  Models of universal quantum computation in which the required interactions
between register (computational) qubits are mediated by some ancillary system
are highly relevant to experimental realisations of a quantum computer. We
introduce such a universal model that employs a $d$-dimensional ancillary
qudit. The ancilla-register interactions take the form of controlled
displacements operators, with a displacement operator defined on the periodic
and discrete lattice phase space of a qudit. We show that these interactions
can implement controlled phase gates on the register by utilising geometric
phases that are created when closed loops are traversed in this phase space.
The extra degrees of freedom of the ancilla can be harnessed to reduce the
number of operations required for certain gate sequences. In particular, we see
that the computational advantages of the quantum bus (qubus) architecture,
which employs a field-mode ancilla, are also applicable to this model. We then
explore an alternative ancilla-mediated model which employs a spin-ensemble as
the ancillary system and again the interactions with the register qubits are
via controlled displacement operators, with a displacement operator defined on
the Bloch sphere phase space of the spin coherent states of the ensemble. We
discuss the computational advantages of this model and its relationship with
the qubus architecture.
