Results 1  10
of
20
A functional quantum programming language
 In: Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
, 2005
"... This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are inte ..."
Abstract

Cited by 47 (12 self)
 Add to MetaCart
This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms
Quantum summation with an application to integration
, 2001
"... 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 psummability ( condition and for d integration of functions from Lebesgue spaces Lp [0, 1] ) and analyze their convergence rates. We ..."
Abstract

Cited by 39 (11 self)
 Add to MetaCart
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 psummability ( condition and for d integration of functions from Lebesgue spaces Lp [0, 1] ) 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, Høyer, Mosca, and Tapp (2000) on computing the mean for bounded sequences and complements results of Novak (2001) on integration of functions from Hölder classes.
Addition on a quantum computer
"... A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits. This approach also allows the addition of a classical number to a quantum supe ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits. This approach also allows the addition of a classical number to a quantum superposition without encoding the classical number in the quantum register. This method also allows for massive
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].
Quantum integration in Sobolev classes
, 2001
"... We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W r p ([0, 1]d) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optim ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
We study high dimensional integration in the quantum model of computation. We develop quantum algorithms for integration of functions from Sobolev classes W r 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 Novak on integration of functions from Hölder classes.
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 11 (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
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
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
QML: Quantum data and control
, 2005
"... We introduce the language QML, a functional language for quantum computations on finite types. QML introduces quantum data and control structures, and integrates reversible and irreversible quantum computation. QML is based on strict linear logic, hence weakenings, which may lead to decoherence, hav ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We introduce the language QML, a functional language for quantum computations on finite types. QML introduces quantum data and control structures, and integrates reversible and irreversible quantum computation. QML is based on strict linear logic, hence weakenings, which may lead to decoherence, have to be explicit. We present an operational semantics of QML programs using quantum circuits, and a denotational semantics using superoperators.
Convexity and the separability problem of quantum mechanical density matrices, Linear Algebra Appl
, 2002
"... A finite dimensional quantum mechanical system is modeled by a density ρ, a trace one, positive semidefinite matrix on a suitable tensor product space H [N]. For the system to demonstrate experimentally certain nonclassical behavior, ρ cannot be in S, a closed convex set of densities whose extreme ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
A finite dimensional quantum mechanical system is modeled by a density ρ, a trace one, positive semidefinite matrix on a suitable tensor product space H [N]. For the system to demonstrate experimentally certain nonclassical behavior, ρ cannot be in S, a closed convex set of densities whose extreme points have a specificed tensor product form. Two mathematical problems in the quantum computing literature arise from this context: (1) the determination whether a given ρ is in S and (2) a measure of the “entanglement ” of such a ρ in terms of its distance from S. In this paper we describe these two problems in detail for a linear algebra audience, discuss some recent results from the quantum computing literature, and prove some new results.We emphasize the roles of densities ρ as both operators on the Hilbert space H [N] and also as points in a real Hilbert space M. We are able to compute the nearest separable densities τ0 to ρ0 in particular classes of inseparable densities and we use the Euclidean distance between the two in M to quantify the entanglement of ρ0. We also show the role of τ0 in the construction of separating hyperplanes, socalled entanglement witnesses in the quantum computing literature.