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?