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.

We study the efficiency of algorithms simulating a system evolving with Hamiltonian H = m j=1 H j . We consider high order splitting methods that play a key role in quantum Hamiltonian simulation. We obtain upper bounds on the number of exponentials required to approximate e −iHt with error ε. Moreover, we derive the order of the splitting method that optimizes the cost of the resulting algorithm. We show significant speedups relative to previously known results.

The Brownian bridge has been suggested as an effective method for reducing the quasi-Monte Carlo error for problems in finance. We give an example of a digital option where the Brownian bridge performs worse than the standard discretization. Hence, the Brownian bridge does not offer a consistent advantage in quasi-Monte Carlo integration. We consider integrals of functions of d variables with Gaussian weights such as the ones encountered in the valuation of financial derivatives and in risk management. Under weak assumptions on the class of functions, we study quasi-Monte Carlo methods that are based on different covariance matrix decompositions. We show that different covariance matrix decompositions lead to the same worst case quasi-Monte Carlo error and are, therefore, equivalent.

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.

In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasi-Monte Carlo method with generalized Faure points is very effective for a variety of high dimensional integrals occuring in mathematical finance. In this paper we report test results of this method on Keister's examples of dimension 9 and 25, and also for examples of dimension 60, 80 and 100.For the 25 dimensional integral we achieved accuracy of 10 −2 with less than 500 points while the two methods tested by Keister used more than 220, 000 points. In all of our tests, for n sample points we obtained an empirical convergence rate proportional to n −1 rather than the n −1/2 of Monte Carlo.