Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics

We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume $\log g$ is well defined for this extension. It is known in Complex Dynamics that under the above hypothesis, for the given potential $\beta \,\log g$, where $\beta$ is a real constant, there exists a real analytic eigenfunction $\phi_\beta$ defined on $[0,1]$ (with a complex analytic extension) for the Ruelle operator of $\beta \,\log g$. Under some assumptions we show that $\frac{1}{\beta}\, \log \phi_\beta$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when $\log g(x)=-\log f'(x)$. In that case we relate the involution kernel to the so called scaling function.Comment: 6 figure

Fast Universal Quantum Computation with Railroad-switch Local Hamiltonians

We present two universal models of quantum computation with a time-independent, frustration-free Hamiltonian. The first construction uses 3-local (qubit) projectors, and the second one requires only 2-local qubit-qutrit projectors. We build on Feynman's Hamiltonian computer idea and use a railroad-switch type clock register. The resources required to simulate a quantum circuit with L gates in this model are O(L) small-dimensional quantum systems (qubits or qutrits), a time-independent Hamiltonian composed of O(L) local, constant norm, projector terms, the possibility to prepare computational basis product states, a running time O(L log^2 L), and the possibility to measure a few qubits in the computational basis. Our models also give a simplified proof of the universality of 3-local Adiabatic Quantum Computation.Comment: Added references to work by de Falco et al., and realized that Feynman's '85 paper already contained the idea of a switch in i

Universal 2-local Hamiltonian Quantum Computing

We present a Hamiltonian quantum computation scheme universal for quantum computation (BQP). Our Hamiltonian is a sum of a polynomial number (in the number of gates L in the quantum circuit) of time-independent, constant-norm, 2-local qubit-qubit interaction terms. Furthermore, each qubit in the system interacts only with a constant number of other qubits. The computer runs in three steps - starts in a simple initial product-state, evolves it for time of order L^2 (up to logarithmic factors) and wraps up with a two-qubit measurement. Our model differs from the previous universal 2-local Hamiltonian constructions in that it does not use perturbation gadgets, does not need large energy penalties in the Hamiltonian and does not need to run slowly to ensure adiabatic evolution.Comment: recomputed the necessary number of interactions, new geometric layout, added reference

Calculating fault propagation in functional programs

Techn. Report TR-HASLab:01:2013The production of safety critical software is bound to a number of safety and certification standards in which estimating the risk of failure plays a central role. Yet risk estimation seems to live outside most programmers’ core practice, involving simulation techniques and worst case analysis performed a posteriori. In this paper we propose that risk be constructively handled in functional programming by writing programs which choose between expected and faulty be- haviour and by reasoning about them in a linear algebra extension to the standard algebra of programming. In particular, the paper calculates propagation of faults across standard program transformation techniques known as tupling and fusion, enabling the fault of the whole to be expressed in terms of the faults of its parts.Fundação para a Ciência e a Tecnologia (FCT

Optical computing of quantum revivals

Interference is the mechanism through which waves can be structured into the most fascinating patterns. While for sensing, imaging, trapping, or in fundamental investigations, structured waves play nowadays an important role and are becoming subject of many interesting studies. Using a coherent optical field as a probe, we show how to structure light into distributions presenting collapse and revival structures in its wavefront. These distributions are obtained from the Fourier spectrum of an arrangement of aperiodic diffracting structures. Interestingly, the resulting interference may present quasiperiodic structures of diffraction peaks on a number of distance scales, even though the diffracting structure is not periodic. We establish an analogy with revival phenomena in the evolution of quantum mechanical systems and illustrate this computation numerically and experimentally, obtaining excellent agreement with the proposed theory.Comment: 10 pages, 4 figure

High Fidelity Adiabatic Quantum Computation via Dynamical Decoupling

We introduce high-order dynamical decoupling strategies for open system adiabatic quantum computation. Our numerical results demonstrate that a judicious choice of high-order dynamical decoupling method, in conjunction with an encoding which allows computation to proceed alongside decoupling, can dramatically enhance the fidelity of adiabatic quantum computation in spite of decoherence.Comment: 5 pages, 4 figure