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IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics)

Incoherent Control of Quantum Systems With Wavefunction-Controllable Subspaces via Quantum Reinforcement Learning

Daoyi Dong, Chunlin Chen, Tzyh‐Jong Tarn, Alexander Nikolaevich Pechen, HERSCHEL A. RABITZ

Published Jul 24, 2008• 77 citations
Publisher

Abstract

In this paper, an incoherent control scheme for accomplishing the state control of a class of quantum systems which have wavefunction-controllable subspaces is proposed. This scheme includes the following two steps: projective measurement on the initial state and learning control in the wavefunction-controllable subspace. The first step probabilistically projects the initial state into the wavefunction-controllable subspace. The probability of success is sensitive to the initial state; however, it can be greatly improved through multiple experiments on several identical initial states even in the case with a small probability of success for an individual measurement. The second step finds a local optimal control sequence via quantum reinforcement learning and drives the controlled system to the objective state through a set of suitable controls. In this strategy, the initial states can be unknown identical states, the quantum measurement is used as an effective control, and the controlled system is not necessarily unitarily controllable. This incoherent control scheme provides an alternative quantum engineering strategy for locally controllable quantum systems.

DOI
10.1109/tsmcb.2008.926603

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In this paper, an incoherent control scheme for accomplishing the state control of a class of quantum systems which have wavefunction-controllable subspaces is proposed. This scheme includes the following two steps: projective measurement on the initial state and learning control in the wavefunction-controllable subspace. The first step probabilistically projects the initial state into the wavefunction-controllable subspace. The probability of success is sensitive to the initial state; however, it can be greatly improved through multiple experiments on several identical initial states even in the case with a small probability of success for an individual measurement. The second step finds a local optimal control sequence via quantum reinforcement learning and drives the controlled system to the objective state through a set of suitable controls. In this strategy, the initial states can be unknown identical states, the quantum measurement is used as an effective control, and the controlled system is not necessarily unitarily controllable. This incoherent control scheme provides an alternative quantum engineering strategy for locally controllable quantum systems.
A robust incoherent quantum control scheme via projective measurements plus unitary transformations is proposed for driving a qubit system from an unknown initial mixed state to an arbitrary target pure state. This scheme consists of two main steps: projective measurement on the initial mixed state and optimal control between two pure states. The first step projects the initial state into an eigenstate of the qubit system by projective measurement and guarantees that the proposed scheme is robust to different initial mixed states. The second step finds a set of suitable optimal controls to drive the qubit system from the conditional eigenstate to the target pure state. The connection between the two steps is accomplished by a switching strategy. To accomplish the second step, two approaches are presented in detail. These approaches are time-optimal transition with unbounded control and bang-bang control with minimal switches. The minimal time and minimal number of switches in these approaches can be calculated by simple analytical expressions. The proposed approaches provide two relatively straightforward optimal design methods.
An incoherent control scheme for state control of locally controllable quantum systems is proposed. This scheme includes three steps: (1) amplitude amplification of the initial state by a suitable unitary transformation, (2) projective measurement of the amplified state, and (3) final optimization by a unitary controlled transformation. The first step increases the amplitudes of some desired eigenstates and the corresponding probability of observing these eigenstates, the second step projects, with high probability, the amplified state into a desired eigenstate, and the last step steers this eigenstate into the target state. Within this scheme, two control algorithms are presented for two classes of quantum systems. As an example, the incoherent control scheme is applied to the control of a hydrogen atom by an external field. The results support the suggestion that projective measurements can serve as an effective control and local controllability information can be used to design control laws for quantum systems. Thus, this scheme establishes a subtle connection between control design and controllability analysis of quantum systems and provides an effective engineering approach in controlling quantum systems with partial controllability information.
A hybrid control approach for uncertain quantum systems is proposed using probabilistic fuzzy estimators (PFE) and quantum reinforcement learning (QRL). This hybrid control design involves coherent control with PFE and learning control via QRL. The problems of controlling a quantum system from an initial state to a pointed target state are studied in this paper, where we assume that the initial quantum state is a mixed state and the target quantum state is a controllable pure state within a wavefunction controllable subspace. First, the initial quantum system is controlled coherently with the help of a PFE. When the controlled system is estimated to be likely to collapse to an expected eigen state, trigger the measurement and the quantum system collapses to an eigen state in the wavefuntion controllable subspace with a high probability. Then the quantum system is driven to the target state with admissible controls, where the control sequence is learned and optimized with QRL. An example is presented and analyzed to demonstrate the control process.
A rapid incoherent control scheme is proposed for driving a quantum system from an initial mixed state to a final state having a high fidelity with a target pure state. This scheme consists of two main steps: rapid purification of the initial mixed state and time-optimal control of the conditional state. The first step rapidly purifies the initial mixed state into an almost-pure state (conditional state) through continuous measurements and feedback control. The second step finds a set of suitable time-optimal controls through the control design of a reference model to drive the quantum system from the conditional state to the final state. The switching time between the two steps is determined by the expected purity or fidelity and the resultant state is an almost-pure state having a high fidelity with the target pure state. As an illustrative example, the authors apply the present rapid incoherent control scheme to a spin system. The potential applications of model predictive control to quantum systems are also discussed briefly.
A rapid incoherent control scheme for driving a simple qubit system from an initial mixed state to an arbitrary target pure state is proposed. This scheme consists of two main steps: rapid purification of the initial mixed state and time optimal transition from the conditional state to the target state. The first step rapidly purifies the initial mixed state into an almost-pure state of the simple qubit system through continuous measurements and feedback control by the protocol proposed in Jacobs 2003 Physical Review A 67 030301(R). The second step finds a set of suitable time optimal controls to drive the simple qubit system from the conditional state to the target state. The switching time between the first step and the second step is determined by the expected purity and the resultant state is an almost-pure state with a high fidelity with the target state. This rapid incoherent control scheme provides an alternative quantum engineering strategy for the rapid preparation of arbitrary pure states of a simple qubit system.
In this thesis, we consider two simple but typical control problems and apply deep reinforcement learning to them, i.e., to cool and control a particle which is subject to continuous position measurement in a one-dimensional quadratic potential or in a quartic potential. We compare the performance of reinforcement learning control and conventional control strategies on the two problems, and show that the reinforcement learning achieves a performance comparable to the optimal control for the quadratic case, and outperforms conventional control strategies for the quartic case for which the optimal control strategy is unknown. To our knowledge, this is the first time deep reinforcement learning is applied to quantum control problems in continuous real space. Our research demonstrates that deep reinforcement learning can be used to control a stochastic quantum system in real space effectively as a measurement-feedback closed-loop controller, and our research also shows the ability of AI to discover new control strategies and properties of the quantum systems that are not well understood, and we can gain insights into these problems by learning from the AI, which opens up a new regime for scientific research.