Quantum resources and performance in the initialization-free Bernstein-Vazirani algorithm

Naseri et al. [Phys. Rev. A 106, 062429 (2022); arXiv:2205.13610] studied which quantum resources in initial states are essential for the probabilistic Bernstein-Vazirani (BV) algorithm, defining its performance as the optimal average success probability over all measurements. In this work, we consider a variant of BV algorithm, which is called the initialization-free (IF) BV algorithm, in which an arbitrary ancilla state as the oracle register is allowed, to improve the performance. We derive an explicit formula for the performance of the probabilistic IF-BV algorithm and obtain a necessary and sufficient condition for an initial state to achieve maximal performance. We further prove that, under a suitable ordering assumption on the coefficients of the initial state, the probabilistic IF-BV algorithm outperforms the standard probabilistic BV algorithm.