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We formulate and numerically simulate the single control qubit Shor algorithmfor the case of static imperfections induced by residual couplings betweenqubits. This allows us to study the accuracy of Shor's algorithm with respectto these imperfections using numerical simulations of realistic quantumcomputations with up to $n_q=18$ computational qubits allowing to factornumbers up to N=205193. We confirm that the algorithm remains operational up toa critical coupling strength $\epsilon_c$ which drops only polynomially with$\log_2 N$. The obtained numerical dependence of $\epsilon_c$ on $\log_2 N$ isin a good agreement with the analytical estimates that allows to obtain thescaling for functionality of Shor's algorithm on realistic quantum computerswith a large number of qubits.

Shor's factorization algorithm with a single control qubit and imperfections