Effects of noise on performance of Bernstein-Vazirani algorithm

Abstract

The Bernstein-Vazirani (BV) algorithm offers exceptional accuracy in finding the hidden bit string of a function. We explore how the algorithm performs in real-world situations where noise can potentially interfere with its performance. In order to assess the impact of imperfect equipments, we introduce various forms of glassy disorders into the effect of the Hadamard gates used in the Bernstein-Vazirani circuit. We incorporated disorders of five different forms, viz., Haar-uniform with finite cutoff, spherical Gaussian, discrete circular, spherical Cauchy-Lorentz, and squeezed. We find that the effectiveness of the algorithm decreases with increasing disorder strength in all cases. Additionally, we demonstrate that as the number of bits in the secret string increases, the success probability of correctly guessing the string becomes increasingly insensitive to the type of disorder and instead depends only on the mean and spread of the disorder. We compare our results with the performance of the analogous classical algorithm in the presence of similar noise. When the length of the secret string is small or moderate, the quantum BV algorithm is found to be more efficient compared to the classical algorithm for almost all types of disorders under consideration, unless the strength of the disorder is very high and the disorder follows a discrete circular distribution. However, if we move to extremely large secret strings, the success probability of the disordered BV algorithm merges with the success probability of the disordered classical algorithm for all considered disorders having arbitrary strengths. The limit on the length of the string after which the efficiency of the quantum algorithm becomes equivalent to the classical algorithm depends on the amount of disorder and not on the type of disorder.