  The quantum state discrimination problem is to distinguish between
non-orthogonal quantum states. This problem has many applications in quantum
information theory, quantum communication and quantum cryptography. In this
paper a quantum algorithm using weak measurement and partial negation will be
proposed to solve the quantum state discrimination problem using a single copy
of an unknown qubit. The usage of weak measurement makes it possible to
reconstruct the qubit after measurement since the superposition will not be
destroyed due to measurement. The proposed algorithm will be able to determine,
with high probability of success, the state of the unknown qubit and whether it
is encoded in the Hadamard or the computational basis by counting the outcome
of the successive measurements on an auxiliary qubit.
