randomized counterparts. In the third part, quantum branching programs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte

    Exponential Separation of Quantum and Classical Online Space Complexity

    Full text link
    Although quantum algorithms realizing an exponential time speed-up over the best known classical algorithms exist, no quantum algorithm is known performing computation using less space resources than classical algorithms. In this paper, we study, for the first time explicitly, space-bounded quantum algorithms for computational problems where the input is given not as a whole, but bit by bit. We show that there exist such problems that a quantum computer can solve using exponentially less work space than a classical computer. More precisely, we introduce a very natural and simple model of a space-bounded quantum online machine and prove an exponential separation of classical and quantum online space complexity, in the bounded-error setting and for a total language. The language we consider is inspired by a communication problem (the set intersection function) that Buhrman, Cleve and Wigderson used to show an almost quadratic separation of quantum and classical bounded-error communication complexity. We prove that, in the framework of online space complexity, the separation becomes exponential.Comment: 13 pages. v3: minor change

    Unbounded-error One-way Classical and Quantum Communication Complexity

    Full text link
    This paper studies the gap between quantum one-way communication complexity Q(f)Q(f) and its classical counterpart C(f)C(f), under the {\em unbounded-error} setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for {\em any} (total or partial) Boolean function ff, Q(f)=C(f)/2Q(f)=\lceil C(f)/2 \rceil, i.e., the former is always exactly one half as large as the latter. The result has an application to obtaining (again an exact) bound for the existence of (m,n,p)(m,n,p)-QRAC which is the nn-qubit random access coding that can recover any one of mm original bits with success probability p\geq p. We can prove that (m,n,>1/2)(m,n,>1/2)-QRAC exists if and only if m22n1m\leq 2^{2n}-1. Previously, only the construction of QRAC using one qubit, the existence of (O(n),n,>1/2)(O(n),n,>1/2)-RAC, and the non-existence of (22n,n,>1/2)(2^{2n},n,>1/2)-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200
    corecore