Effects of decoherence and imperfections for quantum algorithms

Abstract

We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength $\epsilon$. The critical border $\epsilon_c$ between two phases drops polynomially with the number of qubits $n_q$ as $\epsilon_c \sim n_q^{-3/2}$. In the regular phase $(\epsilon < \epsilon_c)$ the algorithm remains robust against imperfections showing the efficiency gain $\epsilon_c / \epsilon$ for $\epsilon > 2^{-n_q/2}$. In the chaotic phase $(\epsilon > \epsilon_c)$ the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.