An example of the decoherence approach to quantum dissipative chaos

Daniel Braun

Giulio Casati, Davide Rossini

Robert Graham

Todd A. Brun

Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is an extremely well-developed and sophisticated field. By contrast, very little work has been done in looking at quantum versions of systems which classically exhibit {\\it dissipative} chaos. Using the decoherence formalism of Gell-Mann and Hartle, I find a quantum mechanical analog of one such system, the forced damped Duffing oscillator. I demonstrate the classical limit of the system, and discuss its decoherent histories. I show that using decoherent histories, one can define not only the quantum map of an entire density operator, but can find an analog to the Poincar\\'e map of the individual trajectory. Finally, I argue the usefulness of this model as an example of quantum dissipative chaos, as well as of a practical application of the decoherence formalism to an interesting problem.

Wojciech Hubert Zurek

Jae Weon Lee, Dima L. Shepelyansky

Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, the quantum ergodic regime, and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates, and per gate dissipation rate induced by external decoherence. In the limit of strong dissipation the quantum algorithm generates a quantum attractor which may have a complex or simple structure. We also compare the effects of dissipative decoherence with the effects of static imperfections.