Suppose there exists an algorithm that takes as input an arbitrary unitary matrix and produces as output a quantum circuit representing that matrix. Then in theory that algorithm could construct any quantum circuit. This would be quite useful.
Furthermore, since any computable algorithm may be implemented as a quantum circuit, the hypothetical circuit constructing algorithm could in principle construct any such algorithm as a quantum circuit. This seems similar, though not identical, to the idea of Turing completeness.
Intuitively it seems bizarre that such an algorithm could exist. However, I am not able to think of something that disproves this. Has such an algorithm's existence been proven/disproven?