  It is straightforward to give a sum-over-paths expression for the transition
amplitudes of a quantum circuit as long as the gates in the circuit are
balanced, where to be balanced is to have all nonzero transition amplitudes of
equal magnitude. Here we consider the question of whether, for such circuits,
the relative phases of different discrete-time paths through the configuration
space can be defined in terms of a classical action, as they are for
continuous-time paths. We show how to do so for certain kinds of quantum
circuits, namely, Clifford circuits where the elementary systems are
continuous-variable systems or discrete systems of odd-prime dimension. These
types of circuit are distinguished by having phase-space representations that
serve to define their classical counterparts. For discrete systems, the
phase-space coordinates are also discrete variables. We show that for each gate
in the generating set, one can associate a symplectomorphism on the phase-space
and to each of these one can associate a generating function, defined on two
copies of the configuration space. For discrete systems, the latter association
is achieved using tools from algebraic geometry. Finally, we show that if the
action functional for a discrete-time path through a sequence of gates is
defined using the sum of the corresponding generating functions, then it yields
the correct relative phases for the path-sum expression. These results are
likely to be relevant for quantizing physical theories where time is
fundamentally discrete, characterizing the classical limit of discrete-time
quantum dynamics, and proving complexity results for quantum circuits.
