Invariance of success probability in Grover's quantum search under local noise with memory

Abstract

We analyze the robustness of Grover's quantum search algorithm performed by a quantum register under a possibly time-correlated noise acting locally on the qubits. We model the noise as originating from an arbitrary but fixed unitary evolution, $U$, of some noisy qubits. The noise can occur with some probability in the interval between any pair of consecutive noiseless Grover evolutions. Although each run of the algorithm is a unitary process, the noise model leads to decoherence when all possible runs are considered. We derive a set of unitary $U$'s, called the 'good noises,' for which the success probability of the algorithm at any given time remains unchanged with varying the non-trivial total number ($m$) of noisy qubits in the register. The result holds irrespective of the presence of any time-correlations in the noise. We show that only when $U$ is either of the Pauli matrices $\sigma_x$ and $\sigma_z$ (which give rise to $m$-qubit bit-flip and phase-damping channels respectively in the time-correlation-less case), the algorithm's success probability stays unchanged when increasing or decreasing $m$. In contrast, when $U$ is the Pauli matrix $\sigma_y$ (giving rise to $m$-qubit bit-phase flip channel in the time-correlation-less case), the success probability at all times stays unaltered as long as the parity (even or odd) of the total number $m$ remains the same. This asymmetry between the Pauli operators stems from the inherent symmetry-breaking existing within the Grover circuit. We further show that the positions of the noisy sites are irrelevant in case of any of the Pauli noises. The results are illustrated in the cases of time-correlated and time-correlation-less noise. We find that the former case leads to a better performance of the noisy algorithm. We also discuss physical scenarios where our chosen noise model is of relevance.