Sharp Error Bounds on Quantum Boolean Summation in Various Settings

We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using $M - 1$ queries is $3\pi /(4M)$ with probability $8/\pi^2$, which improves the error bound $\pi M^{-1} + \pi^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/\pi^2]$ and show they are sharp for large $M$ and $NM^{-1}$. In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. This bound is optimal, as recently shown in [10]. For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to M^{-1}^.Comment: 32 pages, 2 figure

Probabilistic setting of information-based complexity

We study the probabilistic (E, b)-complexity for linear problems equipped with Gaussian measures. The probabilistic (E, S)-complexity, comp@“˜(e, 6), is understood as the minimal cost required to compute approximations with error at most e on a set of measure at least 1 - 6. We find estimates of comp@(e, 6) in terms of eigenvalues of the correlation operator of the Gaussian measure over elements which we want to approximate. In particular, we study the approximation and integration problems. The approximation problem is studied for functions of d variables which are continuous after r times differentiation with respect to each variable. For the Wiener measure placed on rth derivatives, the probabilistic comp@(e, S) is estimated by ,((~/,)“(r+~)(ln(~/~))(d-“˜“r+““r‘+~‘), where a = 1 for the lower bound and a = 0.5 for the upper bound. The integration problem is studied for the same class of functions with d = 1. In this case, compPmb(e,6 ) = @((m/E)“@““˜)

Average Case Complexity of Multivariate Integration

We study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical Wiener sheet measure. To derive the average case complexity one needs to obtain optimal sample points. We prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years. This relation enables us to show that optimal sample points can be derived from Hammersley points. Extending the result of Roth and using the recent result of Wasilkowski, we conclude that the average case complexity is θ(ε-1(lnε-1)(d-1)/2

Tractability of multivariate analytic problems

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions

Generalized Tractability for Multivariate Problems: Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complexity, 23, 262-295 (2007). We study linear tensor product problems for which we can compute linear in- formation which is given by arbitrary continuous linear functionals. We want to approximate an operator Sd given as the d-fold tensor product of a compact linear operator S1 for d = 1,2,..., with ∥S1∥ = 1 and S1 has at least two positive singular values. Let n(ε,Sd) be the minimal number of information evaluations needed to approximate Sd to within ε ∈ [0, 1]. We study generalized tractability by verifying when n(ε, Sd) can be bounded by a multiple of a power of T (ε−1, d) for all (ε−1, d) ∈ Ω ⊆ [1, ∞) × N. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T (ε−1, d) whose multiple bounds n(ε, Sd). We also study weak tractability, i.e., when limε−1+d→∞,(ε−1,d)∈Ω ln n(ε,Sd)/(ε−1 +d) = 0. In our previous paper, we studied generalized tractability for proper subsets Ω of [1, ∞) × N, whereas in this paper we take the unrestricted domain Ωuno = [1, ∞) × N. We consider the three cases for which we have only finitely many positive singular values of S1, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S1 decay slightly faster that logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S1 and mostly limiting properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T (x, y) = x1+ln y . We also study tractability functions T of product form, T(x,y) = f1(x)f2(x). Assume that ai = lim infx→∞(ln ln fi(x))/(ln ln x) is finite for i = 1, 2. Then generalized tractability takes place iff ai >1 and (a1 −1)(a2 −1)≥1, and if (a1−1)(a2−1) = 1 then we need to assume one more condition given in the paper. If (a1 −1)(a2 −1) > 1 then the exponent of tractability is zero, and if (a1 − 1)(a2 − 1) = 1 then the exponent of tractability is finite. It is interesting to add that for T being of the product form, the tractability conditions as well as the exponent of tractability depend only on the second singular eigenvalue of S1 and they do not depend on the rate of their decay. Finally, we compare the results obtained in this paper for the unrestricted domain Ω uno with the results from our previous paper obtained for the restricted domain Ω res =[1,∞)×{1,2,...,d∗}∪ [1, ε−1) × N with d∗ ≥ 1 and ε ∈ (0, 1). In general, the tractability results 00 are quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, for instance, for polynomial tractability T(x,y) = xy. We may also have generalized tractability for both domains with different or with the same exponents of tractability

Generalized Tractability for Multivariate Problems: Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complexity, 23, 262-295 (2007). We study linear tensor product problems for which we can compute linear in- formation which is given by arbitrary continuous linear functionals. We want to approximate an operator Sd given as the d-fold tensor product of a compact linear operator S1 for d = 1,2,..., with ∥S1∥ = 1 and S1 has at least two positive singular values. Let n(ε,Sd) be the minimal number of information evaluations needed to approximate Sd to within ε ∈ [0, 1]. We study generalized tractability by verifying when n(ε, Sd) can be bounded by a multiple of a power of T (ε−1, d) for all (ε−1, d) ∈ Ω ⊆ [1, ∞) × N. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T (ε−1, d) whose multiple bounds n(ε, Sd). We also study weak tractability, i.e., when limε−1+d→∞,(ε−1,d)∈Ω ln n(ε,Sd)/(ε−1 +d) = 0. In our previous paper, we studied generalized tractability for proper subsets Ω of [1, ∞) × N, whereas in this paper we take the unrestricted domain Ωuno = [1, ∞) × N. We consider the three cases for which we have only finitely many positive singular values of S1, or they decay exponentially or polynomially fast. Weak tractability holds for these three cases, and for all linear tensor product problems for which the singular values of S1 decay slightly faster that logarithmically. We provide necessary and sufficient conditions on the function T such that generalized tractability holds. These conditions are obtained in terms of the singular values of S1 and mostly limiting properties of T. The tractability conditions tell us how fast T must go to infinity. It is known that T must go to infinity faster than polynomially. We show that generalized tractability is obtained for T (x, y) = x1+ln y . We also study tractability functions T of product form, T(x,y) = f1(x)f2(x). Assume that ai = lim infx→∞(ln ln fi(x))/(ln ln x) is finite for i = 1, 2. Then generalized tractability takes place iff ai >1 and (a1 −1)(a2 −1)≥1, and if (a1−1)(a2−1) = 1 then we need to assume one more condition given in the paper. If (a1 −1)(a2 −1) > 1 then the exponent of tractability is zero, and if (a1 − 1)(a2 − 1) = 1 then the exponent of tractability is finite. It is interesting to add that for T being of the product form, the tractability conditions as well as the exponent of tractability depend only on the second singular eigenvalue of S1 and they do not depend on the rate of their decay. Finally, we compare the results obtained in this paper for the unrestricted domain Ω uno with the results from our previous paper obtained for the restricted domain Ω res =[1,∞)×{1,2,...,d∗}∪ [1, ε−1) × N with d∗ ≥ 1 and ε ∈ (0, 1). In general, the tractability results 00 are quite different. We may have generalized tractability for the restricted domain and no generalized tractability for the unrestricted domain which is the case, for instance, for polynomial tractability T(x,y) = xy. We may also have generalized tractability for both domains with different or with the same exponents of tractability

Estimating a largest eigenvector by polynomial algorithms with a random start

In 7 and 8 the power and Lanczos algorithms with random start for estimating the largest eigenvalue of an n x n large symmetric positive definite matrix were analyzed. In this paper we continue this study by estimating an eigenvector corresponding to the largest eigenvalue. We analyze polynomial algorithms using Krylov information for two error criteria the randomized error and the randomized residual error