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As far as I understood from a series of papers, minimizing the T-count in Clifford+T circuits is essential for fault-tolerant quantum computing:

While techniques such as magic state distillation and injection allow for fault-tolerant implementation of T gates, they typically require an order of magnitude more resources than Clifford gates

For example see here and here.

But do I understand correctly that while minimizing T-count (incl. in Clifford+T circuits) does not give a significant gain for the current IBM open quantum systems (real hardware, not simulations!), in particular, due to the fact that at the moment T gate (like T†, U1/P and RZ gates)

can be implemented virtually in hardware via framechanges (i.e. at zero error and duration)

See the documentation

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    $\begingroup$ I don't know the specifics of IBM hardware (hence this not being an answer), but yes, my general understanding is the T-gate optimisation is for fault-tolerant scenarios. At the level of physical qubits, you can directly implement any unitary, so why would you go for a long sequence of Hadamard + T to approximately synthesise it? I don't know a reason. $\endgroup$
    – DaftWullie
    Nov 27 '20 at 7:54
  • $\begingroup$ @DaftWullie: On IBM Q you can use U3 which allows to implement any one qubit operation, so you really do not have to do decomposition to H, S and T gates. However, only $U1(\theta)$ gate (in fact $Rz(\theta)$ gate up to global phase) and $Rx(\pi/2)$ are physically implemented one qubit gates. So, some decomposition to basic gate set is necessary and I would expect that $Rz$ for arbitrary angle is non-Clifford gate and hence number of such gates matters. $\endgroup$ Nov 27 '20 at 8:27
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    $\begingroup$ T gates, or any phase gate, is a virtual gate that comes for free today. This is different than the fault tolerant regime. $\endgroup$ Nov 27 '20 at 16:22
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    $\begingroup$ A small addition: IBM Q systems not the only quantum computers in which the Z-rotation gate is implemented virtually on the hardware level, there are others, for example, QSCOUT: "Single-qubit Z gates executed virtually by adjusting the reference clocks of individual qubits". Accordingly, there apparently also does not require the minimization of the T-count. $\endgroup$ Mar 28 at 9:20

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