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arXiv:2308.15805 · Automatica

Design of coherent passive quantum equalizers using robust control theory

Valery Ugrinovskii, M. R. James

Published Feb 27, 2024• 3 citations
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Abstract

The paper develops a methodology for the design of coherent equalizing filters for quantum communication channels. Given a linear quantum system model of a quantum communication channel, the aim is to obtain another quantum system which, when coupled with the original system, mitigates degrading effects of the environment. The main result of the paper is a systematic equalizer synthesis algorithm which relies on methods of state–space robust control design via semidefinite programming.

DOI
10.1016/j.automatica.2024.111599

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The paper develops a methodology for the design of coherent equalizing filters for quantum communication channels. Given a linear quantum system model of a quantum communication channel, the aim is to obtain another quantum system which, when coupled with the original system, mitigates degrading effects of the environment. The main result of the paper is a systematic equalizer synthesis algorithm which relies on methods of state–space robust control design via semidefinite programming.
This paper introduces a H∞-like methodology of coherent filtering for equalization of passive linear quantum systems to help mitigate degrading effects of quantum communication channels. For such systems, which include a wide range of linear quantum optical devices and signals, we seek to find a near optimal equalizing filter which is itself a passive quantum system. The problem amounts to solving an optimization problem subject to constraints dictated by the requirement for the equalizer to be physically realizable. By formulating these constraints in the frequency domain, we show that the problem admits a convex H∞-like formulation. This allows us to derive a set of suboptimal coherent equalizers using J-spectral factorization. An additional semidefinite relaxation combined with the Nevanlinna–Pick interpolation is shown to lead to a tractable algorithm for the design of a near optimal coherent equalizer.
The coherent equalization problem consists in designing a quantum system acting as a mean-square near-optimal filter for a given quantum communication channel. The paper develops an improved method for the synthesis of transfer functions for such equalizing filters, based on a linear quantum system model of the channel. The method draws on a connection with the two-disk problem of H ∞ control for classical (i.e., non-quantum) linear uncertain systems. Compared with the previous methods, the proposed method applies to a broader class of linear quantum communication channels.
The coherent equalization problem involves designing a quantum system that serves as a mean-square near-optimal filter for a given quantum communication channel. Previous work on coherent equalization has typically assumed that the system parameters are known without uncertainty. In this work, we address the problem of coherent passive equalization with uncertainty in the Hamiltonian. Utilizing an S-procedure method, we remodel the passive quantum system with uncertainty as a scaled closed-loop state-space representation with parameter dependencies. We then propose a two-disk synthesis approach for coherent passive equalizers. Numerical examples based on a single optical cavity system and on two-cavity systems with beam splitters demonstrate the effectiveness of the proposed two-disk synthesis method for coherent equalization of passive quantum systems.
Realizing coherent quantum computation requires precise and robust manipulation of quantum systems through quantum control protocols. Most quantum control techniques rely on heuristic methods for designing the driving pulses that steer the system towards a target state. Such methods are often based on brute-force optimization and offer limited understanding of the solution landscape. In contrast, quantum algorithms offer a rich body of analytical methods with rigorous error guarantees for implementing unitary and non-unitary transformations, which suggests a promising direction for developing new approaches to quantum control. Among various such algorithms, quantum signal processing (QSP) has emerged as a powerful framework for quantum algorithm design, implementation, and optimization. However, its potential for quantum control remains largely unexplored. In this work, we establish QSP-Control, an analytical framework for quantum control of qubit-oscillator dynamics. We focus on dispersively coupled qubit-oscillator systems and employ the QSP formalism to mitigate unwanted nonlinear effects arising from cross-Kerr interactions. In addition, we develop constructions for precise manipulation of Fock states by designing Fock-state-selective operators, based on structural parallels between the Jaynes-Cummings interaction and QSP. These findings demonstrate how several practically relevant problems in quantum control can be mapped to forms amenable to QSP, offering both a systematic design framework and an interpretable perspective on quantum control.
The coherent robust control problem for a class of linear quantum passive systems with model uncertainties is considered in this study, where both the plant and the controller are described by quantum stochastic differential equations (QSDEs). In the framework of language, the model uncertainties are translated into the uncertainties in the Hamiltonian, the coupling operators, and the scattering matrices. Based on the quantum bounded real lemma, the robust controller design of the original uncertain plant is transformed into the controller design of a new scaled system without model uncertainties. Further, the latter is transformed into the solving problem of a couple of Riccati equations and therefore the numerical solutions of the coefficient matrices of QSDEs associated with the controller dynamics are obtained. Finally, numerical simulations on an optical system coupled to three optical channels are performed to verify the effectiveness of the method proposed in this study.
The paper considers the problem of equalization of passive linear quantum systems. While our previous work was concerned with the analysis and synthesis of passive equalizers, in this paper we analyze coherent quantum equalizers whose annihilation (respectively, creation) operator dynamics in the Heisenberg picture are driven by both quadratures of the channel output field. We show that the characteristics of the input field must be taken into consideration when choosing the type of the equalizing filter. In particular, we show that for thermal fields allowing the filter to process both quadratures of the channel output may not improve mean square accuracy of the input field estimate, in comparison with passive filters. This situation changes when the input field is `squeezed'.