  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.
