Statistical properties of eigenvalues for an operating quantum computer with static imperfections

We investigate the transition to quantum chaos, induced by static imperfections, for an operating quantum computer that simulates efficiently a dynamical quantum system, the sawtooth map. For the different dynamical regimes of the map, we discuss the quantum chaos border induced by static imperfections by analyzing the statistical properties of the quantum computer eigenvalues. For small imperfection strengths the level spacing statistics is close to the case of quasi-integrable systems while above the border it is described by the random matrix theory. We have found that the border drops exponentially with the number of qubits, both in the ergodic and quasi-integrable dynamical regimes of the map characterized by a complex phase space structure. On the contrary, the regime with integrable map dynamics remains more stable against static imperfections since in this case the border drops only algebraically with the number of qubits. Received 19 June 2002 / Received in final form 30 September 2002 Published online 17 Decembre 2002 RID="a" ID="a"e-mail: dima@irsamc.ups-tlse.fr RID="b" ID="b"UMR 5626 du CNRS     

Squeezed states and quantum chaos

We examine the dynamics of a wave packet that initially corresponds to a coherent state in the model of a quantum rotator excited by a periodic sequence of kicks. This model is the main model of quantum chaos and allows for a transition from regular behavior to chaotic in the classical limit. By doing a numerical experiment we study the generation of squeezed states in quasiclassical conditions and in a time interval when quantum-classical correspondence is well-defined. We find that the degree of squeezing depends on the degree of local instability in the system and increases with the Chirikov classical stochasticity parameter. We also discuss the dependence of the degree of squeezing on the initial width of the packet, the problem of stability and observability of squeezed states in the transition to quantum chaos, and the dynamics of disintegration of wave packets in quantum chaos. Zh. éksp. Teor. Fiz. 113, 111–127 (January 1998)     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Eigenstates of an operating quantum computer: hypersensitivity to static imperfections

We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits n_q, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with n_q. Above this threshold the quantum eigenstate entropy grows linearly with n_q but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in n_q. Received 7 March 2002/ Received in final form 3 May 2002 Published online 19 July 2002     

Efficient classical simulation of continuous variable quantum information processes

We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurements can be simulated efficiently on a classical computer.     

Exponential speedup with a single bit of quantum information: measuring the average fidelity decay

We present an efficient quantum algorithm to measure the average fidelity decay of a quantum map under perturbation using a single bit of quantum information. Our algorithm scales only as the complexity of the map under investigation. Thus for those maps admitting an efficient gate decomposition, it provides an exponential speedup over known classical procedures. Fidelity decay is important in the study of complex dynamical systems, where it is conjectured to be a signature of eigenvector statistics. Our result also illustrates the role of chaos in the process of decoherence.     

Implementation of quantum search algorithm using classical Fourier optics

We report on an experiment on Grover's quantum search algorithm showing that classical waves can search a N-item database as efficiently as quantum mechanics can. The transverse beam profile of a short laser pulse is processed iteratively as the pulse bounces back and forth between two mirrors. We directly observe the sought item being found in approximately square root[N] iterations, in the form of a growing intensity peak on this profile. Although the lack of quantum entanglement limits the size of our database, our results show that entanglement is neither necessary for the algorithm itself, nor for its efficiency.     

Closed path integrals and the quantum action

We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with parameters different from the classical action. We present numerical results for the harmonic oscillator with weak perturbation, the quartic potential, and the double well potential. The quantum action is relevant for quantum chaos and quantum instantons.     

Lyapunov exponents from quantum dynamics

We extract classical Lyapunov exponents from the time dependence of quantum mechanical expectation values. Classical chaos is revealed as a quantum transient with a liftetime ～? ln ?. Our strategy is shown to work for the example of a periodically kicked top.     

Regular and chaotic quantum dynamics in atom-diatom reactive collisions

A new microirreversible 3D theory of quantum multichannel scattering in the three-body system is developed. The quantum approach is constructed on the generating trajectory tubes which allow taking into account influence of classical nonintegrability of the dynamical quantum system. When the volume of classical chaos in phase space is larger than the quantum cell in the corresponding quantum system, quantum chaos is generated. The probability of quantum transitions is constructed for this case. The collinear collision of the Li + (FH) → (LiF) + H system is used for numerical illustration of a system generating quantum (wave) chaos. The text was submitted by the authors in English.     

