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Efficient quantum computing of complex dynamics

Giulio Casati

Center for Nonlinear and Complex Systems,
University of Insubria, Como, Italy

In interacting many body systems such as nuclei, complex atoms, quantum dots, and quantum spin glasses, the interaction leads to quantum chaos characterized by ergodicity of eigenstates and level spacing statistics as in Random Matrix Theory. In this regime, a quantum computer eigenstate is composed by an exponentially large number of quantum register states and the computer operability is destroyed. Here we model an isolated quantum computer as a two-dimensional lattice of qubits (spin halves) with fluctuations in individual qubit energies and residual short-range inter-qubit couplings. We show that above a critical inter-qubit coupling strength, quantum chaos sets in and this results in the interaction induced dynamical thermalization and the occupation numbers well described by the Fermi-Dirac distribution. This thermalization destroys the noninteracting qubit structure and sets serious requirements for the quantum computer operability. We then construct a quantum algorithm which uses the number of qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics. The numerical study of the effect of static imperfections in the quantum computer hardware shows that the main elements of the phase space structures are accurately reproduced up to a time scale which is polynomial in the number of qubits. The errors generated by these imperfections are more significant than the errors of random noise in gate operations.

G. Benenti, G. Casati and D.L. Shepelyansky 2001 Eur. Phys. J. D. 17 265.
G. Benenti,G. Casati, S. Montangero and D. L. Shepelyansky 2001 Phys. Rev. Lett. 87 227901
G. Benenti and G. Casati Quantum-classical Correspondence in perturbed chaotic Systems. To appear in Phys Rev E.
G. Benenti, G. Casati, S. Montangero and D.L. Shepelyansky Eigenstates of Operative Quantum Computer: Hypersensitivity to Static Imperfections. Preprint

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