On Fri, Dec 7, 2018 at 2:54 PM Mike Stay <metaweta@gmail.com> wrote:
So how can I represent a simple harmonic oscillator using one (more more) qbits? Wouldn't a harmonic oscillator "DO" something even if isolated from the rest of the universe? Wouldn't there be some time-variation of the "state" -- i.e., a periodicity?
A quantum harmonic oscillator assumes a quadratic potential V(x) = (x^2)/2. The nth eigenvector of the system has energy proportional to (n+1/2). In the Schrodinger picture, the states with energy E change phase by exp(-iEt/ℏ). In the Heisenberg picture, that phase is absorbed into the Hamiltonian operator. To get the particles in the QHO to change energy levels, you perturb H = (p^2 + x^2)/2 for some period of time, which multiplies the state by a unitary matrix as above.
Oops! I missed the "by using qubits". You can't: qubits have a finite spectrum while a QHO has an infinite spectrum. k qubits will have dimension 2^k, but the Hilbert space of a QHO has a dimension for each natural number. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com