A generalization of Bernstein–Vazirani algorithm with multiple secret keys and a probabilistic oracle

Abstract

A probabilistic version of the Bernstein-Vazirani problem (which is a generalization of the original Bernstein-Vazirani problem) and a quantum algorithm to solve it are proposed. The problem involves finding one or more secret keys from a set of multiple secret keys (encoded in binary form) using a quantum oracle. From a set of multiple unknown keys, the proposed quantum algorithm is capable of (a) obtaining any key (with certainty) using a single query to the probabilistic oracle and (b) finding all keys with a high probability (approaching 1 in the limiting case). In contrast, a classical algorithm will be unable to find even a single bit of a secret key with certainty (in the general case). Owing to the probabilistic nature of the oracle, a classical algorithm can only be useful in obtaining limiting probability distributions of $ 0 $ and $ 1 $ for each bit-position of secret keys (based on multiple oracle calls) and this information can further be used to infer some estimates on the distribution of secret keys based on combinatorial considerations. For comparison, it is worth noting that a classical algorithm can be used to exactly solve the original Bernstein-Vazirani problem (involving a deterministic oracle and a single hidden key containing $n$ bits) with a query complexity of $\mathcal{O}(n)$. An interesting class of problems similar to the probabilistic version of the Bernstein-Vazirani problem can be construed, where quantum algorithms can provide efficient solutions with certainty or with a high degree of confidence and classical algorithms would fail to do so.