  There are various gate sets that can be used to describe a quantum
computation. A particularly popular gate set in the literature on quantum
computing consists of arbitrary single-qubit gates and 2-qubit CNOT gates. A
CNOT gate is however not always the natural multi-qubit interaction that can be
implemented on a given physical quantum computer, necessitating a compilation
step that transforms these CNOT gates to the native gate set. A particularly
interesting case where compilation is necessary is for ion trap quantum
computers, where the natural entangling operation can act on more than 2 qubits
and can even act globally on all qubits at once. This calls for an entirely
different approach to constructing efficient circuits. In this paper we study
the problem of converting a given circuit that uses 2-qubit gates to one that
uses global gates. Our three main contributions are as follows. First, we find
an efficient algorithm for transforming an arbitrary circuit consisting of
Clifford gates and arbitrary phase gates into a circuit consisting of
single-qubit gates and a number of global interactions proportional to the
number of non-Clifford phases present in the original circuit. Second, we find
a general strategy to transform a global gate that targets all qubits into one
that targets only a subset of the qubits. This approach scales linearly with
the number of qubits that are not targeted, in contrast to the exponential
scaling reported in (Maslov & Nam, N. J. Phys. 2018). Third, we improve on the
number of global gates required to synthesise an arbitrary n-qubit Clifford
circuit from the 12n-18 reported in (Maslov & Nam, N. J. Phys. 2018) to 6n-8.
