Scaling and Universality of the Complexity of Analog Computation

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We analyze numerically various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.Comment: 22 pages, latex, 12 eps fig

Limits on Fundamental Limits to Computation

An indispensable part of our lives, computing has also become essential to industries and governments. Steady improvements in computer hardware have been supported by periodic doubling of transistor densities in integrated circuits over the last fifty years. Such Moore scaling now requires increasingly heroic efforts, stimulating research in alternative hardware and stirring controversy. To help evaluate emerging technologies and enrich our understanding of integrated-circuit scaling, we review fundamental limits to computation: in manufacturing, energy, physical space, design and verification effort, and algorithms. To outline what is achievable in principle and in practice, we recall how some limits were circumvented, compare loose and tight limits. We also point out that engineering difficulties encountered by emerging technologies may indicate yet-unknown limits.Comment: 15 pages, 4 figures, 1 tabl

Introduction to Quantum Algorithms for Physics and Chemistry

In this introductory review, we focus on applications of quantum computation to problems of interest in physics and chemistry. We describe quantum simulation algorithms that have been developed for electronic-structure problems, thermal-state preparation, simulation of time dynamics, adiabatic quantum simulation, and density functional theory.Comment: 44 pages, 5 figures; comments or suggestions for improvement are welcom

Can One Trust Quantum Simulators?

Various fundamental phenomena of strongly-correlated quantum systems such as high-$T_c$ superconductivity, the fractional quantum-Hall effect, and quark confinement are still awaiting a universally accepted explanation. The main obstacle is the computational complexity of solving even the most simplified theoretical models that are designed to capture the relevant quantum correlations of the many-body system of interest. In his seminal 1982 paper [Int. J. Theor. Phys. 21, 467], Richard Feynman suggested that such models might be solved by "simulation" with a new type of computer whose constituent parts are effectively governed by a desired quantum many-body dynamics. Measurements on this engineered machine, now known as a "quantum simulator," would reveal some unknown or difficult to compute properties of a model of interest. We argue that a useful quantum simulator must satisfy four conditions: relevance, controllability, reliability, and efficiency. We review the current state of the art of digital and analog quantum simulators. Whereas so far the majority of the focus, both theoretically and experimentally, has been on controllability of relevant models, we emphasize here the need for a careful analysis of reliability and efficiency in the presence of imperfections. We discuss how disorder and noise can impact these conditions, and illustrate our concerns with novel numerical simulations of a paradigmatic example: a disordered quantum spin chain governed by the Ising model in a transverse magnetic field. We find that disorder can decrease the reliability of an analog quantum simulator of this model, although large errors in local observables are introduced only for strong levels of disorder. We conclude that the answer to the question "Can we trust quantum simulators?" is... to some extent.Comment: 20 pages. Minor changes with respect to version 2 (some additional explanations, added references...

Quantum computing and the entanglement frontier - Rapporteur talk at the 25th Solvay Conference

Quantum information science explores the frontier of highly complex quantum states, the "entanglement frontier". This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the day when well controlled quantum systems can perform tasks surpassing what can be done in the classical world. One way to achieve such "quantum supremacy" would be to run an algorithm on a quantum computer which solves a problem with a super-polynomial speedup relative to classical computers, but there may be other ways that can be achieved sooner, such as simulating exotic quantum states of strongly correlated matter. To operate a large scale quantum computer reliably we will need to overcome the debilitating effects of decoherence, which might be done using "standard" quantum hardware protected by quantum error-correcting codes, or by exploiting the nonabelian quantum statistics of anyons realized in solid state systems, or by combining both methods. Only by challenging the entanglement frontier will we learn whether Nature provides extravagant resources far beyond what the classical world would allow