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arXiv:2501.13885 · Annales Henri Poincaré

Quantum Model Reduction for Continuous-Time Quantum Filters

Tommaso Grigoletto, Clément Pellegrini, Francesco Ticozzi

Published Sep 17, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract The use of quantum stochastic models is widespread in dynamical reduction, simulation of open systems, feedback control and adaptive estimation. In many applications only part of the information contained in the filter’s state is actually needed to reconstruct the target observable quantities; thus, filters of smaller dimensions could be in principle implemented to perform the same task. In this work, we propose a systematic method to find, when possible, reduced-order quantum filters that are capable of exactly reproducing the evolution of expectation values of interest. In contrast with existing reduction techniques, the reduced model we obtain is exact and in the form of a Belavkin filtering equation, ensuring physical interpretability. This is attained by leveraging tools from the theory of both minimal realization and non-commutative conditional expectations. The proposed procedure is tested on prototypical examples, laying the groundwork for applications in quantum trajectory simulation and quantum feedback control.

DOI
10.1007/s00023-025-01622-7

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Abstract The use of quantum stochastic models is widespread in dynamical reduction, simulation of open systems, feedback control and adaptive estimation. In many applications only part of the information contained in the filter’s state is actually needed to reconstruct the target observable quantities; thus, filters of smaller dimensions could be in principle implemented to perform the same task. In this work, we propose a systematic method to find, when possible, reduced-order quantum filters that are capable of exactly reproducing the evolution of expectation values of interest. In contrast with existing reduction techniques, the reduced model we obtain is exact and in the form of a Belavkin filtering equation, ensuring physical interpretability. This is attained by leveraging tools from the theory of both minimal realization and non-commutative conditional expectations. The proposed procedure is tested on prototypical examples, laying the groundwork for applications in quantum trajectory simulation and quantum feedback control.
Leveraging an algebraic approach built on minimal realizations and conditional expectations in quantum probability, we propose a method to reduce the dimension of quantum filters in discrete-time, while maintaining the correct distributions on the measurement outcomes and the expectations of some relevant observable. The method is presented for general quantum systems whose dynamics depend on measurement outcomes, hinges on a system-theoretic observability analysis, and is tested on prototypical examples.
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.
In practical applications, quantum systems are inevitably subject to significant uncertainties, including unknown initial states, imprecise physical parameters, and unmodeled environmental noise, all of which pose major challenges to robust quantum feedback control. This paper proposes a feedback stabilization strategy based on a reduced quantum filter that achieves robustness against time-varying Hamiltonian perturbations and additional dissipative effects, without requiring prior knowledge of the initial state or exact system parameters. The proposed filter estimates only O(N) real variables corresponding to the diagonal elements of the system density matrix in a quantum non-demolition basis in contrast to the O(N^2) variables required by a full stochastic master equation, where N is the Hilbert space dimension. This dimensionality reduction substantially simplifies real-time computation and feedback implementation while preserving both convergence and robustness guarantees. Rigorous analysis further establishes global exponential stability of the target subspace. The results provide a scalable framework for robust and efficient measurement-based feedback control applicable to high-dimensional perturbed open quantum systems.
A general approach to obtain reduced models for a wide class of discrete-time quantum systems is proposed. The obtained models not only reproduce exactly the output of a given quantum model, but are also guaranteed to satisfy physical constraints, namely complete positivity and preservation of total probability. A fundamental framework for exact model reduction of quantum systems is constructed leveraging on algebraic methods, as well as novel results on quantum conditional expectations in finite-dimensions. The proposed reduction algorithm is illustrated and tested on prototypical examples, including the quantum walk realizing Grover’s algorithm.
Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.
This manuscript presents an advanced generalization of the Belavkin quantum filter, applying it to Boson and Fermion noise via the Hudson-Parthasarathy quantum stochastic calculus. It focuses on estimating quantum system observables and states in noisy conditions, highlighting the transition from non-commutative to commutative observables post-filtering. The work introduces an algorithm for constructing quantum filter coefficients by solving an infinite series of linear algebraic equations, enabling real-time filtering in the observable domain. This development offers significant insights into quantum noise analysis and state estimation, with potential applications in quantum computing and information processing. The manuscript also explores the integration of classical noisy measurements with quantum non-demolition measurements, creating a hybrid estimation framework that enhances quantum filtering capabilities. The work has potential applications in estimating the time varying state of a quantum gravitational wave from measurements on electromagnetic noise fields with which the wave interacts.