This paper shows how to use quantum computers to simulate nonlinear and non-Hamiltonian classical dynamics by using the Koopman-von Neumann approach, which reformulates the classical system as an equivalent Schrodinger equation with a Hermitian Hamiltonian and a unitary evolution operator on phase space.

arXiv:2003.09980Physical Review ResearchOct 19, 2020109 citations

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arXiv:quant-ph/0210198 · International Journal of Theoretical Physics

Nonlinear Reformulation of Heisenberg's Dynamics

Martin Ziegler, Benno Fuchssteiner

Published Jul 1, 2005• 2 citations

arXiv HTML PDF

DOI: 10.1007/s10773-005-7048-9

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g

, the relative phase

ϕ

, and a dynamical phase

θ

resulting from the interaction time. The optimal operating point for sensing the relative phase (dynamical phase) is found to be

ϕ= π

(

θ=0

) with quantum Fisher information exhibiting Heisenberg scaling

E^{2}

(logarithmically modified Heisenberg scaling

\left({E\over \ln E}\right)^{2}

). Compared to the predictions of the circuit-based model, we find in that in the Hamiltonian model: 1. the optimal operating points occur for a non-vacuum state inside the interferometer, and 2. measurement of the total photon number operator does not provide an estimate of the relative or dynamical phase with precision that saturates the quantum Cramer-Rao bound, whereas an observable based on weighted shift operators becomes optimal as

g

increases. The results indicate a first-principles approach for describing general optical quantum sensors containing multiple optical downconversion processes.