Sampling the time evolution of mixed quantum-classical systems

Quantum mechanics is not logically closed with respect to the classical world. Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians. The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom. Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism. In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity. Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here

Quantum Computation

In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics.Comment: 77 pages, figures included in the ps file. To appear in: Annual Reviews of Computational Physics, ed. Dietrich Stauffer, World Scientific, vol VI, 1998. The paper can be down loaded also from http://www.math.ias.edu/~doria

Classical simulation of short-time quantum dynamics

Recent progress in the development of quantum technologies has enabled the direct investigation of dynamics of increasingly complex quantum many-body systems. This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage. Here we present classical algorithms for approximating the dynamics of local observables and nonlocal quantities such as the Loschmidt echo, where the evolution is governed by a local Hamiltonian. For short times, their computational cost scales polynomially with the system size and the inverse of the approximation error. In the case of local observables, the proposed algorithm has a better dependence on the approximation error than algorithms based on the Lieb-Robinson bound. Our results use cluster expansion techniques adapted to the dynamical setting, for which we give a novel proof of their convergence. This has important physical consequences besides our efficient algorithms. In particular, we establish a novel quantum speed limit, a bound on dynamical phase transitions, and a concentration bound for product states evolved for short times.Comment: 23 pages, 5 figures, comments welcom

Can biological quantum networks solve NP-hard problems?

There is a widespread view that the human brain is so complex that it cannot be efficiently simulated by universal Turing machines. During the last decades the question has therefore been raised whether we need to consider quantum effects to explain the imagined cognitive power of a conscious mind. This paper presents a personal view of several fields of philosophy and computational neurobiology in an attempt to suggest a realistic picture of how the brain might work as a basis for perception, consciousness and cognition. The purpose is to be able to identify and evaluate instances where quantum effects might play a significant role in cognitive processes. Not surprisingly, the conclusion is that quantum-enhanced cognition and intelligence are very unlikely to be found in biological brains. Quantum effects may certainly influence the functionality of various components and signalling pathways at the molecular level in the brain network, like ion ports, synapses, sensors, and enzymes. This might evidently influence the functionality of some nodes and perhaps even the overall intelligence of the brain network, but hardly give it any dramatically enhanced functionality. So, the conclusion is that biological quantum networks can only approximately solve small instances of NP-hard problems. On the other hand, artificial intelligence and machine learning implemented in complex dynamical systems based on genuine quantum networks can certainly be expected to show enhanced performance and quantum advantage compared with classical networks. Nevertheless, even quantum networks can only be expected to efficiently solve NP-hard problems approximately. In the end it is a question of precision - Nature is approximate.Comment: 38 page

The Origins of Computational Mechanics: A Brief Intellectual History and Several Clarifications

The principle goal of computational mechanics is to define pattern and structure so that the organization of complex systems can be detected and quantified. Computational mechanics developed from efforts in the 1970s and early 1980s to identify strange attractors as the mechanism driving weak fluid turbulence via the method of reconstructing attractor geometry from measurement time series and in the mid-1980s to estimate equations of motion directly from complex time series. In providing a mathematical and operational definition of structure it addressed weaknesses of these early approaches to discovering patterns in natural systems. Since then, computational mechanics has led to a range of results from theoretical physics and nonlinear mathematics to diverse applications---from closed-form analysis of Markov and non-Markov stochastic processes that are ergodic or nonergodic and their measures of information and intrinsic computation to complex materials and deterministic chaos and intelligence in Maxwellian demons to quantum compression of classical processes and the evolution of computation and language. This brief review clarifies several misunderstandings and addresses concerns recently raised regarding early works in the field (1980s). We show that misguided evaluations of the contributions of computational mechanics are groundless and stem from a lack of familiarity with its basic goals and from a failure to consider its historical context. For all practical purposes, its modern methods and results largely supersede the early works. This not only renders recent criticism moot and shows the solid ground on which computational mechanics stands but, most importantly, shows the significant progress achieved over three decades and points to the many intriguing and outstanding challenges in understanding the computational nature of complex dynamic systems.Comment: 11 pages, 123 citations; http://csc.ucdavis.edu/~cmg/compmech/pubs/cmr.ht

Simulating chemistry using quantum computers

The difficulty of simulating quantum systems, well-known to quantum chemists, prompted the idea of quantum computation. One can avoid the steep scaling associated with the exact simulation of increasingly large quantum systems on conventional computers, by mapping the quantum system to another, more controllable one. In this review, we discuss to what extent the ideas in quantum computation, now a well-established field, have been applied to chemical problems. We describe algorithms that achieve significant advantages for the electronic-structure problem, the simulation of chemical dynamics, protein folding, and other tasks. Although theory is still ahead of experiment, we outline recent advances that have led to the first chemical calculations on small quantum information processors.Comment: 27 pages. Submitted to Ann. Rev. Phys. Che

Quantum and Classical Tradeoffs

We propose an approach for quantifying a quantum circuit's quantumness as a means to understand the nature of quantum algorithmic speedups. Since quantum gates that do not preserve the computational basis are necessary for achieving quantum speedups, it appears natural to define the quantumness of a quantum circuit using the number of such gates. Intuitively, a reduction in the quantumness requires an increase in the amount of classical computation, hence giving a ``quantum and classical tradeoff''. In this paper we present two results on this direction. The first gives an asymptotic answer to the question: ``what is the minimum number of non-basis-preserving gates required to generate a good approximation to a given state''. This question is the quantum analogy of the following classical question, ``how many fair coins are needed to generate a given probability distribution'', which was studied and resolved by Knuth and Yao in 1976. Our second result shows that any quantum algorithm that solves Grover's Problem of size n using k queries and l levels of non-basis-preserving gates must have k*l=\Omega(n)

The Quantum PCP Conjecture

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics, the global nature of entanglement and its topological properties, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column from Volume 44 Issue 2, June 201