Suppose I have two quantum circuits $A_n,B_n$ that I have already found to approximate the operations $U,V$ within some error $\epsilon_n$ and each with an overall circuit depth $\ell_n$ using $n$ iterations of the Solovay-Kitaev algorithm.
If I then desired another approximation to the product $A_n B_n$, is there a clever way I can use my knowledge of $A_n,B_n$ to more easily compute a circuit $C$ of equal depth as $A_n,B_n$ while perhaps accepting a greater additive error of $||A_nB_n - UV ||<||C - UV ||$?
A basic first attempt could be to use the initial guesses $(A_0,B_0)$ for $A_n,B_n$ with depth $\ell_0$ as the starting point for another call to the SK algorithm. The overall circuit depth will not be doubled but will still be greater than $\ell_n$. Since I am really interested in if an exactly equal depth circuit can be found, this is insufficient. In this direction however a good place to start I think is to not to directly attempt a good approximation to $A_nB_n$ but rather settle for an easily determined initial value for $C_0$, saving a lookup in the computed $\epsilon$-net at the least when computing an approximation using Solovay-Kitaev.
