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13
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 ..."
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Cited by 17 (6 self)
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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].
THE POWER OF STANDARD INFORMATION FOR MULTIVARIATE APPROXIMATION IN THE RANDOMIZED SETTING
, 2006
"... We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted L2norm. We consider functions with an arbitrarily large number of variables, d, and we focus on the randomized setting with algorit ..."
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Cited by 9 (3 self)
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We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted L2norm. We consider functions with an arbitrarily large number of variables, d, and we focus on the randomized setting with algorithms using standard information consisting of function values at randomly chosen points. We prove that standard information in the randomized setting is as powerful as linear information in the worst case setting. Linear information means that algorithms may use arbitrary continuous linear functionals, and by the power of information we mean the speed of convergence of the nth minimal errors, i.e., of the minimal errors among all algorithms using n function evaluations. Previously, it was only known that standard information in the randomized setting is no more powerful than the linear information in the worst case setting. We also study (strong) tractability of multivariate approximation in the
Strang splitting for the time dependent Schrödinger equation on sparse grids
"... Abstract. The timedependent Schrödinger equation is discretized in space by a sparse grid pseudospectral method. The Strang splitting for the resulting evolutionary problem features first or second order convergence in time, depending on the smoothness of the potential and of the initial data. In ..."
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Cited by 9 (1 self)
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Abstract. The timedependent Schrödinger equation is discretized in space by a sparse grid pseudospectral method. The Strang splitting for the resulting evolutionary problem features first or second order convergence in time, depending on the smoothness of the potential and of the initial data. In contrast to the full grid case, where the frequency domain is the working place, the proof of the sufficient conditions for the convergence is done in the space realm.
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 ..."
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Cited by 8 (1 self)
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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.
The SturmLiouville eigenvalue problem and NPcomplete problems in the quantum setting with queries, in preparation
"... We show how a number of NPcomplete as well as NPhard problems can be reduced to the SturmLiouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of th ..."
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Cited by 4 (1 self)
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We show how a number of NPcomplete as well as NPhard problems can be reduced to the SturmLiouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of the SturmLiouville operator. We show that the number of power queries as well the number of qubits needed to solve the problems studied in this paper is a low degree polynomial. The implementation of power queries by a polynomial number of elementary quantum gates is an open issue. If this problem is solved positively for the power queries used for the SturmLiouville eigenvalue problem then a quantum computer would be a very powerful computation device allowing us to solve NPcomplete problems in polynomial time. 1
On the power of quantum algorithms for vector valued mean computation, Monte Carlo
 Methods and Applications 10 (2004
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Quantum algorithms and complexity for certain continuous and related discrete problems
, 2005
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Numerical Analysis on a Quantum Computer
"... We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bound ..."
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Cited by 1 (0 self)
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We give a short introduction to quantum computing and its relation to numerical analysis. We survey recent research on quantum algorithms and quantum complexity theory for two basic numerical problems – high dimensional integration and approximation. Having matching upper and lower complexity bounds for the quantum setting, we are in a position to compare them with those for the classical deterministic and randomized setting, previously obtained in informationbased complexity theory. This enables us to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.
unknown title
, 2006
"... The SturmLiouville eigenvalue problem and NPcomplete problems in the quantum setting with queries ..."
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The SturmLiouville eigenvalue problem and NPcomplete problems in the quantum setting with queries