  We analyse the optimal times for implementing unitary quantum gates in a
constrained finite dimensional controlled quantum system. The family of
constraints studied is that the permitted set of (time dependent) Hamiltonians
is the unit ball of a norm induced by an inner product on su(n). We also
consider a generalisation of this to arbitrary norms. We construct a Randers
metric, by applying a theorem of Shen on Zermelo navigation, the geodesics of
which are the time optimal trajectories compatible with the prescribed
constraint. We determine all geodesics and the corresponding time optimal
Hamiltonian for a specific constraint on the control i.e. k (Tr(Hc(t)^2) = 1
for any given value of k > 0. Some of the results of Carlini et. al. are
re-derived using alternative methods. A first order system of differential
equations for the optimal Hamiltonian is obtained and shown to be of the form
of the Euler Poincare equations. We illustrate that this method can form a
methodology for determining which physical substrates are effective at
supporting the implementation of fast quantum computation.
