  This paper is concerned with quadratic-exponential moments (QEMs) for dynamic
variables of quantum stochastic systems with position-momentum type canonical
commutation relations. The QEMs play an important role for statistical
``localisation'' of the quantum dynamics in the form of upper bounds on the
tail probability distribution for a positive definite quadratic function of the
system variables. We employ a randomised representation of the QEMs in terms of
the moment-generating function (MGF) of the system variables, which is averaged
over its parameters using an auxiliary classical Gaussian random vector. This
representation is combined with a family of weighted $L^2$-norms of the MGF,
leading to upper bounds for the QEMs of the system variables. These bounds are
demonstrated for open quantum harmonic oscillators with vacuum input fields and
non-Gaussian initial states.
