New Results on Deterministic Pricing of Financial Derivatives

Monte Carlo simulation is widely used to price complex financial instruments Recent theoretical results and extensive computer testing indicate that deterministic methods may be far superior in speed and confidence. Simulations using the Sobol or Faure points are examples of deterministic methods. For the sake of brevity we refer to a deterministic method using the name of the sequence of points which the method uses (e.g. Sobol method.

Divide and conquer approach to quantum Hamiltonian simulation

We show a divide and conquer approach for simulating quantum mechanical systems on quantum computers. We can obtain fast simulation algorithms using Hamiltonian structure. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under very mild assumptions

A regularized solution of shape from shadows

We present a regularized solution to the shape from shadows problem. In this problem the shadows cast on an unknown surface yield data that can be used for the reconstruction of this surface. In the simulation presented here we assume that the data can now be perturbed by noise. It is shown that the regularized approach produces a solution that can handle noisy information while being very similar to the solution obtained by the approximation theoretic approaches used in earlier work. We provide implementation runs where the performance of the algorithm in recovering unknown surfaces is tested. Furthermore, we study the visual effects of smoothing on the various reconstructions

Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation

We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schroedinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its success probability and the size of the coarse grid.Comment: 4 page

Quantum algorithm and circuit design solving the Poisson equation

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 \varepsilon. We assume we are given a supersposition 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 \varepsilon^{-1}. We present quantum circuit modules together with performance guarantees which can be also used for other problems.Comment: 30 pages, 9 figures. This is the revised version for publication in New Journal of Physic