We consider a formalism of quantum compiler functions – functions that map unitary matrices to corresponding gate-circuit approximations – and prove the infeasibility of digitally computing such functions. Since the gate-circuit model of quantum computing emerged, much research has been conducted to find algorithmic solutions to the quantum compiler problem. The renowned Solovay-Kitaev theorem proves the existence of quantum compiler functions that provide low-complexity gatecircuit approximations to arbitrary unitary matrices, which is indispensable for the practical feasibility of gate-based quantum computing. However, the mere existence of such functions does not imply their realizability by means of an algorithm – a constructive procedure executed by a Turing machine. In fact, no algorithm for computing any quantum compiler function is known today. The present article demonstrates that no such algorithm can exist. We prove that no quantum compiler function can satisfy Banach-Mazur computability, which is a formalization of algorithmic feasibility with (mathematically) weak requirements. In consequence, there definitely does not exist a Turing machine that computes any quantum compiler function in the above sense, nor can there exist a constructive proof of the existence of any such function. Furthermore, we discuss proposed methods of quantum compiling and analyze them in the context of our results.