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.

中文翻译：

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高斯量子态的参数随机化、复辛因式分解和二次指数泛函

本文结合概率和代数技术来计算高斯态量子变量二次形式的算子指数（及其乘积）的量子期望。这种二次指数泛函 (QEF) 类似于具有二次哈密顿量的量子统计力学配分函数，并且还用作线性量子随机系统的量子风险敏感过滤和控制问题的性能标准。我们采用复辛矩阵与量子谐振子的系统变量的二次指数函数之间的李代数对应。根据辅助经典高斯分布，复辛因式分解与准特征或矩生成函数的参数随机化一起使用。