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 evaluations of the right-hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in ε −1 . We present quantum circuit modules together with performance guarantees which can also be used for other problems.

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 vanquish this curse. In particular, we study a ground state eigenvalue problem and exhibit a quantum algorithm that achieves relative error ε using a number of qubits C ′ d log ε −1 with total cost (number of queries plus other quantum operations) Cdε −(3+δ) , where δ > 0 is arbitrarily small and C and C ′ are independent of d and ε.

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 computing the square root, the natural logarithm, and arbitrary fractional powers. We provide performance guarantees in terms of their worst-case accuracy and cost. We further illustrate their performance by providing tests comparing them to the respective floating point implementations found in widely used numerical software.

In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε. The cost of the algorithm is polynomial in d and ε −1 , while the number of qubits is polynomial in d and log ε −1 . In addition, we present an algorithm for preparing a quantum state that overlaps within 1 − δ, δ ∈ (0, 1), with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε. The cost of the algorithm is polynomial in d, ε −1 and δ −1 , while the number of qubits is polynomial in d, log ε −1 and log δ −1 .

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a b s t r a c tIt 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 → ∞, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.

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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 computing the square root, the natural logarithm, and arbitrary fractional powers. We provide performance guarantees in terms of their worst-case accuracy and cost. We further illustrate their performance by providing tests comparing them to the respective floating point implementations found in widely used numerical software.