  In this Letter, we construct the quantum algorithms for the Simon problem and
the period-finding problem, which do not require initializing the auxiliary
qubits involved in the process of functional evaluation but are as efficient as
the original algorithms. In these quantum algorithms, one can use any
arbitrarily mixed state as the auxiliary qubits, and furthermore can recover
the state of the auxiliary qubits to the original one after completing the
computations. Since the recovered state can be employed in any other
computations, we obtain that a single preparation of the auxiliary qubits in an
arbitrarily mixed state is sufficient to implement the iterative procedure in
the Simon algorithm or the period-finding algorithm.
