A quantum algorithm begins with a register of qubits in an initial state, a unitary operator (the algorithm) manipulates the state of those qubits, and then the state of the qubits is read out (or at least some information about the state on a single run of the algorithm).
It seems to me that a quantum computer answers the question of the unitary acts on the quantum state. This is "just" a matter of linear algebra. It strikes me, then, that quantum computers can be seen as linear algebra calculators.
Why then do we need quantum mechanics? Can we not find a classical system which implements linear algebra operations and use this to implement the algorithms which have been designed for quantum computers? Of course classical digital computers will not suffice, these machines are based on binary processing of information rather than the manipulation of vectors in a high dimensional space.
Question: Are there any candidates for classical linear algebra solvers (classical analog computers) which could implement the "quantum computer" algorithms whiles enjoying a similar speedup over digital classical computers?
Question 2: Perhaps I'm over simplifying by reducing a quantum computer to being simply a linear algebra solver. Is this the case? What complexity am I glossing over?