Robustness of the quantum Fourier transform with respect to static gate defects

Abstract

The quantum Fourier transform (QFT) is one of the most widely used quantum algorithms, ranging from its primary role in finding the periodicity hidden in a quantum state to its use in constructing a quantum adder. Testing how the QFT performs under more realistic conditions, we find that the QFT, when used for period finding, shows extraordinary robustness with respect to static gate defects. For instance, replacing all rotation angles $\ensuremath{\pi}/{2}^{j}$ of the controlled rotation gates in the QFT circuit by $\ensuremath{\pi}(1+r)/{2}^{j}$, where $r$ is a uniformly distributed random variable taking values in the range $[\ensuremath{-}1,1]$, effectively resulting in a QFT with random gates, the QFT performs well above the expected random result. However, it is important to keep the ${2}^{j}$ terms in the denominators of the rotation angles, resulting in random, but hierarchically random, gates. Relaxing this hierarchical structure of the QFT circuit, we find that the performance of the QFT deteriorates significantly. This observation indicates that the hierarchical structure of the quantum circuit of the QFT is more important for the observed robustness in performance than the precise actions of individual gates. In addition to the specific example of the QFT circuit studied here, this observation also corroborates our experience with more general and more complex quantum circuits. Thus, backed by our detailed numerical and analytical results, we may condense the results of our research into the following general principle: The topology of a quantum circuit is more important than the precise actions of its gates.