Performance scaling of the quantum Fourier transform with defective rotation gates

Abstract

Addressing Landauer's question concerning the influence of static gate defects on quantum information processor performance, we investigate analytically and numerically the case of the quantum Fourier transform (QFT) with defective controlled rotation (CROT) gates. Two types of defects are studied, separately and in combination: systematic and random. Analytical scaling laws of QFT performance are derived with respect to the number of qubits $n$, the size $\delta$ of systematic defects, and the size $\epsilon$ of random defects. The 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. Overall, we observe that the CROT gates may exhibit static and random defects of the order of 30\% and larger, and still result in satisfactory QFT performance. Thus we answer Landauer's question in the case of the QFT: far from being lethal, the QFT can tolerate tremendous static gate defects and still perform its function. 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.