Noise resistance of adiabatic quantum computation using random matrix theory

Besides the traditional circuit-based model of quantum computation, several quantum algorithms based on a continuous-time Hamiltonian evolution have recently been introduced, including for instance continuous-time quantum walk algorithms as well as adiabatic quantum algorithms. Unfortunately, very little is known today on the behavior of these Hamiltonian algorithms in the presence of noise. Here, we perform a fully analytical study of the resistance to noise of these algorithms using perturbation theory combined with a theoretical noise model based on random matrices drawn from the Gaussian Orthogonal Ensemble, whose elements vary in time and form a stationary random process.Comment: 9 pages, 3 figure

Effects of Noise, Correlations and errors in the preparation of initial states in Quantum Simulations

In principle a quantum system could be used to simulate another quantum system. The purpose of such a simulation would be to obtain information about problems which cannot be simulated with a classical computer due to the exponential increase of the Hilbert space with the size of the system and which cannot be measured or controlled in an actual experiment. The system will interact with the surrounding environment, with the other particles in the system and be implemented using imperfect controls making it subject to noise. It has been suggested that noise does not need to be controlled to the same extent as it must be for general quantum computing. However the effects of noise in quantum simulations and how to treat them are not completely understood. In this paper we study an existing quantum algorithm for the one-dimensional Fano-Anderson model to be simulated using a liquid-state NMR device. We calculate the evolution of different initial states in the original model, and then we add interacting spins to simulate a more realistic situation. We find that states which are entangled with their environment, and sometimes correlated but not necessarily entangled have an evolution which is described by maps which are not completely positive. We discuss the conditions for this to occur and also the implications.Comment: Revtex 4-1, 14 pages, 21 figures, version 2 has typos corrected and acknowledgement adde

Discrete single-photon quantum walks with tunable decoherence

Quantum walks have a host of applications, ranging from quantum computing to the simulation of biological systems. We present an intrinsically stable, deterministic implementation of discrete quantum walks with single photons in space. The number of optical elements required scales linearly with the number of steps. We measure walks with up to 6 steps and explore the quantum-to-classical transition by introducing tunable decoherence. Finally, we also investigate the effect of absorbing boundaries and show that decoherence significantly affects the probability of absorption.Comment: Published version, 5 pages, 4 figure

Qudit versions of the qubit "pi-over-eight" gate

When visualised as an operation on the Bloch sphere, the qubit "pi-over-eight" gate corresponds to one-eighth of a complete rotation about the vertical axis. This simple gate often plays an important role in quantum information theory, typically in situations for which Pauli and Clifford gates are insufficient. Most notably, when it supplements the set of Clifford gates then universal quantum computation can be achieved. The "pi-over-eight" gate is the simplest example of an operation from the third level of the Clifford hierarchy (i.e., it maps Pauli operations to Clifford operations under conjugation). Here we derive explicit expressions for all qudit (d-level, where d is prime) versions of this gate and analyze the resulting group structure that is generated by these diagonal gates. This group structure differs depending on whether the dimensionality of the qudit is two, three or greater than three. We then discuss the geometrical relationship of these gates (and associated states) with respect to Clifford gates and stabilizer states. We present evidence that these gates are maximally robust to depolarizing and phase damping noise, in complete analogy with the qubit case. Motivated by this and other similarities we conjecture that these gates could be useful for the task of qudit magic-state distillation and, by extension, fault-tolerant quantum computing. Very recent, independent work by Campbell, Anwar and Browne confirms the correctness of this intuition, and we build upon their work to characterize noise regimes for which noisy implementations of these gates can (or provably cannot) supplement Clifford gates to enable universal quantum computation.Comment: Version 2 changed to reflect improved distillation routines in arXiv:1205.3104v2. Minor typos fixed. 12 Pages,2 Figures,3 Table

Design of a five-axis ultra-precision micro-milling machine—UltraMill. Part 2: Integrated dynamic modelling, design optimisation and analysis

Using computer models to predict the dynamic performance of ultra-precision machine tools can help manufacturers to substantially reduce the lead time and cost of developing new machines. However, the use of electronic drives on such machines is becoming widespread, the machine dynamic performance depending not only on the mechanical structure and components but also on the control system and electronic drives. Bench-top ultra-precision machine tools are highly desirable for the micro-manufacturing of high-accuracy micro-mechanical components. However, the development is still at the nascent stage and hence lacks standardised guidelines. Part 2 of this two-part paper proposes an integrated approach, which permits analysis and optimisation of the entire machine dynamic performance at the early design stage. Based on the proposed approach, the modelling and simulation process of a novel five-axis bench-top ultra-precision micro-milling machine tool—UltraMill—is presented. The modelling and simulation cover the dynamics of the machine structure, the moving components, the control system and the machining process and are used to predict the entire machine performance of two typical configurations

Controllability and universal three-qubit quantum computation with trapped electron states

We show how to control and perform universal three-qubit quantum computation with trapped electron quantum states. The three qubits are the electron spin, and the first two quantum states of the cyclotron and axial harmonic oscillators. We explicitly show how the universal gates can be performed. As an example of a non-trivial quantum algorithm, we outline the implementation of the Deutsch-Jozsa algorithm in this system.Comment: 4 pages, 1 figure. Typos corrected. The original publication is available at http://www.springerlink.co

Estimating the functional form for the density dependence from life history data

Two contrasting approaches to the analysis of population dynamics are currently popular: demographic approaches where the associations between demographic rates and statistics summarizing the population dynamics are identified; and time series approaches where the associations between population dynamics, population density, and environmental covariates are investigated. In this paper, we develop an approach to combine these methods and apply it to detailed data from Soay sheep (Ovis aries). We examine how density dependence and climate contribute to fluctuations in population size via age- and sex-specific demographic rates, and how fluctuations in demographic structure influence population dynamics. Density dependence contributes most, followed by climatic variation, age structure fluctuations and interactions between density and climate. We then simplify the density-dependent, stochastic, age-structured demographic model and derive a new phenomenological time series which captures the dynamics better than previously selected functions. The simple method we develop has potential to provide substantial insight into the relative contributions of population and individual-level processes to the dynamics of populations in stochastic environments

Distance-based degrees of polarization for a quantum field

It is well established that unpolarized light is invariant with respect to any SU(2) polarization transformation. This requirement fully characterizes the set of density matrices representing unpolarized states. We introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We use two different candidates of distance, namely the Hilbert-Schmidt and the Bures metric, showing that they induce fundamentally different degrees of polarization. We apply these notions to relevant field states and we demonstrate that they avoid some of the problems arising with the classical definition.Comment: 8 pages, 1 eps figur

Hitting Time of Quantum Walks with Perturbation

The hitting time is the required minimum time for a Markov chain-based walk (classical or quantum) to reach a target state in the state space. We investigate the effect of the perturbation on the hitting time of a quantum walk. We obtain an upper bound for the perturbed quantum walk hitting time by applying Szegedy's work and the perturbation bounds with Weyl's perturbation theorem on classical matrix. Based on the definition of quantum hitting time given in MNRS algorithm, we further compute the delayed perturbed hitting time (DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the upper bound for DPQHT is actually greater than the difference between the square root of the upper bound for a perturbed random walk and the square root of the lower bound for a random walk.Comment: 9 page