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arXiv:2204.00479 · Physical Review A

Unified collision model of coherent and measurement-based quantum feedback

Alfred Harwood, Matteo Brunelli, Alessio Serafini

Published Oct 16, 2023• 4 citations
arXivHTMLPDF

Abstract

We introduce a general framework, based on collision models and discrete completely positive maps, to describe on an equal footing coherent and measurement-based feedback control of quantum mechanical systems. We apply our framework to prominent tasks in quantum control, ranging from cooling to Hamiltonian control. Unlike other proposed comparisons, where coherent feedback always proves superior, we find that either measurements or coherent manipulations of the controller can be advantageous depending on the task at hand. Measurement-based feedback is typically superior in cooling, while coherent feedback is better at assisting quantum operations. Furthermore, we show that both coherent and measurement-based feedback loops allow one to simulate arbitrary Hamiltonian evolutions, and discuss their respective effectiveness in this regard.

DOI
10.1103/physreva.108.042413

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We introduce a general framework, based on collision models and discrete completely positive maps, to describe on an equal footing coherent and measurement-based feedback control of quantum mechanical systems. We apply our framework to prominent tasks in quantum control, ranging from cooling to Hamiltonian control. Unlike other proposed comparisons, where coherent feedback always proves superior, we find that either measurements or coherent manipulations of the controller can be advantageous depending on the task at hand. Measurement-based feedback is typically superior in cooling, while coherent feedback is better at assisting quantum operations. Furthermore, we show that both coherent and measurement-based feedback loops allow one to simulate arbitrary Hamiltonian evolutions, and discuss their respective effectiveness in this regard.
Feedback plays an essential role in classical control theory, which has been investigated in depth. However, in quantum control theory, the study of feedback is still in its infancy. The paper continues to explore the fundamental characteristics of quantum feedback control for a typical coherent model based on the results of [3] by highlighting the significant differences between quantum feedback control and classical feedback control due to quantum measurement. The paper compares the control performance of quantum feedback control and quantum open-loop control in the cases with and without environment-induced decoherence. It concludes that in some sense, the quantum feedback control is still superior to the quantum open-loop control despite the complicated tradeoff in the feedback case, and further points out that how the quantum measurement-induced-collapse can play a positive role in the quantum feedback control.
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 paper first briefly introduces the current research on quantum control. Then the quantum open-loop control (no measurement introduced) and the closed-loop control (measurement introduced) was compared through a standard model of coherent control. When using quantum closed-loop control, we need measurement which inevitably brings in the reduction noise. This is different from the classical control systems. On the other hand, measurement-based feedback can reduce the influence of the system uncertainties. Thus, there is a tradeoff between the noise influence brought in by the measurement and the information got by the measurement. The paper demonstrate that in the case without decoherence, the quantum closed-loop control is superior to the open-loop control in dealing with the uncertainty in some sense. In the case with decoherence, we discuss the controllability of the quantum control through two performance indices.
In the conventional picture of quantum feedback control, sensors perform measurements on the system, a classical controller processes the results of the measurements, and actuators supply semiclassical potentials to alter the behavior of the quantum system. In this picture, the sensors tend to destroy coherence in the process of making measurements, and although the controller can use the actuators to act coherently on the quantum system, it is processing and feeding back classical information. This paper proposes an alternative method for quantum feedback control, in which the sensors, controller, and actuators are quantum systems that interact coherently with the system to be controlled. In this picture, the controller gets, processes, and feeds back quantum information. Controllers that operate using such quantum feedback loops can perform tasks such as entanglement transfer that are not possible using classical feedback. Necessary and sufficient conditions are presented for Hamiltonian quantum systems to be controllable and observable using both classical and quantum feedback.
Abstract We compare the performance of continuous coherent feedback, implemented using an ideal single-qubit controller, to that of continuous measurement-based feedback (MBF) for the task of controlling the state of a single qubit. We take as the basic dynamical resource the ability to couple the system to a quadrature of a traveling-wave field (for example, a transmission line) via a system observable, and the fundamental limitation is the maximum rate that is available for this coupling. We focus on the question of the best achievable control given ideal controllers. To obtain a fair comparison we acknowledge that the amplification involved in measurement-based control allows the controller to use macroscopic fields to apply feedback forces to the system, so it is natural to allow these feedback forces to be much larger than the mesoscopic coupling to the transmission line that mediates both the measurement for measurement-based control and the coupling to the mesoscopic controller for coherent control. Interestingly our numerical results indicate that under the above platform for comparison, coherent feedback can achieve exactly the performance of MBF given ideal controllers. We also discuss two control mechanisms available to coherent feedback.
Feedback is the core concept in cybernetics and its effective use has made great success in but not limited to the fields of engineering, biology, and computer science. When feedback is used to quantum systems, two major types of feedback control protocols including coherent feedback control (CFC) and measurement-based feedback control (MFC) have been developed. In this paper, we compare the two types of quantum feedback control protocols by focusing on the real-time information used in the feedback loop and the capability in dealing with parameter uncertainty. An equivalent relationship is established between quantum CFC and non-selective quantum MFC in the form of operator-sum representation. Using several examples of quantum feedback control, we show that quantum MFC can theoretically achieve better performance than quantum CFC in stabilizing a quantum state and dealing with Hamiltonian parameter uncertainty. The results enrich understanding of the relative advantages between quantum MFC and quantum CFC, and can provide useful information in choosing suitable feedback protocols for quantum systems.