  This paper combines probabilistic and algebraic techniques for computing
quantum expectations of operator exponentials (and their products) of quadratic
forms of quantum variables in Gaussian states. Such quadratic-exponential
functionals (QEFs) resemble quantum statistical mechanical partition functions
with quadratic Hamiltonians and are also used as performance criteria in
quantum risk-sensitive filtering and control problems for linear quantum
stochastic systems. We employ a Lie-algebraic correspondence between complex
symplectic matrices and quadratic-exponential functions of system variables of
a quantum harmonic oscillator. The complex symplectic factorizations are used
together with a parametric randomization of the quasi-characteristic or
moment-generating functions according to an auxiliary classical Gaussian
distribution. This reduces the QEF to an exponential moment of a quadratic form
of classical Gaussian random variables with a complex symmetric matrix and is
applicable to recursive computation of such moments.
