We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query, complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: P/sub /spl par///sup NP[2k]//spl sube/EQP/sub /spl par///sup NP[k]/; P/sub /spl par///sup NP[2k+1-2]//spl sube/EQP/sup NP[k]/; FP/sub /spl par///sup NP[2k=1-2]//spl sube/FEQP/sup NP[2k]/; FP/sub /spl par///sup NP/spl sube/FEQP(NP[Olog n)]/. For sets A that are many-one complete for PSPACE or EXP we show that Fp/sup A//spl sube/FEQP/sup A[1]/. Sets A that are many-one complete for PP have the property that FP/sub /spl par///sup A//spl sube/FEQP/sup A[1]/. In general we prove that for any set A there is a set X such that FP/sup A//spl sube/FEQP/sup X[1]/, establishing that no set is superterse in the quantum setting.
We combine the classical notions and techniques for bounded query classes with those developed in quantum computing. We give strong evidence that quantum queries to an oracle in the class NP does indeed reduce the query, complexity of decision problems. Under traditional complexity assumptions, we obtain an exponential speedup between the quantum and the classical query complexity of function classes. For decision problems and function classes we obtain the following results: P/sub /spl par///sup NP[2k]//spl sube/EQP/sub /spl par///sup NP[k]/; P/sub /spl par///sup NP[2k+1-2]//spl sube/EQP/sup NP[k]/; FP/sub /spl par///sup NP[2k=1-2]//spl sube/FEQP/sup NP[2k]/; FP/sub /spl par///sup NP/spl sube/FEQP(NP[Olog n)]/. For sets A that are many-one complete for PSPACE or EXP we show that Fp/sup A//spl sube/FEQP/sup A[1]/. Sets A that are many-one complete for PP have the property that FP/sub /spl par///sup A//spl sube/FEQP/sup A[1]/. In general we prove that for any set A there is a set X such that FP/sup A//spl sube/FEQP/sup X[1]/, establishing that no set is superterse in the quantum setting.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.
This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.
Studies the quantum complexity of the static set membership problem: given a subset S (|S|/spl les/n) of a universe of size m(/spl Gt/n), store it as a table, T:(0,1)/sup r//spl rarr/(0,1), of bits so that queries of the form 'is x in S?' can be answered. The goal is to use a small table and yet answer queries using a few bit probes. This problem was considered by H. Buhrman et al. (2000), who showed lower and upper bounds for this problem in the classical deterministic and randomised models. In this paper, we formulate this problem in the "quantum bit-probe model". We assume that access to the table T is provided by means of a black-box (oracle) unitary transform O/sub T/ that takes the basis state (y,b) to the basis state |y,b/spl oplus/T(y)>. The query algorithm is allowed to apply O/sub T/ on any superposition of basis states. We show tradeoff results between the space (defined as 2/sup r/) and the number of probes (oracle calls) in this model. Our results show that the lower bounds shown by Buhrman et al. for the classical model also hold (with minor differences) in the quantum bit-probe model. These bounds almost match the classical upper bounds. Our lower bounds are proved using linear algebraic arguments.
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(logn) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(logn) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(logn) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(logn) NP queries, then BPPNP[o(n)] contains a PNP-complete problem if the algorithm is classical and FBQPNP[o(n)] contains an FPNP-complete problem if the algorithm is quantum.
We consider the problem where /spl pi/ is an unknown permutation on (0, 1,..., 2/sup n/-1), /spl gamma//sub 0//spl isin/(0, 1,..., 2/sup n/-1), and the goal is to determine the minimum r>0 such that /spl pi//sup r/(y/sub 0/)=y/sub 0/. Information about /spl pi/ is available only via queries that yield /spl pi//sup x/(y) from any x/spl isin/(0, 1,..., 2/sup n/-1) and /spl gamma//sub /spl isin//(0, 1,..., 2/sup n/-1) (where m is polynomial in n). The resource under consideration is the number of these queries (hence our model of computation is the decision tree). We show that the number of queries necessary to solve the problem in the classical probabilistic bounded error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.
We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1) show that the workspace of a quantum computer can be limited to at most n+k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model; 2) give an algorithm that on input the truth table of a partial Boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries; 3) use semidefinite programming duality to formulate a dual SDP P/spl circ/(f, t, /spl epsi/) that is feasible if and only if f cannot be evaluated within error /spl epsi/ by a t-step quantum query algorithm. Using this SDP, we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations; 4) give an interpretation of a generalized form of branching in quantum computation.