  Quantum circuit synthesis describes the process of converting arbitrary
unitary operations into a gate sequence of a fixed universal gate set, usually
defined by the operations native to a given hardware platform. Most current
synthesis algorithms are designed to synthesize towards a set of single qubit
rotations and an additional entangling two qubit gate, such as CX, CZ, or the
Molmer Sorensen gate. However, with the emergence of neutral atom based
hardware and their native support for gates with more than two qubits,
synthesis approaches tailored to these new gate sets become necessary. In this
work, we present an approach to synthesize multi controlled phase gates using
ZX calculus. By representing quantum circuits as graph like ZX diagrams, one
can utilize the distinct graph structure of diagonal gates to identify multi
controlled phase gates inherently present in some quantum circuits even if none
were explicitly defined in the original circuit. We evaluate the approach on a
wide range of benchmark circuits and compare them to the standard Qiskit
synthesis regarding its circuit execution time for neutral atom based hardware
with native support of multi controlled gates. Our results show possible
advantages for current state of the art hardware and represent the first exact
synthesis algorithm supporting arbitrary sized multi controlled phase gates.
