  It is shown in the paper that the unitary quantum dynamics in quantum
mechanics is the universal quantum driving force to speed up a quantum
computation. This assertion supports strongly in theory that the unitary
quantum dynamics is the fundamental and universal principle in nature. On the
other hand, the symmetric structure of Hilbert space of a composite quantum
system is the quantum-computing resource that is not owned by classical
computation. A new quantum-computing speedup theory is set up on the basis of
the unitary quantum dynamics. Both the unitary quantum dynamics and the
symmetric structure and property of the Hilbert space of the quantum system are
mainly responsible for an exponential quantum-computing speedup for a general
efficient quantum algorithm. The inherent importance for the unitary quantum
dynamics to speed up a quantum computation lies in the unique ability of the
unitary quantum dynamics to build the effective interaction between the
symmetric structure of the Hilbert space of the quantum system and the
mathematical symmetric structure of a problem to be solved on the quantum
system. This unique ability could result in an essential difference of
computational power between quantum and classical computations by combining the
symmetric structure and property of the Hilbert space. The new
quantum-computing speedup theory also provides reasonable mechanisms for
exponential quantum-computing speedup for the existing efficient quantum
algorithms based on the quantum parallel principle. These existing quantum
algorithms including the hidden-subgroup-problem quantum algorithms and
conventional quantum search algorithms have the common character that the
symmetric structure of the Hilbert space does not have any effective effect on
these quantum algorithms. This could be the main reason why these quantum
algorithms are quite special and considered to be semiclassical.
