# What is the shortest sequence of decomposition a given single-qubit unitary gate

Given a single-qubit unitary matrix, can we find the shortest sequence of Clifford + T gates that correspond to that unitary?

According to Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates , which is Solovay-Kitaev decomposition, I learned single-qubit decomposition may need $$O(\log^{3.97} (1/\delta))$$ clifford+T gates with the accuracy $$\delta$$.

And later many optimization is worked on it. For example: Synthesis of unitaries with Clifford+T circuits

So I want to know if there exists a shortest sequence of Clifford + T gates that correspond to decompose any single-qubit unitary into Clifford+T? If it existed, what is commonly used in current compiler?

In other words, this is the best that can be done. It yields the shortest possible sequence. This best is $$O(\log\frac{1}{\delta})$$ Hadamard + T.