Demonstration of essentiality of entanglement in a Deutsch-like quantum algorithm

Quantum algorithms can be used to efficiently solve certain classically intractable problems by exploiting quantum parallelism. However, the effectiveness of quantum entanglement in quantum computing remains a question of debate. This study presents a new quantum algorithm that shows entanglement could provide advantages over both classical algorithms and quantum algo- rithms without entanglement. Experiments are implemented to demonstrate the proposed algorithm using superconducting qubits. Results show the viability of the algorithm and suggest that entanglement is essential in obtaining quantum speedup for certain problems in quantum computing. The study provides reliable and clear guidance for developing useful quantum algorithms.     

Review: Characterizing and quantifying quantum chaos with quantum tomography

We explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. The tomographic record consists of a time series of expectation values of a Hermitian operator evolving under the application of the Floquet operator of a quantum map that possesses (or lacks) time-reversal symmetry. We find that the rate of information gain, and hence the fidelity of quantum state reconstruction, depends on the symmetry class of the quantum map involved. Moreover, we find an increase in information gain and hence higher reconstruction fidelities when the Floquet maps employed increase in chaoticity. We make predictions for the information gain and show that these results are well described by random matrix theory in the fully chaotic regime. We derive analytical expressions for bounds on information gain using random matrix theory for different classes of maps and show that these bounds are realized by fully chaotic quantum systems.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Kicked-rotor quantum resonances in position space: application to situations of experimental interest

We discuss a model for quantum computing with initially mixed states. Although such a computer is known to be less powerful than a quantum computer operating with pure (entangled) states, it may efficiently solve some problems for which no efficient classical algorithms are known. We suggest a new implementation of quantum computation with initially mixed states in which an algorithm realization is achieved by means of optimal basis independent transformations of qubits.     

The periodically kicked quantum spin

It is shown that the same kind of deterministic chaos that occurs in classical systems can occur in certain quantum mechanical, many-body systems. The example of the physical realization of the periodically kicked quantum spin (PKQS) is considered in detail. The quantum mechanical equations of motion for this system can be converted into the three-dimensional PKQS map, which exhibits deterministic chaos and Arnold diffusion. Although the case of quantum spin s= 1/2 is assumed, it is shown that the same map results for s=1 (but not for s>/=3/2), and for a suitably chosen classical particle with orbital angular momentum. A simple generalization of the PKQS model gives rise to stochastic webs on the surface of the unit sphere very similar to the Zaslavsky stochastic webs in a plane.     

Quantum computing of quantum chaos and imperfection effects

We study numerically the imperfection effects in the quantum computing of the kicked rotator model in the regime of quantum chaos. It is shown that there are two types of physical characteristics: for one of them the quantum computation errors grow exponentially with the number of qubits in the computer, while for the other the growth is polynomial. A certain similarity between classical and quantum computing errors is also discussed.     

Gravity-sensitive quantum dynamics in cold atoms

We subject a falling cloud of cold cesium atoms to periodic kicks from a sinusoidal potential created by a vertical standing wave of laser light. By controllably accelerating the potential, we show quantum accelerator mode dynamics to be highly sensitive to the effective gravitational acceleration when close to specific, resonant values. This quantum sensitivity to a control parameter is reminiscent of that associated with classical chaos and promises techniques for precision measurement.     

Experimental implementation of an adiabatic quantum optimization algorithm

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.     

Random quantum states: recent developments and applications

We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrödinger equations describing open quantum systems.     

Open quantum dots: II. Probing the classical to quantum transition

Open quantum dots provide a natural system in which to study both classical and quantum features of transport. From the classical point of view these dots possess a mixed phase space which yields families of closed, regular orbits as well as an expansive sea of chaos. An important question concerns the manner in which these classical states evolve into the set of quantum states that populate the dot in the quantum limit. In the reverse direction, the manner in which the quantum states evolve to the classical world is governed strongly by Zurek's decoherence theory. This was discussed from the quantum perspective in an earlier review?(Ferry et?al 2011 Semicond. Sci. Technol. 26 043001). Here, we discuss the nature of the various classical states, how they are formed, how they progress to the quantum world, and the signatures that they create in magnetotransport and general conductance studies of these dots.