  In August 2015 the cryptographic world was shaken by a sudden and surprising
announcement by the US National Security Agency NSA concerning plans to
transition to post-quantum algorithms. Since this announcement post-quantum
cryptography has become a topic of primary interest for several standardization
bodies. The transition from the currently deployed public-key algorithms to
post-quantum algorithms has been found to be challenging in many aspects. In
particular the problem of evaluating the quantum-bit security of such
post-quantum cryptosystems remains vastly open. Of course this question is of
primarily concern in the process of standardizing the post-quantum
cryptosystems. In this paper we consider the quantum security of the problem of
solving a system of {\it $m$ Boolean multivariate quadratic equations in $n$
variables} (\MQb); a central problem in post-quantum cryptography. When $n=m$,
under a natural algebraic assumption, we present a Las-Vegas quantum algorithm
solving \MQb{} that requires the evaluation of, on average, $O(2^{0.462n})$
quantum gates. To our knowledge this is the fastest algorithm for solving
\MQb{}.
