Frequency-domain computation of quadratic-exponential cost functionals for linear quantum stochastic systems

TL;DR

This paper develops frequency-domain formulas to compute the asymptotic growth rate of quadratic-exponential cost functionals in linear quantum stochastic systems, enhancing robustness analysis and control design.

Contribution

It introduces frequency-domain expressions for the QEF growth rate using spectral functions, enabling efficient analysis of quantum systems' risk-sensitive performance.

Findings

Derived frequency-domain formulas for QEF growth rate

Applied homotopy and contour integration techniques

Provided numerical example with a two-mode oscillator

Abstract

This paper is concerned with quadratic-exponential functionals (QEFs) as risk-sensitive performance criteria for linear quantum stochastic systems driven by multichannel bosonic fields. Such costs impose an exponential penalty on quadratic functions of the quantum system variables over a bounded time interval, and their minimization secures a number of robustness properties for the system. We use an integral operator representation of the QEF, obtained recently, in order to compute its asymptotic infinite-horizon growth rate in the invariant Gaussian state when the stable system is driven by vacuum input fields. The resulting frequency-domain formulas express the QEF growth rate in terms of two spectral functions associated with the real and imaginary parts of the quantum covariance kernel of the system variables. We also discuss the computation of the QEF growth rate using homotopy and contour integration techniques and provide two illustrations including a numerical example with a two-mode oscillator.