  Quantum computations usually take place under the control of the classical
world. We introduce a Classically-controlled Quantum Turing Machine (CQTM)
which is a Turing Machine (TM) with a quantum tape for acting on quantum data,
and a classical transition function for a formalized classical control. In
CQTM, unitary transformations and measurements are allowed. We show that any
classical TM is simulated by a CQTM without loss of efficiency. The gap between
classical and quantum computations, already pointed out in the framework of
measurement-based quantum computation is confirmed. To appreciate the
similarity of programming classical TM and CQTM, examples are given.
