Marcos Saraceno

Quantum computers in phase space

We represent both the states and the evolution of a quantum computer in phase space using the discrete Wigner function. We study properties of the phase space representation of quantum algorithms: apart from analyzing important examples, such as the Fourier transform and Grovers search, we examine the conditions for the existence of a direct correspondence between quantum and classical evolutions in phase space. Finally, we describe how to measure directly the Wigner function in a given phase-space point by means of a tomographic method that, itself, can be interpreted as a simple quantum algorithm.

Interpretation of tomography and spectroscopy as dual forms of quantum computation

It is important to be able to determine the state of a quantum system and to measure properties of its evolution. State determination can be achieved using tomography, in which the system is subjected to a series of experiments, whereas spectroscopy can be used to probe the energy spectrum associated with the systems evolution. Here we show that, for a quantum system whose state or evolution can be modelled on a quantum computer, tomography and spectroscopy can be interpreted as dual forms of quantum computation. Specifically, we find that the phase estimation algorithm (which underlies a quantum computers ability to perform efficient simulations and to factorize large numbers) can be adapted for tomography or spectroscopy. This is analogous to the situation encountered in scattering experiments, in which it is possible to obtain information about both the state of the scatterer and its interactions. We provide an experimental demonstration of the tomographic application by performing a measurement of the Wigner function (a phase space distribution) of a quantum system. For this purpose, we use three qubits formed from spin-1/2 nuclei in a quantum computation involving liquid-state nuclear magnetic resonance.

Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem

We analyze and further develop a method to represent the quantum state of a system of n qubits in a phase-space grid of NxN points (where N=2{sup n}). The method, which was recently proposed by Wootters and co-workers (Gibbons et al., Phys. Rev. A 70, 062101 (2004).), is based on the use of the elements of the finite field GF(2{sup n}) to label the phase-space axes. We present a self-contained overview of the method, we give insights into some of its features, and we apply it to investigate problems which are of interest for quantum-information theory: We analyze the phase-space representation of stabilizer states and quantum error-correction codes and present a phase-space solution to the so-called mean king problem.

Discrete Wigner functions and the phase space representation of quantum computers

We show how to represent the state and the evolution of a quantum computer (or any system with an N-dimensional Hilbert space) in phase space. For this purpose we use a discrete version of the Wigner function which, for arbitrary N, is defined in a phase space grid of 2N×2N points. We compute such Wigner function for states which are relevant for quantum computation. Finally, we discuss properties of quantum algorithms in phase space and present the phase space representation of Grovers quantum search algorithm.

Decoherence for classically chaotic quantum maps.

We study the behavior of an open quantum system, with an N-dimensional space of states, whose density matrix evolves according to a nonunitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (bakers map and Harpers map are the two examples we consider). A nonunitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large N) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail by means of numerical and analytic tools. The existence of a regime where the entropy grows with a rate that is independent of the strength of the diffusion coefficient is demonstrated. The nature of the processes that determine the production of entropy is analyzed.

Quantum algorithms for phase-space tomography

We present efficient circuits that can be used for the phase-space tomography of quantum states. The circuits evaluate individual values or selected averages of the Wigner, Kirkwood, and Husimi distributions. These quantum gate arrays can be programmed by initializing appropriate computational states. The Husimi circuit relies on a subroutine that is also interesting in its own right: the efficient preparation of a coherent state, which is the ground state of the Harper Hamiltonian.

Localization of Resonance Eigenfunctions on Quantum Repellers

We introduce a new phase space representation for open quantum systems. This is a very powerful tool to help advance in the study of the morphology of their eigenstates. We apply it to two different versions of a paradigmatic model, the baker map. This allows us to show that the long-lived resonances are strongly scarred along the shortest periodic orbits that belong to the classical repeller. Moreover, the shape of the short-lived eigenstates is also analyzed. Finally, we apply an antiunitary symmetry measure to the resonances that allows us to quantify their localization on the repeller.

Noise models for superoperators in the chord representation

We study many-qubit generalizations of quantum noise channels that can be written as an incoherent sum of translations in phase space, for which the chord representation results specially useful. Physical descriptions in terms of the spectral properties of the superoperator and the action in phase space are provided. A very natural description of decoherence leading to a preferred basis is achieved with diffusion along a phase space line. The numerical advantages of using the chord representation are illustrated in the case of coarse-graining noise.

SPECTRAL APPROACH TO CHAOS AND QUANTUM-CLASSICAL CORRESPONDENCE IN QUANTUM MAPS

Correspondence in quantum chaotic systems is lost in short time scales. Introducing some noise we study the spectrum of the resulting coarse-grained propagator of density matrices. Some different methods to compute the spectrum are reviewed. The relationship between the eigenvalues of the coarse-grained superoperator and the classical Ruelle–Pollicott resonances is remarked on. As a consequence, classical decay rates in quantum time-dependent quantities appear.

Spectral properties and classical decays in quantum open systems

We study the relationship between the spectral properties of diffusive open quantum maps and the classical spectrum of Ruelle-Pollicott resonances. The leading resonances determine the asymptotic time regime for several quantities of interest--the linear entropy, the Loschmidt echo, and the correlations of the initial state. A numerical method that allows an efficient calculation of the leading spectrum is developed using a truncated basis adapted to the dynamics.