How could the system for computing exist before the algorithm that runs it? It is accepted by the mainstream that a simple algorithm could transform a normal computer into a quantum computer.
No. There's no difference, in this regard, between a classical computer and a quantum computer-the states they describe are, just, different. An algorithm is different from its implementation for some computer, incidentally. Just like there exist many implementations for the same algorithm in classical computing, the same holds for quantum computing.