  The parametric deformations of quasienergies and eigenvectors of unitary
operators are applied to the design of quantum adiabatic algorithms. The
conventional, standard adiabatic quantum computation proceeds along
eigenenergies of parameter-dependent Hamiltonians. By contrast, discrete
adiabatic computation utilizes adiabatic passage along the quasienergies of
parameter-dependent unitary operators. For example, such computation can be
realized by a concatenation of parameterized quantum circuits, with an
adiabatic though inevitably discrete change of the parameter. A design
principle of adiabatic passage along quasienergy is recently proposed: Cheon's
quasienergy and eigenspace anholonomies on unitary operators is available to
realize anholonomic adiabatic algorithms [Tanaka and Miyamoto, Phys. Rev. Lett.
98, 160407 (2007)], which compose a nontrivial family of discrete adiabatic
algorithms. It is straightforward to port a standard adiabatic algorithm to an
anholonomic adiabatic one, except an introduction of a parameter |v>, which is
available to adjust the gaps of the quasienergies to control the running time
steps. In Grover's database search problem, the costs to prepare |v> for the
qualitatively different, i.e., power or exponential, running time steps are
shown to be qualitatively different. Curiously, in establishing the equivalence
between the standard quantum computation based on the circuit model and the
anholonomic adiabatic quantum computation model, it is shown that the cost for
|v> to enlarge the gaps of the eigenvalue is qualitatively negligible.
