  This paper extends the energy-based version of the stochastic linearization
method, known for classical nonlinear systems, to open quantum systems with
canonically commuting dynamic variables governed by quantum stochastic
differential equations with non-quadratic Hamiltonians. The linearization
proceeds by approximating the actual Hamiltonian of the quantum system by a
quadratic function of its observables which corresponds to the Hamiltonian of a
quantum harmonic oscillator. This approximation is carried out in a mean square
optimal sense with respect to a Gaussian reference quantum state and leads to a
self-consistent linearization procedure where the mean vector and quantum
covariance matrix of the system observables evolve in time according to the
effective linear dynamics. We demonstrate the proposed Hamiltonian-based
Gaussian linearization for the quantum Duffing oscillator whose Hamiltonian is
a quadro-quartic polynomial of the momentum and position operators. The results
of the paper are applicable to the design of suboptimal controllers and filters
for nonlinear quantum systems.
