Quantum algorithm and circuit design solving the Poisson equation

New Journal of Physics (Impact Factor: 3.56). 01/2013; 15:013021. DOI: 10.1088/1367-2630/15/1/013021
Source: arXiv


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.

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    • "Moreover, since we know the eigenvalues and eigenvectors of L 0 , in certain cases one might be able to simulate its evolution explicitly without relying on an oracle. For instance when L 0 = −∆, a quantum algorithm and circuit implementing the evolution of the discretized Laplacian, is shown in [8]. That paper deals with the solution of the Poisson equation with Dirichlet boundary conditions. "
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