# a question about quantum gate decomposition on simulator or emulator

I have read a paper about "approximated decomposition" of a unitary single gate (Solovay-Kitaev algorithm) which told us a any unitary single gate can be decomposed into {Hadamard, Phase} with any accuracy $$\epsilon$$ . The number of {Hadamard, Phase} grows exponentially as the $$\epsilon$$ becomes lower and the result is a better approximation of the gate. However, in simulator or emulator such as project Q and Qiskit, the $$U3$$ gate seems exacted decomposition which means only one parameterized $$U3$$ gate can realize any unitary single gate.

So if we can realize "exact unitary single gate" in quantum computer, why still so many people focus on the "approximated one" and try to optimized the sequence after decomposition?

The restricted gate set of, for example, $$\{H,T\}$$, is more relevant when you start talking about error corrected quantum computation. It may be that when you act on individual, physical, qubits, you can implement and arbitrary single-qubit rotation.
However, when you encode in an error correcting code, and you want to implement a gate directly on the logical qubit, the set of logical gates that you can enact may be restricted. For example, it is common to restrict to a set of gates comprising the Clifford gates + $$T$$. This is because, for many codes, the Clifford gates can be implemented with relatively low cost. The fault-tolerant threshold (the limit of error rate that you need to be under in order to have properly managed errors) is primarily dependent on the implementation of any other gates. So, we keep things as simple as possible and implement a single extra gate that we try to do as well as possible. Typically, that's the $$T$$ gate.