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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].
Sharp Error Bounds on Quantum Boolean Summation in Various Settings, submitted for publication
"... 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 ..."
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Cited by 6 (3 self)
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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 ..."
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Cited by 5 (2 self)
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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
On the power of quantum algorithms for vector valued mean computation, Monte Carlo
 Methods and Applications 10 (2004
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VORONOI DIAGRAMS FOR QUANTUM STATES AND ITS APPLICATION TO A NUMERICAL ESTIMATION OF A QUANTUM CHANNEL CAPACITY
, 2007
"... This version is slightly chaged from the original thesis. In the original thesis, the title and the abstract are also written in Japanese. They are omitted to comply with arXiv’s In quantum information theory, a geometric approach, known as “quantum information geometry, ” has been considered as a p ..."
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This version is slightly chaged from the original thesis. In the original thesis, the title and the abstract are also written in Japanese. They are omitted to comply with arXiv’s In quantum information theory, a geometric approach, known as “quantum information geometry, ” has been considered as a powerful method. In this thesis, we give a computational geometric interpretation to the geometric structure of a quantum system. Especially we introduce the concept of the Voronoi diagram and the smallest enclosing ball problem to the space of quantum states. With those tools in computational geometry, we analyze the adjacency structure of a point set in the quantum state space. Additionally, as an application, we show an effective method to compute the capacity of a quantum channel. In the first part of this thesis, we show some coincidences of Voronoi diagrams in a quantum state space with respect to some distances. That helps us to reinterpret the structure of the space of quantum pure states as a subspace of the whole space. More properly, we investigate the Voronoi diagrams with respect to the divergence, FubiniStudy distance, Bures distance, geodesic distance and Euclidean distance.