Efficient Quantum Algorithms for State Measurement and Linear Algebra Applications

We present an algorithm for measurement of $k$-local operators in a quantum state, which scales logarithmically both in the system size and the output accuracy. The key ingredients of the algorithm are a digital representation of the quantum state, and a decomposition of the measurement operator in a basis of operators with known discrete spectra. We then show how this algorithm can be combined with (a) Hamiltonian evolution to make quantum simulations efficient, (b) the Newton-Raphson method based solution of matrix inverse to efficiently solve linear simultaneous equations, and (c) Chebyshev expansion of matrix exponentials to efficiently evaluate thermal expectation values. The general strategy may be useful in solving many other linear algebra problems efficiently.Comment: 17 pages, 3 figures (v2) Sections reorganised, several clarifications added, results unchange

Euclidean asymptotic expansions of Green functions of quantum fields (II) Combinatorics of the asymptotic operation

The results of part I (hep-ph/9612284) are used to obtain full asymptotic expansions of Feynman diagrams renormalized within the MS-scheme in the regimes when some of the masses and external momenta are large with respect to the others. The large momenta are Euclidean, and the expanded diagrams are regarded as distributions with respect to them. The small masses may be equal to zero. The asymptotic operation for integrals is defined and a simple combinatorial techniques is developed to study its exponentiation. The asymptotic operation is used to obtain the corresponding expansions of arbitrary Green functions. Such expansions generalize and improve upon the well-known short-distance operator-product expansions, the decoupling theorem etc.; e.g. the low-energy effective Lagrangians are obtained to all orders of the inverse heavy mass. The obtained expansions possess the property of perfect factorization of large and small parameters, which is essential for meaningful applications to phenomenology. As an auxiliary tool, the inversion of the R-operation is constructed. The results are valid for arbitrary QFT models.Comment: one .sty + one .tex (LaTeX 2.09) + one .ps (GSview) = 46 pp. Many fewer misprints than the journal versio

The Kentucky Noisy Monte Carlo Algorithm for Wilson Dynamical Fermions

We develop an implementation for a recently proposed Noisy Monte Carlo approach to the simulation of lattice QCD with dynamical fermions by incorporating the full fermion determinant directly. Our algorithm uses a quenched gauge field update with a shifted gauge coupling to minimize fluctuations in the trace log of the Wilson Dirac matrix. The details of tuning the gauge coupling shift as well as results for the distribution of noisy estimators in our implementation are given. We present data for some basic observables from the noisy method, as well as acceptance rate information and discuss potential autocorrelation and sign violation effects. Both the results and the efficiency of the algorithm are compared against those of Hybrid Monte Carlo. PACS Numbers: 12.38.Gc, 11.15.Ha, 02.70.Uu Keywords: Noisy Monte Carlo, Lattice QCD, Determinant, Finite Density, QCDSPComment: 30 pages, 6 figure

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

Basis functions for solution of non-homogeneous wave equation

In this note we extend the Differential Transfer Matrix Method (DTMM) for a second-order linear ordinary differential equation to the complex plane. This is achieved by separation of real and imaginary parts, and then forming a system of equations having a rank twice the size of the real-valued problem. The method discussed in this paper also successfully removes the problem of dealing with essential singularities, which was present in the earlier formulations. Then we simplify the result for real-valued problems and obtain a new set of basis functions, which may be used instead of the WKB solutions. These basis functions not only satisfy the initial conditions perfectly, but also, may approach the turning points without the divergent behavior, which is observed in WKB solutions. Finally, an analytical transformation in the form of a matrix exponential is presented for improving the accuracy of solutions.Comment: to appear in Proc. SPIE, vol. 861

Sudakov logs and power corrections for selected event shapes

I summarize the results of recent studies analyzing perturbative and nonperturbative effects of soft gluon radiation on the distributions of the C-parameter and of the class of angularities, by means of dressed gluon exponentiation.Comment: Talk given at DIS 2004, Strbske Pleso, Slovakia, 14-18 April 200