# Quantum algorithm and circuit design solving the Poisson equation

### Full-text

Sabre Kais, Jul 04, 2015 Available from:- [Show abstract] [Hide abstract]

**ABSTRACT:**Quantum computers have the potential of solving certain problems exponentially faster than classical computers. Recently, Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving linear systems of equations: given an $N\times{N}$ matrix $A$ and a vector $\vec b$, find the vector $\vec x$ that satisfies $A\vec x = \vec b$. It has been shown that using the algorithm one could obtain the solution encoded in a quantum state $|x$ using $O(\log{N})$ quantum operations, while classical algorithms require at least O(N) steps. If one is not interested in the solution $\vec{x}$ itself but certain statistical feature of the solution ${x}|M|x$ ($M$ is some quantum mechanical operator), the quantum algorithm will be able to achieve exponential speedup over the best classical algorithm as $N$ grows. Here we report a proof-of-concept experimental demonstration of the quantum algorithm using a 4-qubit nuclear magnetic resonance (NMR) quantum information processor. For all the three sets of experiments with different choices of $\vec b$, we obtain the solutions with over 96% fidelity. This experiment is a first implementation of the algorithm. Because solving linear systems is a common problem in nearly all fields of science and engineering, we will also discuss the implication of our results on the potential of using quantum computers for solving practical linear systems.Physical Review A 02/2013; 89(2). DOI:10.1103/PhysRevA.89.022313 · 2.99 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We develop a method for approximate synthesis of single--qubit rotations of the form $e^{-i f(\phi_1,\ldots,\phi_k)X}$ that is based on the Repeat-Until-Success (RUS) framework for quantum circuit synthesis. We demonstrate how smooth computable functions $f$ can be synthesized from two basic primitives. This synthesis approach constitutes a manifestly quantum form of arithmetic that differs greatly from the approaches commonly used in quantum algorithms. The key advantage of our approach is that it requires far fewer qubits than existing approaches: as a case in point, we show that using as few as $3$ ancilla qubits, one can obtain RUS circuits for approximate multiplication and reciprocals. We also analyze the costs of performing multiplication and inversion on a quantum computer using conventional approaches and find that they can require too many qubits to execute on a small quantum computer, unlike our approach. - [Show abstract] [Hide abstract]

**ABSTRACT:**The affine Weyl groups with their corresponding four types of orbit functions are considered. Two independent admissible shifts, which preserve the symmetries of the weight and the dual weight lattices, are classified. Finite subsets of the shifted weight and the shifted dual weight lattices, which serve as a sampling grid and a set of labels of the orbit functions, respectively, are introduced. The complete sets of discretely orthogonal orbit functions over the sampling grids are found and the corresponding discrete Fourier transforms are formulated. The eight standard one-dimensional discrete cosine and sine transforms form special cases of the presented transforms.Journal of Mathematical Physics 11/2014; 55(11). DOI:10.1063/1.4901230 · 1.18 Impact Factor