# Why do we care about the number of $T$ gates in a quantum circuit?

When reading this question and quickly reading some of the linked papers, I wondered why was the number of $$T$$ gates specified along the number of controlled-$$X$$ gates.

I've often read that implementing a controlled-$$X$$ gate was more costly than implementing single-qubit gates, though I never got the explanation as to why this is true. Assuming this however, I can understand why this number is of importance.

Concerning the number of $$T$$ gates however, what's the motive behind it? My intuition is that it gives an idea of how deep the circuit is, since $$T$$ is required to precisely approximate the desired gates, but if it is the case, why bother with the number of $$T$$ gates rather than with the circuit depth directly?

If you calculate a fault-tolerant threshold for a concatenated scheme, you can roughly think of this as taking each gate in your gate set, specifying how each of them is made on the physical qubits (including a round of error correction before & after), and (roughly speaking) evaluating the size (=depth$$\times$$width, think of it like the number of places where an error could occur) of that circuit. The fault tolerant threshold is determined by the gate in the set that has the largest size.