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ℋ₂ Model Reduction for Linear Quantum Systems

Guangpu Wu, Shibei Xue, Zhang, Guofeng, Ian Richard Petersen

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this paper, an ℋ2 model reduction method for linear quantum systems is presented, which can obtain a physically realizable model with a reduced order for closely approximating the original system. The model reduction problem is formulated as an optimization problem, which minimizes the ℋ2 norm of the difference between the transfer functions of the original and reduced models. Different from classical model reduction, physical realizability conditions for guaranteeing that the reduced-order system is also a quantum system should be taken as non-convex constraints in the optimization. Based on the special properties of quantum systems, we derive necessary conditions for ℋ2 optimality. Moreover, we employ a matrix inequality approach to transform nonlinear inequality constraints into readily solvable linear matrix inequalities (LMIs) and nonlinear equality constraints, so that it can be solved by a lifting variables approach. We emphasize that different from existing works, which only introduces a criterion to evaluate the performance after model reduction, we guide our method to obtain an optimal reduced model with respect to the ℋ2 norm. Finally, a comparison with the results in the existing works validates the efficacy of the proposed method.

DOI
10.1109/tac.2026.3689010

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In this paper, an ℋ2 model reduction method for linear quantum systems is presented, which can obtain a physically realizable model with a reduced order for closely approximating the original system. The model reduction problem is formulated as an optimization problem, which minimizes the ℋ2 norm of the difference between the transfer functions of the original and reduced models. Different from classical model reduction, physical realizability conditions for guaranteeing that the reduced-order system is also a quantum system should be taken as non-convex constraints in the optimization. Based on the special properties of quantum systems, we derive necessary conditions for ℋ2 optimality. Moreover, we employ a matrix inequality approach to transform nonlinear inequality constraints into readily solvable linear matrix inequalities (LMIs) and nonlinear equality constraints, so that it can be solved by a lifting variables approach. We emphasize that different from existing works, which only introduces a criterion to evaluate the performance after model reduction, we guide our method to obtain an optimal reduced model with respect to the ℋ2 norm. Finally, a comparison with the results in the existing works validates the efficacy of the proposed method.
H2\mathscr {H}_{2}H2​ Model Reduction for Linear Quantum Systems
In this paper, an H2\mathscr{H}_2H2​ norm-based model reduction method for linear quantum systems is presented, which can obtain a physically realizable model with a reduced order for closely approximating the original system. The model reduction problem is described as an optimization problem, whose objective is taken as an H2\mathscr{H}_2H2​ norm of the difference between the transfer function of the original system and that of the reduced one. Different from classical model reduction problems, physical realizability conditions for guaranteeing that the reduced-order system is also a quantum system should be taken as nonlinear constraints in the optimization. To solve the optimization problem with such nonlinear constraints, we employ a matrix inequality approach to transform nonlinear inequality constraints into readily solvable linear matrix inequalities (LMIs) and nonlinear equality constraints, so that the optimization problem can be solved by a lifting variables approach. We emphasize that different from existing work, which only introduces a criterion to evaluate the performance after model reduction, we guide our method to obtain an optimal reduced model with respect to the H2\mathscr{H}_2H2​ norm. In addition, the above approach for model reduction is extended to passive linear quantum systems. Finally, examples of active and passive linear quantum systems validate the efficacy of the proposed method.
H2\mathscr{H}_2H2​ Model Reduction for Augmented Model of Linear Non-Markovian Quantum Systems
An augmented system model provides an effective way to model non-Markovian quantum systems, which is useful in filtering and control for this class of systems. However, since a large number of ancillary quantum oscillators representing internal modes of a non-Markovian environment directly interact with the principal system in these models, the dimension of the augmented system may be very large causing significant computational burden in designing filters and controllers. In this context, this paper proposes an H2\mathscr{H}_2H2​ model reduction method for the augmented model of linear non-Markovian quantum systems. We first establish necessary and sufficient conditions for the physical realizability of the augmented model of linear non-Markovian quantum systems, which are more stringent than those for Markovian quantum systems. However, these physical realizability conditions of augmented system model pose non-convex constrains in the optimization problem of model reduction, which makes the problem different from the corresponding classical model reduction problem. To solve the problem, we derive necessary conditions for determining the input matrix in the reduced model, with which a theorem for designing the system matrix of the ancillary system in the reduced system is proved. Building on this, we convert the nonlinear equality constraints into inequality constraints so that a semidefinite programming algorithm can be developed to solve the optimization problem for model reduction. A numerical example of a two-mode linear quantum system driven by three internal modes of a non-Markovian environment validates the effectiveness of our method.
