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Chinese Control Conference (CCC)

Rapid Preparation of Bell States Based on an Exponential Quantum Projection Filter

Shumin Zhou, Gan Li, Daoyi Dong, Sen Kuang

Published Jul 25, 2022• 0 citations
Publisher

Abstract

The exponential quantum projection filtering algorithm is applied in measurement-based quantum feedback control and a specific switching control law is designed to realize the rapid preparation of a target Bell state. First, we use an exponential projection filter to obtain an approximate estimate of the system state, which can reduce the calculation time spent in the measurement process. Then, the quantum state space is partitioned into two subsets to separate the target state from other equilibrium states, two Lyapunov functions are constructed and the corresponding control laws are designed in the two subsets. It is shown that the switching control scheme in this paper ensures that the system trajectory crosses the boundary of the two subsets no more than twice from any initial state, which effectively reduces the time that the system converges to the target Bell state. Numerical simulation on a two-qubit system verifies the effectiveness of the proposed control scheme.

DOI
10.23919/ccc55666.2022.9901567

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The exponential quantum projection filtering algorithm is applied in measurement-based quantum feedback control and a specific switching control law is designed to realize the rapid preparation of a target Bell state. First, we use an exponential projection filter to obtain an approximate estimate of the system state, which can reduce the calculation time spent in the measurement process. Then, the quantum state space is partitioned into two subsets to separate the target state from other equilibrium states, two Lyapunov functions are constructed and the corresponding control laws are designed in the two subsets. It is shown that the switching control scheme in this paper ensures that the system trajectory crosses the boundary of the two subsets no more than twice from any initial state, which effectively reduces the time that the system converges to the target Bell state. Numerical simulation on a two-qubit system verifies the effectiveness of the proposed control scheme.
For stochastic quantum systems with measurement feedback, this article proposes a rapid switching control scheme based on state space partition and realizes the rapid stabilization of an eigenstate of an observable operator. Meanwhile, we apply the proposed scheme to the preparation of typical entangled states in multiqubit systems. In view of the convergence obstacle caused by the symmetric structure of the state space, especially in the case with degenerate observable operators, we first partition the state space into a subset containing the target state and its complement to distinguish the target state from its antipodal points, and then design the corresponding control laws in these two subsets, respectively, by using different Lyapunov functions. The interaction Hamiltonians are also constructed to drive the system state to the desired subset first, and further to the target state. In particular, the control law designed in the undesired subset guarantees the strictly monotonic descent of the corresponding Lyapunov function, which makes the system trajectory switch between the two subsets at most twice and has the potential to speed up the convergence process. We also prove the stability of the closed-loop system with the proposed switching control law based on the stochastic Lyapunov stability theory. By applying the proposed switching control scheme to a three-qubit system, we achieve the preparation of a GHZ state and a W state.
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Bell states are the maximally entangled states in two-qubit quantum systems, and play a significant role in quantum computation and quantum communication. Feedback control of stochastic quantum systems usually suffers from time-delay problems caused by the computation time for filter states and the control input. This paper addresses the preparation of Bell states in two-qubit systems with a constant delay time. A Lyapunov method is used to design a switching control law and a constant is introduced to compensate for the computation time of the estimated states and the feedback control input, thereby stabilizing the target Bell state globally. Numerical results show the effectiveness of the proposed control law.
In this paper, we consider a two-qubit system undergoing continuous-time measurements. In presence of multiple channels, we provide sufficient conditions on the continuous feedback control law ensuring almost sure exponential convergence to a predetermined Bell state. This is obtained by applying stochastic tools, Lyapunov methods and geometric control tools. With one channel, we establish asymptotic convergence towards a predetermined Bell state. In both cases, we provide explicit expressions of feedback control laws satisfying the above-mentioned conditions. Finally, we demonstrate the effectiveness of our methodology through numerical simulations.
Abstract. This paper considers a simplified model of open quantum systems undergoing imperfect measurements obtained via a projection filter approach. We use this approximate filter in the feedback stabilization problem, specifically in the case of quantum nondemolition (QND) measurements. The feedback design relies on the structure of the exponential family utilized for the projection process. We demonstrate that the introduced feedback guarantees exponential convergence of the original filter equation toward a predefined target state, corresponding to an eigenstate of the measurement operator.
The stabilization of quantum states is a fundamental problem for realizing various quantum technologies. Measurement-based-feedback strategies have demonstrated powerful performance, and the construction of quantum control signals using measurement information has attracted great interest. However, the interaction between quantum systems and the environment is inevitable, especially when measurements are introduced, which leads to decoherence. To mitigate decoherence, it is desirable to stabilize quantum systems faster, thereby reducing the time of interaction with the environment. In this paper, we utilize information obtained from measurement and apply deep reinforcement learning (DRL) algorithms, without explicitly constructing specific complex measurement-control mappings, to rapidly drive random initial quantum state to the target state. The proposed DRL algorithm has the ability to speed up the convergence to a target state, which shortens the interaction between quantum systems and their environments to protect coherence. Simulations are performed on two-qubit and three-qubit systems, and the results show that our algorithm can successfully stabilize random initial quantum system to the target entangled state, with a convergence time faster than traditional methods such as Lyapunov feedback control and several DRL algorithms with different reward functions. Moreover, it exhibits robustness against imperfect measurements and delays in system evolution.