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arXiv:2605.07152

Symplectic H2 Model Reduction for High-Dimensional Linear Quantum Systems

Alfo Borzi, Guofeng Zhang

quant-pheess.SYmath.NAmath.OCSubmitted May 8, 2026• Announced May 11, 2026
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Abstract

The $\mathcal{H}_2$ 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 $\mathcal{H}_2$ model reduction of linear quantum systems can be achieved while strictly preserving the underlying physical structure.

Comments
28 pages, 9 figures. Comments are welcome,

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H2\mathscr {H}_{2}H2​ Model Reduction for Linear Quantum Systems
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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.
We consider finite-dimensional many-body quantum systems described by time-independent Hamiltonians and Markovian master equations, and present a systematic method for constructing smaller-dimensional, reduced models that e x a c t l y reproduce the time evolution of a set of initial conditions or observables of interest. Our approach exploits Krylov operator spaces and their extension to operator algebras, and may be used to obtain reduced linear models of minimal dimension, well-suited for simulation on classical computers, or reduced quantum models that preserve the structural constraints of physically admissible quantum dynamics, as required for simulation on quantum computers. Notably, we prove that the reduced quantum-dynamical generator is still in Lindblad form. By introducing a new type of observable-dependent symmetries , we show that our method provides a non-trivial generalization of techniques that leverage symmetries, unlocking new reduction opportunities. We quantitatively benchmark our method on paradigmatic open many-body systems of relevance to condensed-matter and quantum-information physics. In particular, we demonstrate how our reduced models can quantitatively describe decoherence dynamics in central-spin systems coupled to structured environments, magnetization transport in boundary-driven dissipative spin chains, and unwanted error dynamics on information encoded in a noiseless quantum code.
This paper presents a model reduction method for the class of linear quantum stochastic systems often encountered in quantum optics and related fields. The approach is proposed on the basis of an interpolatory projection ensuring that specific input-output responses of the original and the reduced-order systems are matched at multiple selected points (or frequencies). Importantly, the physical realizability property of the original quantum system imposed by the laws of quantum mechanics is preserved under our tangential interpolatory projection. Error bounds are established for the proposed model reduction method and an avenue to select interpolation points is proposed. A passivity preserving model reduction method is also presented. Examples of both active and passive systems are provided to illustrate the merits of our proposed approach.
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