  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.
