Quadratic-exponential functionals of Gaussian quantum processes

TL;DR

This paper develops a mathematical framework for analyzing exponential moments of quadratic functions of Gaussian quantum processes, with applications to control and performance evaluation of quantum harmonic oscillators.

Contribution

It introduces a novel randomised representation of quadratic-exponential functionals using Karhunen-Loeve expansion, applicable to general quantum states and enabling frequency-domain analysis.

Findings

Derived a frequency-domain formula for the QEF rate.

Established a differential equation for the QEF rate with respect to risk sensitivity.

Applied the QEF to large deviations and worst-case cost bounds in quantum control.

Abstract

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.