Explain Like I'm Five

"I should really get back around to responding to that comment about time-evolution in quantum systems"

3 months ago

"hmm..."

Well, better late than never.

I understand quantum computing, though, and that evolution of a circuit is a unitary (linear) operator/matrix. So, wouldn’t continuous evolution be a one-parameter Lie subgroup of the unitary operators over your Hilbert space?

I am in quite the opposite situation; my experience is in the raw physics without much of the application to quantum computing. From what I understand, though, I think this is largely correct.

In general, observables (such as the Hamiltonian) are Hermitian (self-adjoint), which is neither a superset nor a subset of unitary operators. You are not, of course, restricted to only applying observables to your quantum state (In fact, you could apply any operator you want to your quantum state; the physical meaning behind most operators involves fundamentally changing the system, but it's not strictly forbidden to do this). We require observables to be represented by Hermitian operators because the eigenvalues of Hermitian matrices are always real (since the value you observe is an eigenvalue of the observables' matrix representation, you don't want any of the eigenvalues to be non-physical complex numbers).

I had to look up why specifically quantum circuits require unitary operators, and I found this Stack Exchange response, which describes how unitary operators are used to find the time-varying component of the wavefunction that solves Schrödinger's equation. I think we were kinda describing the same thing, ultimately: continuous evolution of a quantum system is dictated by the Hamiltonian (as shown by it's presence in Schrödinger's equation), and the time-varying component of the solution is unitary (the non-time varying component is a linear combination of eigenstates of the Hamiltonian whose associated eigenvalues need not be a root of unity).

Basically, when you say:

So, wouldn’t continuous evolution be a one-parameter Lie subgroup of the unitary operators over your Hilbert space?

... the answer is 'yes,' but the Hilbert space itself is defined by the eigenstates of the Hamiltonian, which itself could be changing in time,^†^ meaning that a complete description of the time evolution requires slightly more careful consideration.

In your example of a standing wave on a string, unitary operators would take you from one mode to another, but if the length of string is changing, those unitary operators are changing too.

^†^A time-varying Hamiltonian implies a time-varying energy in the system. This sounds like breaking energy conservation (and it kinda is), but it is used to describe any system that is being pumped from the outside.