In augmented system model of non-Markovian quantum systems, ancillary oscillators are introduced to capture the internal modes of a non-Markovian environment, which would greatly increase the dimension of the augmented system model. In this paper, we focus on a single-oscillator principal system and propose a model reduction method for an augmented system model of a non-Markovian quantum oscillator, where the reduced model should satisfy physical realizability conditions for linear quantum systems. We consider the model reduction problem as minimizing the H2{{\mathcal{H}}_2}H2​ norm between the original augmented system model and the reduced one under constraints including nonlinear physical realizability conditions. We relax the nonlinear constraints so as to convert the H2{{\mathcal{H}}_2}H2​ optimization problem to a linear matrix inequality problem which can be solved using a matrix lifting variable approach. Finally, an example is given to show the effectiveness of our approach.
The H2\mathcal{H}_2H2​ model reduction problem for high-dimensional linear quantum systems is studied under the constraint of physical realizability (PR). This constraint requires preservation of the canonical commutation relations and the quantum input-output structure, and therefore prevents the direct use of standard projection methods. A symplectic Petrov-Galerkin framework is presented, in which reduced-order models automatically satisfy the PR identities by construction. Within this framework, a symplectic variant of the iterative rational Krylov algorithm is developed and referred to as Quantum IRKA (Q-IRKA). At each iteration, an enriched tangential rational Krylov pool is generated from shifted linear solves. A symplectic basis is then extracted by a Gram-Schmidt-type procedure, paired with symplectic conjugates, and normalized so that the reduced trial space satisfies the canonical symplectic constraint. The interpolation points are updated from selected mirror images of the poles of the current reduced-order model, while the reduced-order matrices are obtained exclusively by structure-preserving projection. Numerical experiments on low-channel oscillator-chain systems and on a bosonic Kitaev-chain-inspired benchmark show that Q-IRKA is effective for large-scale linear quantum systems. Symplecticity and PR are preserved to machine precision, and accurate reduced-order models are obtained with moderate computational cost. The results also show that reduction quality depends substantially on dissipation geometry, channel placement, heterogeneity, and reduced order. These findings indicate that scalable H2\mathcal{H}_2H2​ model reduction of linear quantum systems can be achieved while strictly preserving the underlying physical structure.
A coherent quantum controller is itself a quantum system that is required to be physically realizable. Thus, additional non-linear and linear constraints must be imposed on the coefficients of a physically realizable quantum controller, which differs the quantum Linear Quadratic Gaussian (LQG) design from the standard LQG problem. The purpose of this paper is to propose one numerical procedure based on extended linear matrix inequality (LMI) approach and new physical realizability conditions proposed in [14] to design a coherent quantum controller. The extended synthesis linear matrix inequalities are, in addition to new analysis tools, less conservative in comparison to the conventional counterparts since the optimization variables related to the system parameters in extended LMIs are independent of the symmetric Lyapunov matrix. These features may be useful in the optimal design of quantum optical networks. For comparison, we apply our numerical procedure to the same example given in [9].
This paper derives conditions under which co-diagonalization of the controllability and observability Gramians can be performed in a physically consistent manner for the class of linear quantum stochastic systems that are common in the fields of quantum optics and related areas like quantum nano- and optomechanical systems. Such a class is essentially a quantum analogue of classical (non-quantum) linear stochastic systems. However, unlike their classical counterparts, the laws of quantum mechanics impose nonlinear equality constraints on the system matrices of linear quantum stochastic systems, the so-called physical realizability constraints. Therefore, a meaningful model reduction procedure must yield a physically realizable reduced model satisfying these constraints. Previous work has shown that model reduction by subsystem truncation preserves physical realizability, but did not propose a method for truncating a subsystem. One candidate approach for this truncation is a quantum adaptation of the well-known balanced truncation method for classical linear time-invariant systems. It is shown in this paper that balanced realization for linear quantum stochastic systems is only possible under a strongly restrictive necessary and sufficient condition. However, the paper also derives less restrictive necessary and sufficient conditions for other realizations with simultaneously diagonal controllability and observability Gramians, and introduces the notion of a quasi-balanced realization. An example of the application of the results is provided to demonstrate quasi-balanced truncation in the linear quantum stochastic setting.