Probabilistic Computability and Choice

We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a probabilistic Turing machine that receives a random binary sequence as auxiliary input. The machine can take advantage of this random sequence, but it always has to produce a correct result or to stop the computation after finite time if the random advice is not successful. With positive probability the random advice has to be successful. We characterize the class of Las Vegas computable functions in the Weihrauch lattice with the help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among other things we prove an Independent Choice Theorem that implies that Las Vegas computable functions are closed under composition. In a case study we show that Nash equilibria are Las Vegas computable, while zeros of continuous functions with sign changes cannot be computed on Las Vegas machines. However, we show that the latter problem admits randomized algorithms with weaker failure recognition mechanisms. The last mentioned results can be interpreted such that the Intermediate Value Theorem is reducible to the jump of Weak Weak K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These examples also demonstrate that Las Vegas computable functions form a proper superclass of the class of computable functions and a proper subclass of the class of non-deterministically computable functions. We also study the impact of specific lower bounds on the success probabilities, which leads to a strict hierarchy of classes. In particular, the classical technique of probability amplification fails for computations on infinite objects. We also investigate the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication

Phase Structure of the Random-Plaquette Z_2 Gauge Model: Accuracy Threshold for a Toric Quantum Memory

We study the phase structure of the random-plaquette Z_2 lattice gauge model in three dimensions. In this model, the "gauge coupling" for each plaquette is a quenched random variable that takes the value \beta with the probability 1-p and -\beta with the probability p. This model is relevant for the recently proposed quantum memory of toric code. The parameter p is the concentration of the plaquettes with "wrong-sign" couplings -\beta, and interpreted as the error probability per qubit in quantum code. In the gauge system with p=0, i.e., with the uniform gauge couplings \beta, it is known that there exists a second-order phase transition at a certain critical "temperature", T(\equiv \beta^{-1}) = T_c =1.31, which separates an ordered(Higgs) phase at T<T_c and a disordered(confinement) phase at T>T_c. As p increases, the critical temperature T_c(p) decreases. In the p-T plane, the curve T_c(p) intersects with the Nishimori line T_{N}(p) at the certain point (p_c, T_{N}(p_c)). The value p_c is just the accuracy threshold for a fault-tolerant quantum memory and associated quantum computations. By the Monte-Carlo simulations, we calculate the specific heat and the expectation values of the Wilson loop to obtain the phase-transition line T_c(p) numerically. The accuracy threshold is estimated as p_c \simeq 0.033.Comment: 24 pages, 14 figures, some clarification

A highly optimized vectorized code for Monte Carlo simulations of SU(3) lattice gauge theories

New methods are introduced for improving the performance of the vectorized Monte Carlo SU(3) lattice gauge theory algorithm using the CDC CYBER 205. Structure, algorithm and programming considerations are discussed. The performance achieved for a 16(4) lattice on a 2-pipe system may be phrased in terms of the link update time or overall MFLOPS rates. For 32-bit arithmetic, it is 36.3 microsecond/link for 8 hits per iteration (40.9 microsecond for 10 hits) or 101.5 MFLOPS

Regge gravity on general triangulations

We investigate quantum gravity in four dimensions using the Regge approach on triangulations of the four-torus with general, non-regular incidence matrices. We find that the simplicial lattice tends to develop spikes for vertices with low coordination numbers even for vanishing gravitational coupling. Different to the regular, hypercubic lattices almost exclusively used in previous studies, we find now that the observables depend on the measure. Computations with nonvanishing gravitational coupling still reveal the existence of a region with well-defined expectation values. However, the phase structure depends on the triangulation. Even with additional higher- order terms in the action the critical behavior of the system changes with varying (local) coordination numbers.Comment: uuencoded postscript file, 16 page

An Algorithmic Approach to Quantum Field Theory

The lattice formulation provides a way to regularize, define and compute the Path Integral in a Quantum Field Theory. In this paper we review the theoretical foundations and the most basic algorithms required to implement a typical lattice computation, including the Metropolis, the Gibbs sampling, the Minimal Residual, and the Stabilized Biconjugate inverters. The main emphasis is on gauge theories with fermions such as QCD. We also provide examples of typical results from lattice QCD computations for quantities of phenomenological interest.Comment: 44 pages, to be published in IJMP

SU(2) Lattice Gauge Theory Simulations on Fermi GPUs

In this work we explore the performance of CUDA in quenched lattice SU(2) simulations. CUDA, NVIDIA Compute Unified Device Architecture, is a hardware and software architecture developed by NVIDIA for computing on the GPU. We present an analysis and performance comparison between the GPU and CPU in single and double precision. Analyses with multiple GPUs and two different architectures (G200 and Fermi architectures) are also presented. In order to obtain a high performance, the code must be optimized for the GPU architecture, i.e., an implementation that exploits the memory hierarchy of the CUDA programming model. We produce codes for the Monte Carlo generation of SU(2) lattice gauge configurations, for the mean plaquette, for the Polyakov Loop at finite T and for the Wilson loop. We also present results for the potential using many configurations ($50\ 000$) without smearing and almost $2\ 000$ configurations with APE smearing. With two Fermi GPUs we have achieved an excellent performance of $200 \times$ the speed over one CPU, in single precision, around 110 Gflops/s. We also find that, using the Fermi architecture, double precision computations for the static quark-antiquark potential are not much slower (less than $2 \times$ slower) than single precision computations.Comment: 20 pages, 11 figures, 3 tables, accepted in Journal of Computational Physic

Efficient generation and optimization of stochastic template banks by a neighboring cell algorithm

Placing signal templates (grid points) as efficiently as possible to cover a multi-dimensional parameter space is crucial in computing-intensive matched-filtering searches for gravitational waves, but also in similar searches in other fields of astronomy. To generate efficient coverings of arbitrary parameter spaces, stochastic template banks have been advocated, where templates are placed at random while rejecting those too close to others. However, in this simple scheme, for each new random point its distance to every template in the existing bank is computed. This rapidly increasing number of distance computations can render the acceptance of new templates computationally prohibitive, particularly for wide parameter spaces or in large dimensions. This work presents a neighboring cell algorithm that can dramatically improve the efficiency of constructing a stochastic template bank. By dividing the parameter space into sub-volumes (cells), for an arbitrary point an efficient hashing technique is exploited to obtain the index of its enclosing cell along with the parameters of its neighboring templates. Hence only distances to these neighboring templates in the bank are computed, massively lowering the overall computing cost, as demonstrated in simple examples. Furthermore, we propose a novel method based on this technique to increase the fraction of covered parameter space solely by directed template shifts, without adding any templates. As is demonstrated in examples, this method can be highly effective..Comment: PRD accepte