Quantum Fourier transform performance scaling; defective rotation gates

We investigate analytically and numerically the quantum Fourier transform (QFT) with defective controlled rotation (CROT) gates. We find that the QFT can tolerate systematic and random defects up to $30\%$ and still perform its function. Analytical scaling laws of QFT performance are derived with respect to the number of qubits $n$, the size $\delta$ of systematic defects, and size $\epsilon$ of random defects. Our analytical results are in excellent agreement with numerical simulations. In addition, we present an unexpected result: The performance of the defective QFT does not deteriorate with increasing $n$, but approaches a constant that scales in $\epsilon$. We derive an analytical formula that accurately reproduces the $\epsilon$ scaling of the performance plateaus. The extraordinary robustness of the QFT with respect to static gate defects displayed in our numerical and analytical calculations should be a welcome boon for laboratory and industrial realizations of quantum circuitry.