Why is it so difficult to implement a T gate?

Question

Much effort has gone into optimizing T gates (distillation, factories, and numerous circuit optimizations).

I am aware that non-Clifford gates are necessary to attain quantum speedup, and the T gate, which is basically $R_Z\left(\frac{\pi}{4}\right)$, is probably the simplest non-Clifford gate.

Why is it so difficult to implement a T gate? I have seen references to the error correction being more difficult (non-transversal?)

The Pauli gates ($\pi$ rotation around the Bloch axes) are considered simple. Is that due to simpler error correction as well?

In super-conducting qubits, after they are calibrated, they are manipulated with pulses and the parameters (duration, frequency, phase, amplitude) of the pulse control the rotation. Why is it harder to create a $R_Z\left(\frac{\pi}{4}\right)$ rotation than a $R_Z\left(\pi\right)$ rotation? (and might it be easier in different qubit implementations? e.g., silicon, photonics, ...)?

Answer 1

A few misconceptions here.

The $T$ gate is necessary to obtain universality, not speed-up.

Also, running a $T$ gate is not harder than other single-qubit gates. On contrary, it is among the most reliables.

The complexity is not given by the gate itself, but by the domain space that we can reach with combination of Clifford and non-Clifford gates.

Since we need to define and implement fault-tolerant computation, the distinction between Clifford and non-Clifford becomes convenient. As we can make Clifford gates resistant to noise by means of very efficient correction schemes. While fault-tolerant non-Clifford gates are achieved by means of distillation.