The Edge of Quantum Chaos

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with $[1+(q-1) (t/\tau)^2]^{1/(1-q)}$ (with the nonextensive entropic index $q$ and $\tau$ depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.Comment: 4 pages, 4 figures, revised version in press PR

Efficient Algorithms for Universal Quantum Simulation

A universal quantum simulator would enable efficient simulation of quantum dynamics by implementing quantum-simulation algorithms on a quantum computer. Specifically the quantum simulator would efficiently generate qubit-string states that closely approximate physical states obtained from a broad class of dynamical evolutions. I provide an overview of theoretical research into universal quantum simulators and the strategies for minimizing computational space and time costs. Applications to simulating many-body quantum simulation and solving linear equations are discussed

Near integrable systems

A two-dimensional circular quantum billiard with unusual boundary conditions introduced by Berry and Dennis (\emph{J Phys A} {\bf 41} (2008) 135203) is considered in detail. It is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with Neumann boundary conditions. Deviations from strong localization are also discussed. These results agree well with numerical calculations.Comment: 27 pages, 10 figure

Testing integrability with a single bit of quantum information

We show that deterministic quantum computing with a single bit (DQC1) can determine whether the classical limit of a quantum system is chaotic or integrable using O(N) physical resources, where $N$ is the dimension of the Hilbert space of the system under study. This is a square root improvement over all known classical procedures. Our study relies strictly on the random matrix conjecture. We also present numerical results for the nonlinear kicked top.Comment: Minor changes taking into account Howard Wiseman's comment: quant-ph/0305153. Accepted for publication in Phys. Rev.

Efficient quantum algorithms for simulating sparse Hamiltonians

We present an efficient quantum algorithm for simulating the evolution of a sparse Hamiltonian H for a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and |H| is bounded by a constant, we may select any positive integer $k$ such that the simulation requires O((\log^*n)t^{1+1/2k}) accesses to matrix entries of H. We show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.Comment: 9 pages, 2 figures, substantial revision

Quantum chaos, random matrix theory, and statistical mechanics in two dimensions - a unified approach

We present a theory where the statistical mechanics for dilute ideal gases can be derived from random matrix approach. We show the connection of this approach with Srednicki approach which connects Berry conjecture with statistical mechanics. We further establish a link between Berry conjecture and random matrix theory, thus providing a unified edifice for quantum chaos, random matrix theory, and statistical mechanics. In the course of arguing for these connections, we observe sum rules associated with the outstanding counting problem in the theory of braid groups. We are able to show that the presented approach leads to the second law of thermodynamics.Comment: 23 pages, TeX typ

On the relationship between continuous- and discrete-time quantum walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio

Towards Real-Time Crowd Simulation Under Uncertainty Using an Agent-Based Model and an Unscented Kalman Filter

Agent-based modelling (ABM) is ideally suited to simulating crowds of people as it captures the complex behaviours and interactions between individuals that lead to the emergence of crowding. Currently, it is not possible to use ABM for real-time simulation due to the absence of established mechanisms for dynamically incorporating real-time data. This means that, although models are able to perform useful offline crowd simulations, they are unable to simulate the behaviours of crowds in real time. This paper begins to address this drawback by demonstrating how a data assimilation algorithm, the Unscented Kalman Filter (UKF), can be used to incorporate pseudo-real data into an agent-based model at run time. Experiments are conducted to test how well the algorithm works when a proportion of agents are tracked directly under varying levels of uncertainty. Notably, the experiments show that the behaviour of unobserved agents can be inferred from the behaviours of those that are observed. This has implications for modelling real crowds where full knowledge of all individuals will never be known. In presenting a new approach for creating real-time simulations of crowds, this paper has important implications for the management of various environments in global cities, from single buildings to larger structures such as transportation hubs, sports stadiums, through to entire city regions

Lifetime Risks for Cardiovascular Disease Mortality by Cardiorespiratory Fitness Levels Measured at Ages 45, 55, and 65 Years in Men The Cooper Center Longitudinal Study

ObjectivesThe purpose of this study was to determine the association between fitness and lifetime risk for cardiovascular disease (CVD).BackgroundHigher levels of traditional risk factors are associated with marked differences in lifetime risks for CVD. However, data are sparse regarding the association between fitness and the lifetime risk for CVD.MethodsWe followed up 11,049 men who underwent clinical examination at the Cooper Institute in Dallas, Texas, before 1990 until the occurrence of CVD death, non-CVD death, or attainment of age 90 years (281,469 person-years of follow-up, median follow-up 25.3 years, 1,106 CVD deaths). Fitness was measured by the Balke protocol and categorized according to treadmill time into low, moderate, and high fitness, with further stratification by CVD risk factor burden. Lifetime risk for CVD death determined by the National Death Index was estimated for fitness levels measured at ages 45, 55, and 65 years, with non-CVD death as the competing event.ResultsDifferences in fitness levels (low fitness vs. high fitness) were associated with marked differences in the lifetime risks for CVD death at each index age: age 45 years, 13.7% versus 3.4%; age 55 years, 34.2% versus 15.3%; and age 65 years, 35.6% versus 17.1%. These associations were strongest among persons with CVD risk factors.ConclusionsA single measurement of low fitness in mid-life was associated with higher lifetime risk for CVD death, particularly among persons with a high burden of CVD risk factors