A lot of modern quantum computation has this idea of "block-encoding" matrices; loosely, this is encoding a non-unitary matrix into the top left corner of a larger unitary matrix. This technique is referenced a lot, but I don't see how it can be implemented with quantum circuits.
Suppose we were given an arbitrary matrix, $C$. This matrix can be non-unitary and non-square. As a result, it cannot be represented by a quantum circuit. How is it possible that we can "block-encode" this into a quantum circuit (a universal gate set acted on by tensor product operations) in a reasonable amount of steps?
Mathematically, the technique makes sense when the spectral norm is less than $1$. But I don't see how it makes sense in practice and complexity-wise.