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arXiv:2608.16608

Discretisation of quantum feedback networks for implementation on a quantum computer

Luc Bouten, John Gough

quant-phSubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations
v1 · Aug 17, 2026v2 · Aug 22, 2026v3 · Aug 30, 2026
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Abstract

Quantum Feedback Network Theory (also known colloquially as the SLH-framework) is a powerful tool for modeling systems in quantum optics. This paper describes a method for discretising SLH-networks such that they can be implemented on a quantum computer. Building on a discretisation theorem for quantum stochastic differential equations, we establish strong convergence, uniformly on compact time intervals, of the discretised system to the continuous SLH-network as the discretisation mesh goes to zero.

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32 pages, 1 figure

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Quantum Feedback Network Theory (also known colloquially as the SLH-framework) is a powerful tool for modeling systems in quantum optics. This paper describes a method for discretising SLH-networks such that they can be implemented on a quantum computer. Building on a discretisation theorem for quantum stochastic differential equations, we establish strong convergence, uniformly on compact time intervals, of the discretised system to the continuous SLH-network as the discretisation mesh goes to zero.
Many emerging quantum technologies demand precise engineering and control over networks consisting of quantum mechanical degrees of freedom connected by propagating electromagnetic fields, or quantum input-output networks. Here we review recent progress in theory and experiment related to such quantum input-output networks, with a focus on the SLH framework, a powerful modeling framework for networked quantum systems that is naturally endowed with properties such as modularity and hierarchy. We begin by explaining the physical approximations required to represent any individual node of a network, e.g. atoms in cavity or a mechanical oscillator, and its coupling to quantum fields by an operator triple (S,L,H). Then we explain how these nodes can be composed into a network with arbitrary connectivity, including coherent feedback channels, using algebraic rules, and how to derive the dynamics of network components and output fields. The second part of the review discusses several extensions to the basic SLH framework that expand its modeling capabilities, and the prospects for modeling integrated implementations of quantum input-output networks. In addition to summarizing major results and recent literature, we discuss the potential applications and limitations of the SLH framework and quantum input-output networks, with the intention of providing context to a reader unfamiliar with the field.
A scheme making use of an isolated feedback loop was recently proposed in \cite{GP_} for creating an arbitrary bilinear Hamiltonian interaction between two multi-mode Linear Quantum Stochastic Systems (LQSSs). In this work we examine the presence of an isolated feedback loop in a general SLH network, and derive the modified Hamiltonian of the network due to the presence of the loop. In the case of a bipartite network with an isolated loop running through both parts, this results in modified Hamiltonians for each subnetwork, as well as a Hamiltonian interaction between them. As in the LQSS case, by engineering appropriate ports in each subnetwork, we may create desired interactions between them. Examples are provided that illustrate the general theory.
We express the rules for forming quantum feedback networks using the Stratonovich form of quantum stochastic calculus rather than the Itō or SLH (J. E. Gough and M. R. James, “Quantum feedback networks: Hamiltonian formulation,” Commun. Math. Phys. 287, 1109 (2009), J. E. Gough and M. R. James, “The Series product and its application to quantum feedforward and feedback networks,” IEEE Trans. Autom. Control 54, 2530 (2009)) form. Remarkably the feedback reduction rule implies that we obtain the Schur complement of the matrix of Stratonovich coupling operators where we short out the internal input/output coefficients.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
The mathematical theory of quantum feedback networks has recently been developed by Gough and James \cite{QFN1} for general open quantum dynamical systems interacting with bosonic input fields. In this article we show, that their feedback reduction formula for the coefficients of the closed-loop quantum stochastic differential equation can be formulated in terms of Belavkin matrices. We show that the reduction formula leads to a non-commutative Mobius transformation based on Belavkin matrices, and establish a ⋆\star⋆-unitary version of the Siegel identities.