  Nonadiabatic holonomic quantum computation has been proposed as a method to
implement quantum logic gates with robustness comparable to that of adiabatic
holonomic gates but with shorter execution times. In this paper, we establish
an isoholonomic inequality for quantum gates, which provides a lower bound on
the lengths of cyclic transformations of the computational space that generate
a specific gate. Then, as a corollary, we derive a nonadiabatic execution time
estimate for holonomic gates. In addition, we demonstrate that under certain
dimensional conditions, the isoholonomic inequality is tight in the sense that
every gate on the computational space can be implemented holonomically and
unitarily in a time-optimal way. We illustrate the results by showing that the
procedures for implementing a universal set of holonomic gates proposed in a
pioneering paper on nonadiabatic holonomic quantum computation saturate the
isoholonomic inequality and are thus time optimal.
