Quantum Chaos & Quantum Computers

The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n. As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.Comment: Lecture at Nobel symposium on "Quantum chaos", June 2000, Sweden; revtex, 10 pages, 9 figure

PageRank model of opinion formation on Ulam networks

We consider a PageRank model of opinion formation on Ulam networks, generated by the intermittency map and the typical Chirikov map. The Ulam networks generated by these maps have certain similarities with such scale-free networks as the World Wide Web (WWW), showing an algebraic decay of the PageRank probability. We find that the opinion formation process on Ulam networks have certain similarities but also distinct features comparing to the WWW. We attribute these distinctions to internal differences in network structure of the Ulam and WWW networks. We also analyze the process of opinion formation in the frame of generalized Sznajd model which protects opinion of small communities.Comment: 7 pages, 6 figures. Updated version for publicatio

Two interacting particles in an effective 2-3d random potential

We study the effect of coherent propagation of two interacting particles in an effective 2-3-d disordered potential. Our numerical data demonstrate that in dimension $d > 2$, interaction can lead to two--particles delocalization below one--particle delocalization border. We also find that the distance between the two delocalized particles (pair size) grows logarithmically with time. As a result pair propagation is subdiffusive.Comment: 15 Latex pages, + 15 figures compressed with uufile

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