Comment on "Stable Quantum Computation of Unstable Classical Chaos"

I think the title and content of the recent Letter by Georgeot and Shepelyanski [PRL 86, 5393 (2001), also quant-ph/0101004)] are not correct. As long as the classical Arnold map is considered, the classical computational algorithm can be made exactly equivalent with the quantum one. The claimed advantage of the Letter's quantum algorithm disappears if we correctly restrict the statistical analysis for the classical Arnold system.Comment: 1 page, PRL version + footnote [2] + refs.[3,5

Quantum computer inverting time arrow for macroscopic systems

A legend tells that once Loschmidt asked Boltzmann on what happens to his statistical theory if one inverts the velocities of all particles, so that, due to the reversibility of Newton's equations, they return from the equilibrium to a nonequilibrium initial state. Boltzmann only replied ``then go and invert them''. This problem of the relationship between the microscopic and macroscopic descriptions of the physical world and time-reversibility has been hotly debated from the XIXth century up to nowadays. At present, no modern computer is able to perform Boltzmann's demand for a macroscopic number of particles. In addition, dynamical chaos implies exponential growth of any imprecision in the inversion that leads to practical irreversibility. Here we show that a quantum computer composed of a few tens of qubits, and operating even with moderate precision, can perform Boltzmann's demand for a macroscopic number of classical particles. Thus, even in the regime of dynamical chaos, a realistic quantum computer allows to rebuild a specific initial distribution from a macroscopic state given by thermodynamic laws.Comment: revtex, 4 pages, 4 figure

Efficient quantum computation of high harmonics of the Liouville density distribution

We show explicitly that high harmonics of the classical Liouville density distribution in the chaotic regime can be obtained efficiently on a quantum computer [1,2]. As was stated in [1], this provides information unaccessible for classical computer simulations, and replies to the questions raised in [3,4].Comment: revtex, 2 pages, 1 figure; related to [1] quant-ph/0101004, [2] quant-ph/0102082, [8] quant-ph/0105149, [4] quant-ph/0110019, [3] quant-ph/011002

Distinguishing humans from computers in the game of go: a complex network approach

We compare complex networks built from the game of go and obtained from databases of human-played games with those obtained from computer-played games. Our investigations show that statistical features of the human-based networks and the computer-based networks differ, and that these differences can be statistically significant on a relatively small number of games using specific estimators. We show that the deterministic or stochastic nature of the computer algorithm playing the game can also be distinguished from these quantities. This can be seen as tool to implement a Turing-like test for go simulators.Comment: 7 pages, 6 figure

Move ordering and communities in complex networks describing the game of go

We analyze the game of go from the point of view of complex networks. We construct three different directed networks of increasing complexity, defining nodes as local patterns on plaquettes of increasing sizes, and links as actual successions of these patterns in databases of real games. We discuss the peculiarities of these networks compared to other types of networks. We explore the ranking vectors and community structure of the networks and show that this approach enables to extract groups of moves with common strategic properties. We also investigate different networks built from games with players of different levels or from different phases of the game. We discuss how the study of the community structure of these networks may help to improve the computer simulations of the game. More generally, we believe such studies may help to improve the understanding of human decision process.Comment: 14 pages, 21 figure

Strange attractor simulated on a quantum computer

We show that dissipative classical dynamics converging to a strange attractor can be simulated on a quantum computer. Such quantum computations allow to investigate efficiently the small scale structure of strange attractors, yielding new information inaccessible to classical computers. This opens new possibilities for quantum simulations of various dissipative processes in nature.Comment: latex 4 pages, 4 figures, research at http://www.quantware.ups-tlse.fr, one fig and discussion adde

Multifractality and intermediate statistics in quantum maps

We study multifractal properties of wave functions for a one-parameter family of quantum maps displaying the whole range of spectral statistics intermediate between integrable and chaotic statistics. We perform extensive numerical computations and provide analytical arguments showing that the generalized fractal dimensions are directly related to the parameter of the underlying classical map, and thus to other properties such as spectral statistics. Our results could be relevant for Anderson and quantum Hall transitions, where wave functions also show multifractality.Comment: 4 pages, 4 figure

Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincar\'e's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfrequency of the mode. We find that there are nested layers, the thinnest and most internal layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.Comment: 42 pages, 20 figures, abstract shortene