Famously, the CCZ gate requires 7 T gates to construct unitarily whereas using an ancilla, measurement, and classically-controlled Clifford gates, it can be constructed with only 4 T gates [1212.5069]. Are there any other diagonal gates over the CNOT+T gate set (i.e 3rd level of the Clifford hierarchy, or equivalently the T, CS, CCZ gate set) with protocols like this which are better than the equivalent unitary construction?
Obviously, outside of the CNOT+T gate set, there are some examples (e.g the 6 T gate construction of CCCZ). Also we can generalize the CCZ construction by applying a change of basis using a CNOT circuit, or by stacking it in various ways. But are there any other truly 'different' (for some definition of different) examples?
Reading through Table 2 in [1904.01124] I don't see any other examples for the states considered there. In [1709.06648] there are constructions based on the CCZ only. Likewise in [1303.6971] there are some gate-teleportation based constructions but they don't save on T-count compared to the unitary versions.