# Approximating the concatenation of two approximate circuits

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-Kiteav 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-Kiteav.