A programming methodology for designing block recursive algorithms

[[abstract]]© 2006 Institute of Electrical and Electronics Engineers - In this paper, we use the tensor product notation as the framework of a programming methodology for designing block recursive algorithms. We first express a computational problem in its matrix form. Next, we formulate a matrix equation for the matrix of the computational problem. Then, we try to find a solution of the matrix equation such that the solution is composed of simple matrices. Finally, we recursively factorize the subproblem to obtain a tensor product formula representing an algorithm for the given problem. In this methodology, the operations of a tensor product formula can be mapped to language constructs of high-level programming languages. That is, we can generate computer programs, including programs for parallel computers and distributed-memory multiprocessors, from tensor product formulas. In this paper, we use the parallel prefix problem and the discrete Fourier transform problem as examples to illustrate the methodology and derive various parallel prefix and fast Fourier transform algorithms.[[fileno]]2030220010001[[department]]資訊工程學

Solving the Klein-Gordon equation using Fourier spectral methods: A benchmark test for computer performance

The cubic Klein-Gordon equation is a simple but non-trivial partial differential equation whose numerical solution has the main building blocks required for the solution of many other partial differential equations. In this study, the library 2DECOMP&FFT is used in a Fourier spectral scheme to solve the Klein-Gordon equation and strong scaling of the code is examined on thirteen different machines for a problem size of 512^3. The results are useful in assessing likely performance of other parallel fast Fourier transform based programs for solving partial differential equations. The problem is chosen to be large enough to solve on a workstation, yet also of interest to solve quickly on a supercomputer, in particular for parametric studies. Unlike other high performance computing benchmarks, for this problem size, the time to solution will not be improved by simply building a bigger supercomputer.Comment: 10 page

General-Purpose Parallel Simulator for Quantum Computing

With current technologies, it seems to be very difficult to implement quantum computers with many qubits. It is therefore of importance to simulate quantum algorithms and circuits on the existing computers. However, for a large-size problem, the simulation often requires more computational power than is available from sequential processing. Therefore, the simulation methods using parallel processing are required. We have developed a general-purpose simulator for quantum computing on the parallel computer (Sun, Enterprise4500). It can deal with up-to 30 qubits. We have performed Shor's factorization and Grover's database search by using the simulator, and we analyzed robustness of the corresponding quantum circuits in the presence of decoherence and operational errors. The corresponding results, statistics and analyses are presented.Comment: 15 pages, 15 figure

Efficient Spherical Harmonic Transforms aimed at pseudo-spectral numerical simulations

In this paper, we report on very efficient algorithms for the spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Even though the complexity of the algorithms implemented in SHTns are in $O(N^3)$ (where N is the maximum harmonic degree of the transform), they perform much better than any third party implementation, including lower complexity algorithms, even for truncations as high as N=1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software.Comment: 8 page

Simulating chemistry efficiently on fault-tolerant quantum computers

Quantum computers can in principle simulate quantum physics exponentially faster than their classical counterparts, but some technical hurdles remain. Here we consider methods to make proposed chemical simulation algorithms computationally fast on fault-tolerant quantum computers in the circuit model. Fault tolerance constrains the choice of available gates, so that arbitrary gates required for a simulation algorithm must be constructed from sequences of fundamental operations. We examine techniques for constructing arbitrary gates which perform substantially faster than circuits based on the conventional Solovay-Kitaev algorithm [C.M. Dawson and M.A. Nielsen, \emph{Quantum Inf. Comput.}, \textbf{6}:81, 2006]. For a given approximation error $\epsilon$, arbitrary single-qubit gates can be produced fault-tolerantly and using a limited set of gates in time which is $O(\log \epsilon)$ or $O(\log \log \epsilon)$; with sufficient parallel preparation of ancillas, constant average depth is possible using a method we call programmable ancilla rotations. Moreover, we construct and analyze efficient implementations of first- and second-quantized simulation algorithms using the fault-tolerant arbitrary gates and other techniques, such as implementing various subroutines in constant time. A specific example we analyze is the ground-state energy calculation for Lithium hydride.Comment: 33 pages, 18 figure