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IEEE Conference on Decision and Control (CDC)

Model Reduction for an Augmented System Model of a Non-Markovian Quantum Oscillator

Guangpu Wu, Hao Liu, Shibei Xue

Published Dec 9, 2025• 0 citations
Publisher

Abstract

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 ${{\mathcal{H}}_2}$ 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 ${{\mathcal{H}}_2}$ 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.

DOI
10.1109/cdc57313.2025.11312200

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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.
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.
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.
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.
This brief presents an augmented Markovian system framework that can be applied to model non-Markovian quantum systems. In this augmented system model, ancillary systems are designed to play the role of internal modes of the non-Markovian environment converting white noise to colored noise. To capture the non-classical mutual influence between a quantum system and non-Markovian environments, direct interactions between a principal system and the ancillary system are introduced. Consequently, we showed that the augmented Markovian formulation can be used to theoretically model the environment for observed non-Markovian behavior in a recent experiment on quantum dots.
In this paper, a quantum filter for estimating the states of a non-Markovian qubit system is presented in an augmented Markovian system framework including both the qubit system of interest and multi-ancillary systems for representing the internal modes of the non-Markovian environment. The colored noise generated by the multi-ancillary systems disturbs the qubit system via a direct interaction. The resulting non-Markovian dynamics of the qubit is determined by a memory kernel function arising from the dynamics of the ancillary system. In principle, colored noise with arbitrary power spectrum can be generated by a combination of Lorentzian noises. Hence, the quantum filter can be constructed for the qubit disturbed by arbitrary colored noise and the conditional state of the qubit system can be obtained by tracing out the multi-ancillary systems.
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.