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

Minimizing Communication for Eigenproblems and the Singular Value Decomposition

Algorithms have two costs: arithmetic and communication. The latter represents the cost of moving data, either between levels of a memory hierarchy, or between processors over a network. Communication often dominates arithmetic and represents a rapidly increasing proportion of the total cost, so we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds were presented on the amount of communication required for essentially all $O(n^3)$-like algorithms for linear algebra, including eigenvalue problems and the SVD. Conventional algorithms, including those currently implemented in (Sca)LAPACK, perform asymptotically more communication than these lower bounds require. In this paper we present parallel and sequential eigenvalue algorithms (for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms that do attain these lower bounds, and analyze their convergence and communication costs.Comment: 43 pages, 11 figure

A Quasi-Random Approach to Matrix Spectral Analysis

Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in $O(\log^2 n)$ parallel time with a total cost of $O(n^{\omega+1})$ Boolean operations. This Boolean complexity matches the best known rigorous $O(\log^2 n)$ parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases $(\mod (2\pi))$ is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in $O(\log^2 n)$ parallel time and $O(n^\omega)$ Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total complexity $n^{\omega+1}$ and not $n^{\omega}$. However, the depth of the implementing circuit is $\log^2(n)$: hence comparable to fastest eigen-decomposition algorithms know

Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues

The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods. We also prove interesting decay properties of the exponential of a generator having a block-triangular, block-Toeplitz structure

The role of topology and mechanics in uniaxially growing cell networks

In biological systems, the growth of cells, tissues, and organs is influenced by mechanical cues. Locally, cell growth leads to a mechanically heterogeneous environment as cells pull and push their neighbors in a cell network. Despite this local heterogeneity, at the tissue level, the cell network is remarkably robust, as it is not easily perturbed by changes in the mechanical environment or the network connectivity. Through a network model, we relate global tissue structure (i.e. the cell network topology) and local growth mechanisms (growth laws) to the overall tissue response. Within this framework, we investigate the two main mechanical growth laws that have been proposed: stress-driven or strain-driven growth. We show that in order to create a robust and stable tissue environment, networks with predominantly series connections are naturally driven by stress-driven growth, whereas networks with predominantly parallel connections are associated with strain-driven growth

On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented

Adaptive System Identification using Markov Chain Monte Carlo

One of the major problems in adaptive filtering is the problem of system identification. It has been studied extensively due to its immense practical importance in a variety of fields. The underlying goal is to identify the impulse response of an unknown system. This is accomplished by placing a known system in parallel and feeding both systems with the same input. Due to initial disparity in their impulse responses, an error is generated between their outputs. This error is set to tune the impulse response of known system in a way that every change in impulse response reduces the magnitude of prospective error. This process is repeated until the error becomes negligible and the responses of both systems match. To specifically minimize the error, numerous adaptive algorithms are available. They are noteworthy either for their low computational complexity or high convergence speed. Recently, a method, known as Markov Chain Monte Carlo (MCMC), has gained much attention due to its remarkably low computational complexity. But despite this colossal advantage, properties of MCMC method have not been investigated for adaptive system identification problem. This article bridges this gap by providing a complete treatment of MCMC method in the aforementioned context