We dont know exactly how the universe is doing or how the photons do to themselves, but by doing experiments, scientists figured out the law of computiation of this vague "universe computer" a.k.a laws of physics.
And since scientist's discovery do not violate the actually behavior of the physical world, there established a isomorphism. And the very same law is used in quantum computing, mainly, in chemistry. And everything success and predictable. It is a successful scientific theory. Under this notion, it is nothing wrong to say that the quantum world is doing some kind of computation underneath because it is the current understanding of how the universe works.
To my mind, I think that if you want to say that the universe is a computer, then you can't say it's doing linear algebra, even though the behavior of particles is described by linear algebra.
If you want to model the universe as a computer, than its basic operations are the interactions described in the Standard Model, and the symbols it works on are particles&fields that exist. But the universe computer is not resolving a linear algebra equation to decide what happens when an electron emits a photon. Instead, the electron emitting a photon with some energy etc. is one of the elementary operations of the universe computer.
Coming back to the CPU example, a basic operation in a CPU is setting a bit to 1. That operation is not divisible into any other more elementary operations from the point of view of modeling the physical CPU as a computer. Of course, there are other physical phenomena going down to the SM that are the realization of this basic operation, but those are not part of the modeling: the CPU computer, as a model, works by flipping bits.
Similarly, for QM, the universe computer works by doing one of the possible interactions from the standard model. As far as we know, there is no layer of detail underneath this, even if the interactions of the standard model are indeed linear algebra.
One important way in which saying "the universe is computing linear algebra" is wrong is that, as far as we know, the universe is instantly calculating the solutions of the linear equations - the electron doesn't go into an "emitting" state, then emit a photon with the appropriate values some time later after the computation is finished.
Bascially your assumption is that something that looks and works like an Intel CPU carry out an operation, that can be called a computation, everything else is not.
But sorry this is not the only method to "compute". You are limiting the definition of computiation and try too hard to justify all false assumption that you have been made.
My assumption is that something like a Turing machine, or like a reduction in Lambda calculus, or like a demonstration with the basic laws of predicate logic (all known equivalent models of computation) is a computer.
We dont know exactly how the universe is doing or how the photons do to themselves, but by doing experiments, scientists figured out the law of computiation of this vague "universe computer" a.k.a laws of physics.
And since scientist's discovery do not violate the actually behavior of the physical world, there established a isomorphism. And the very same law is used in quantum computing, mainly, in chemistry. And everything success and predictable. It is a successful scientific theory. Under this notion, it is nothing wrong to say that the quantum world is doing some kind of computation underneath because it is the current understanding of how the universe works.