  This paper is concerned with exponential moments of integral-of-quadratic
functions of quantum processes with canonical commutation relations of
position-momentum type. Such quadratic-exponential functionals (QEFs) arise as
robust performance criteria in control problems for open quantum harmonic
oscillators (OQHOs) driven by bosonic fields. We develop a randomised
representation for the QEF using a Karhunen-Loeve expansion of the quantum
process on a bounded time interval over the eigenbasis of its two-point
commutator kernel, with noncommuting position-momentum pairs as coefficients.
This representation holds regardless of a particular quantum state and employs
averaging over an auxiliary classical Gaussian random process whose covariance
operator is specified by the commutator kernel. This allows the QEF to be
related to the moment-generating functional of the quantum process and computed
for multipoint Gaussian states. For stationary Gaussian quantum processes, we
establish a frequency-domain formula for the QEF rate in terms of the Fourier
transform of the quantum covariance kernel in composition with trigonometric
functions. A differential equation is obtained for the QEF rate with respect to
the risk sensitivity parameter for its approximation and numerical computation.
The QEF is also applied to large deviations and worst-case mean square cost
bounds for OQHOs in the presence of statistical uncertainty with a quantum
relative entropy description.
