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To approximate arbitrary unitary transformations on one or more qubits, onemust perform transformations which are outside of the Clifford group. The gatemost commonly considered for this purpose is the T = diag(1, exp(i \pi/4))gate. As T gates are computationally expensive to perform fault-tolerantly inthe most promising error-correction technologies, minimising the "T-count" (thenumber of T gates) required to realise a given unitary in a Clifford+T circuitis of great interest. We describe techniques to find circuits with reducedT-count in unitary circuits, which develop on the ideas of Heyfron and Campbell[arXiv:1712.01557] with the help of the ZX calculus. Following[arXiv:1712.01557], we reduce the problem to that of minimising the T count ofa CNOT+T circuit. The ZX calculus motivates a further reduction to simplifyinga product of commuting "\pi/4-parity-phase" operations: diagonal unitarytransformations which induce a relative phase of exp(i \pi/4) depending on theoutcome of a parity computation on the standard basis (which motivatedKissinger and van de Wetering [1903.10477] to introduce "phase gadgets"). For anumber of standard benchmark circuits, we show that these techniques -- in somecases supplemented by the TODD subroutine of Heyfron and Campbell[arXiv:1712.01557] -- yield T-counts comparable to or better than the bestpreviously known results.

Techniques to Reduce $\pi/4$-Parity-Phase Circuits, Motivated by the ZX Calculus