Quantum chaos and random matrix theory for fidelity decay in quantum computations with static imperfections

We determine the universal law for fidelity decay in quantum comp tit at ions of complex dynamics in presence of internal static imperfections in a quantum computer. Our approach is based on random matrix theory applied to quantum computations in presence of imperfections. The theoretical predictions are tested and confirmed in extensive numerical simulations of a quantum algorithm for quantum chaos in the dynamical tent map with up to 18 qubits. The theory developed determines the time scales for reliable quantum computations in absence of the quantum error correction codes. These time scales are related to the Heisenberg time, the Thouless time, and the decay time given by Fermi's golden rule which are well-known in the context of mesoscopic systems. The comparison is presented for static imperfection effects and random errors in quantum gates. A new convenient method for the quantum computation of the coarse-grained Wigner function is also proposed.