Phase diagram for the Grover algorithm with static imperfections

Abstract

We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the 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 $\epsilon \gtrsim 2^{-n_q/2}$. In the chaotic phase $(\epsilon > \epsilon_c)$ the algorithm is completely destroyed.