Modern quantum algorithms with provable speedups originate historically from three disparate origins: simulation, search, and factoring.

Today, we can now understand and appreciate all of these as being instances of a single framework, and remarkably, the essence is how the rotations of a single quantum bit can be transformed

non-linearly by a simple sequence of operations. On the face of it, this is physically non-intuitive, because quantum mechanics is linear.

The key is to think not about eigenvalues and closed systems, but instead, about singular values and subsystem dynamics.