Results 1  10
of
40
Quantum approximation I. Embeddings of finite dimensional Lp spaces
 J. COMPLEXITY
, 2003
"... We study approximation of embeddings between finite dimensional Lp spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
We study approximation of embeddings between finite dimensional Lp spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or randomized approximation, while there are other regions where the best possible rates coincide for all three settings. These results serve as a crucial building block for analyzing approximation in function spaces in a subsequent paper [11].
Path integration on a quantum computer
 QUANTUM INFORMATION PROCESSING
, 2002
"... We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j −k with k> 1. For the Wiener measure occurring in many applications we have k = 2. We want to com ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j −k with k> 1. For the Wiener measure occurring in many applications we have k = 2. We want to compute an εapproximation to path integrals whose integrands are at least Lipschitz. We prove: • Path integration on a quantum computer is tractable. • Path integration on a quantum computer can be solved roughly ε −1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. • The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.22ε −1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved.
Tractability of approximation for weighted Korobov spaces on classical and quantum computers
 Found. of Comput. Math
"... We study the approximation problem (or problem of optimal recovery in the L2norm) for weighted Korobov spaces with smoothness parameter α. The weights γj of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The nonnegative smoothness parameter α me ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
We study the approximation problem (or problem of optimal recovery in the L2norm) for weighted Korobov spaces with smoothness parameter α. The weights γj of the Korobov spaces moderate the behavior of periodic functions with respect to successive variables. The nonnegative smoothness parameter α measures the decay of Fourier coefficients. For α = 0, the Korobov space is the L2 space, whereas for positive α, the Korobov space is a space of periodic functions with some smoothness and the approximation problem corresponds to a compact operator. The periodic functions are defined on [0,1] d and our main interest is when the dimension d varies and may be large. We consider algorithms using two different classes of information. The first class Λ all consists of arbitrary linear functionals. The second class Λ std consists of only function values and this class is more realistic in practical computations. We want to know when the approximation problem is tractable. Tractability means that there exists an algorithm whose error is at most ε and whose information cost is bounded by a polynomial in the dimension d and in ε −1. Strong tractability means that
Fast Quantum Algorithms for Handling Probabilistic, Interval, and Fuzzy Uncertainty
, 2003
"... We show how quantum computing can speed up computations related to processing probabilistic, interval, and fuzzy uncertainty. ..."
Abstract

Cited by 12 (9 self)
 Add to MetaCart
We show how quantum computing can speed up computations related to processing probabilistic, interval, and fuzzy uncertainty.
Classical and quantum complexity of the sturm–liouville eigenvalue problem
 Quantum Information Processing
"... We study the approximation of the smallest eigenvalue of a SturmLiouville problem in the classical and quantum settings. We consider a univariate SturmLiouville eigenvalue problem with a nonnegative function q from the class C 2 ([0, 1]) and study the minimal number n(ε) of function evaluations or ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We study the approximation of the smallest eigenvalue of a SturmLiouville problem in the classical and quantum settings. We consider a univariate SturmLiouville eigenvalue problem with a nonnegative function q from the class C 2 ([0, 1]) and study the minimal number n(ε) of function evaluations or queries that are necessary to compute an εapproximation of the smallest eigenvalue. We prove that n(ε) = Θ(ε−1/2) in the (deterministic) worst case setting, and n(ε) = Θ(ε−2/5) in the randomized setting. The quantum setting offers a polynomial speedup with bit queries and an exponential speedup with power queries. Bit queries are similar to the oracle calls used in Grover’s algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix exp ( 1 2iM), where M is an n × n matrix obtained from the standard discretization of the SturmLiouville differential operator.
Quantum Approximation II. Sobolev Embeddings
, 2003
"... A basic problem of approximation theory, the approximation of functions from the Sobolev space W r p ([0, 1]d) in the norm of Lq([0, 1] d), is considered from the point of view of quantum computation. We determine the quantum query complexity of this problem (up to logarithmic factors). It turns out ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
A basic problem of approximation theory, the approximation of functions from the Sobolev space W r p ([0, 1]d) in the norm of Lq([0, 1] d), is considered from the point of view of quantum computation. We determine the quantum query complexity of this problem (up to logarithmic factors). It turns out that in certain regions of the domain of parameters p, q, r, d quantum computation can reach a speedup of roughly squaring the rate of convergence of classical deterministic or randomized approximation methods. There are other regions were the best possible rates coincide for all three settings.
From Monte Carlo to Quantum Computation
 PROCEEDINGS OF THE 3RD IMACS SEMINAR ON MONTE CARLO METHODS MCM2001, SALZBURG, SPECIAL ISSUE OF MATHEMATICS AND COMPUTERS IN SIMULATION
, 2003
"... Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this ty ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
Quantum computing was so far mainly concerned with discrete problems. Recently, E. Novak and the author studied quantum algorithms for high dimensional integration and dealt with the question, which advantages quantum computing can bring over classical deterministic or randomized methods for this type of problem. In this
Fast Quantum Algorithms for Handling Probabilistic and Interval Uncertainty
, 2003
"... this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) "hypothetic" physical devices can compute NPhard problems faster than in exponential time ..."
Abstract

Cited by 7 (7 self)
 Add to MetaCart
this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) "hypothetic" physical devices can compute NPhard problems faster than in exponential time
Sharp Error Bounds on Quantum Boolean Summation in Various Settings
, 2003
"... We study the quantum summation (QS) algorithm of Brassard, Høyer, Mosca and Tapp, see [1], that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worstprobabilistic setting, and present new error bounds in the averageproba ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
We study the quantum summation (QS) algorithm of Brassard, Høyer, Mosca and Tapp, see [1], that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worstprobabilistic setting, and present new error bounds in the averageprobabilistic setting. In particular, in the worstprobabilistic setting, we prove that the error of the QS algorithm using M −1 quantum queries is 3 4πM −1 with probability 8 π2, which improves the error bound πM −1 +π 2 M −2 of [1]. We also present error bounds with probabilities p ∈ ( 1 8 π 2], and show that they are sharp for large M and NM −1.
2006): The quantum query complexity of elliptic PDE
"... The complexity of the following numerical problem is studied in the quantum model of computation: Consider a general elliptic partial differential equation of order 2m in a smooth, bounded domain Q ⊂ R d with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solutio ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The complexity of the following numerical problem is studied in the quantum model of computation: Consider a general elliptic partial differential equation of order 2m in a smooth, bounded domain Q ⊂ R d with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold M ⊆ Q of dimension 0 ≤ d1 ≤ d. With the right hand side belonging to C r (Q), and the error being measured in the L∞(M) norm, we prove that the nth minimal quantum error is (up to logarithmic factors) of order n −min((r+2m)/d1, r/d+1). For comparison, in the classical deterministic setting the nth minimal error is known to be of order n −r/d, for all d1, while in the classical randomized setting it is (up to logarithmic factors) n −min((r+2m)/d1, r/d+1/2). 1