Quantum Summation with an Application to Integration

We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a p-summability condition and for integration of functions from Lebesgue spaces L_p([0,1]^d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Brassard, Hoyer, Mosca, and Tapp (2000) on computing the mean for bounded sequences and complements results of Novak (2001) on integration of functions from Hoelder classes.Comment: 48 pages, paper submitted to the Journal of Complexit

Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Hoelder or Sobolev spaces. First we discuss optimal deterministic and randomized algorithms. Then we add a new aspect, which has not been covered before on conferences about (quasi-) Monte Carlo methods: quantum computation. We give a short introduction into this setting and present recent results of the authors on optimal quantum algorithms for summation and integration. We discuss comparisons between the three settings. The most interesting case for Monte Carlo and quantum integration is that of moderate smoothness k and large dimension d which, in fact, occurs in a number of important applied problems. In that case the deterministic exponent is negligible, so the n^{-1/2} Monte Carlo and the n^{-1} quantum speedup essentially constitute the entire convergence rate. We observe that -- there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if k/d tends to zero; -- there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical algorithms, if k/d is small.Comment: 13 pages, contribution to the 4th International Conference on Monte Carlo and Quasi-Monte Carlo Methods, Hong Kong 200

Randomized and Quantum Algorithms Yield a Speed-Up for Initial-Value Problems

Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration problem, for which a speed-up is shown by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem.Comment: LaTeX v. 2.09, 13 page

Almost Optimal Solution of Initial-Value Problems by Randomized and Quantum Algorithms

We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.Comment: 16 pages, minor presentation change

Average case quantum lower bounds for computing the boolean mean

We study the average case approximation of the Boolean mean by quantum algorithms. We prove general query lower bounds for classes of probability measures on the set of inputs. We pay special attention to two probabilities, where we show specific query and error lower bounds and the algorithms that achieve them. We also study the worst expected error and the average expected error of quantum algorithms and show the respective query lower bounds. Our results extend the optimality of the algorithm of Brassard et al.Comment: 18 page

Improved Bounds on the Randomized and Quantum Complexity of Initial-Value Problems

We deal with the problem, initiated in [8], of finding randomized and quantum complexity of initial-value problems. We showed in [8] that a speed-up in both settings over the worst-case deterministic complexity is possible. In the present paper we prove, by defining new algorithms, that further improvement in upper bounds on the randomized and quantum complexity can be achieved. In the H\"older class of right-hand side functions with r continuous bounded partial derivatives, with r-th derivative being a H\"older function with exponent \rho, the \epsilon-complexity is shown to be O((1/\epsilon)^{1/(r+\rho+1/3)}) in the randomized setting, and O((1/\epsilon)^{1/(r+\rho+1/2)}) on a quantum computer (up to logarithmic factors). This is an improvement for the general problem over the results from [8]. The gap still remaining between upper and lower bounds on the complexity is further discussed for a special problem. We consider scalar autonomous problems, with the aim of computing the solution at the end point of the interval of integration. For this problem, we fill up the gap by establishing (essentially) matching upper and lower complexity bounds. We show that the complexity in this case is of order (1/\epsilon)^{1/(r+\rho+1/2)} in the randomized setting, and (1/\epsilon)^{1/(r+\rho+1)} in the quantum setting (again up to logarithmic factors).Comment: 17 pages, extended version (new section added), to appear in the Journal of Complexit