Quantum Circuit Design for Solving Linear Systems of Equations

Recently, it is shown that quantum computers can be used for obtaining certain information about the solution of a linear system Ax=b exponentially faster than what is possible with classical computation. Here we first review some key aspects of the algorithm from the standpoint of finding its efficient quantum circuit implementation using only elementary quantum operations, which is important for determining the potential usefulness of the algorithm in practical settings. Then we present a small-scale quantum circuit that solves a 2x2 linear system. The quantum circuit uses only 4 qubits, implying a tempting possibility for experimental realization. Furthermore, the circuit is numerically simulated and its performance under different circuit parameter settings is demonstrated.Comment: 7 pages, 3 figures. The errors are corrected. For the general case, discussions are added to account for recent results. The 4x4 example is replaced by a 2x2 one due to recent experimental efforts. The 2x2 example was devised at the time of writing v1 but not included in v1 for brevit

Fourier analysis of the SOR iteration

The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor omega. It is shown that for the standard model problem of Poisson's equation on a rectangle, the optimal omega and corresponding convergence rate can be rigorously obtained by Fourier analysis. The trick is to tilt the space-time grid so that the SOR stencil becomes symmetrical. The tilted grid also gives insight into the relation between convergence rates of several variants

TR-2012007: Solving Linear Systems of Equations with Randomization, Augmentation and Aggregation II

Seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization, augmentation, and aggregation to reduce our task to computations with well conditioned matrices of full rank. Our algorithms avoid pivoting and orthogonalization, preserve matrix structure and sparseness, and in the case of an ill conditioned input perform only a small part of the computations with high accuracy. We extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities. Our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned. Our study can be of independent technical interest: we extend the known results on conditioning of random matrices to randomized preconditioning, estimate the condition numbers of randomly augmented matrices, and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations

Limited memory preconditioners for nonsymmetric systems

This paper presents a class of limited memory preconditioners (LMPs) for solving linear systems of equations with multiple nonsymmetric matrices and multiple right-hand sides. These preconditioners based on limited memory quasi-Newton formulas require a small number k of linearly independent vectors. They may be used to improve an existing first-level preconditioner and are especially worth considering when the solution of a sequence of linear systems with slowly varying left-hand sides is addressed