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It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues {λ i } i∈N of a certain operator. It is known that if λ 1 = 1 and λ 2 ∈ (0, 1) then λ n = o((ln n) −2), as n →… (More)

The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error ε. We assume we are given a superposition of function… (More)

Estimating the ground state energy of a multiparticle system with relative error ε using deterministic classical algorithms has cost that grows exponentially with the number of particles. The problem depends on a number of state variables d that is proportional to the number of particles and suffers from the curse of dimensionality. Quantum computers can… (More)

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem S = {S d } in the average case setting to be weakly tractable but not polynomially tractable. As a result of the tensor product structure , the eigenvalues of the covariance operator of the induced measure in the one dimensional problem… (More)

Quantum algorithms for scientific computing require modules implementing fundamental functions, such as the square root, the logarithm, and others. We require algorithms that have a well-controlled numerical error, that are uniformly scalable and reversible (unitary), and that can be implemented efficiently. We present quantum algorithms and circuits for… (More)

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