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IEEE Transactions on Automatic Control

Feedback Control of Quantum Systems to Stabilize Superposition States

Abolghasem Daeichian

Published Jan 17, 2023• 8 citations
Publisher

Abstract

As a specific trait of quantum mechanics, quantum superposition is exploited in many applications such as quantum computation. This paper proposes a feedback scheme to stabilize the desired quantum superposition states. To this aim, a new factorization is derived for the stochastic master equation of a quantum system which is driven by the field in the superposition of coherent states. Then, a feedback scheme is proposed, which consists of a quantum system driven by the field in the superposition of coherent states and a classical feedback channel proportional to the observations of a Homodyne detector. Afterward, the quantum filter of the closed loop system is attained using the SLH framework. Finally, the factorization of the closed loop quantum filter is obtained and the control law is calculated such that the proposed factorization satisfies the stabilization condition. Finally, as an example, the control method is simulated for a two-level atom and the stabilization of the superposition states is achieved.

DOI
10.1109/tac.2023.3237504

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As a specific trait of quantum mechanics, quantum superposition is exploited in many applications such as quantum computation. This paper proposes a feedback scheme to stabilize the desired quantum superposition states. To this aim, a new factorization is derived for the stochastic master equation of a quantum system which is driven by the field in the superposition of coherent states. Then, a feedback scheme is proposed, which consists of a quantum system driven by the field in the superposition of coherent states and a classical feedback channel proportional to the observations of a Homodyne detector. Afterward, the quantum filter of the closed loop system is attained using the SLH framework. Finally, the factorization of the closed loop quantum filter is obtained and the control law is calculated such that the proposed factorization satisfies the stabilization condition. Finally, as an example, the control method is simulated for a two-level atom and the stabilization of the superposition states is achieved.
Quantum systems can be controlled by other quantum systems in a reversible way, without any information leaking to the outside of the system-controller compound. Such coherent quantum control is deterministic, is less noisy than measurement-based feedback control, and has potential applications in a variety of quantum technologies, including quantum computation, quantum communication, and quantum metrology. Here we introduce a coherent feedback protocol, consisting of a sequence of identical interactions with controlling quantum systems, that steers a quantum system from an arbitrary initial state towards a target state. We determine the broad class of such coherent feedback channels that achieve convergence to the target state, and then stabilize as well as protect it against noise. Our results imply that also weak system-controller interactions can counter noise if they occur with suitably high frequency. We present an example of a control scheme that does not require knowledge of the target state encoded in the controllers, which could be the result of a quantum computation. It thus provides a mechanism for autonomous, purely quantum closed-loop control.
Summary form only given. Decoherence in quantum systems can be loosely defined as loss of purity. Therefore the ultimate success in using feedback to fight decoherence would be to create an arbitrary stable pure state in the presence of dissipation. This goal was realized by Hoffman, Mahler and Hess (1998) for a very simple system: a resonantly driven two-level atom. They showed that by making part of the coherent driving proportional to the homodyne photocurrent can stabilize the state to any point on the bottom half of the sphere. Here we reanalyze their proposal by using the technique of stochastic master equations, allowing their results to be generalized in three ways.
Abstract. No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
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.