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Time evolution of a quantum state is fully described by a one parameter family of unitary operators. What I can't seem to understand is, given some unitary operator acting on some Hilbert space, can it always be realized as a corresponding quantum gate.

So far, I've understood that from the Stone Theorem, any such one-parametric (strongly continuous) family is fully understood in terms of a self-adjoint operator. So my question could be rephrased as: does every self-adjoint operator on a finite-dimensional Hilbert Space describes some physical Hamiltonian?

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It depends on the physical system! In principle the answer is yes, also because in finite dimensional Hilbert spaces every self-adjoint operator is bounded below, differently from the infinite-dimensional case and physically sensible Hamiltonian must be bounded below. However, it is difficult to say if every self-adjoint operator can be endowed with a dynamical meaning. –  Valter Moretti May 13 at 14:52
    
@V.Moretti The answer is almost certainly no, since it is possible to devise grossly non-local Hamiltonians that still satisfy the conditions you give. For example, a finite many-body lattice model that describes particles hopping on a fully connected graph, which could never be realised in nature due to locality. Although of course if you assume that you are allowed, e.g. a quadrillion different lasers and the power output of a quasar, then you can probably engineer any Hamiltonian in the lab in principle... –  Mark Mitchison May 13 at 15:11
    
Sorry, you must be right, but I cannot understand well your point since, probably, I am too theoretically minded, whereas the question is much closer to experimental physics and technology... too far from my research field. –  Valter Moretti May 13 at 15:16
    
Let's ask a simpler question. Say the hilbert space has bounded dimension (by a small constant K). Is there a simple algorithm that takes a unitary operator U as input and outputs (H, t) where U = e^{-iHt/\hbar} –  user3001348 May 13 at 16:23

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