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IEEE Control Systems Letters

Controlling Ensemble States of a Quantum System

Vishesh Kaushik, Navin Khaneja

Published Jan 1, 2025• 4 citations
Publisher

Abstract

We develop a method for controlling the ensemble states of a two-level quantum system on the Bloch sphere. A spectrum of resonance frequencies parametrizes the ensemble. The evolution of the ensemble states under externally applied controls is uniquely determined by their respective resonance frequencies. Therefore, the control fields must be designed to simultaneously evolve the ensemble states across the domain of resonance frequencies. The objective is to evolve the discretized quantum ensemble states from an arbitrary initial dispersion to a targeted final configuration on the Bloch sphere. The proposed method holds potential for broad applicability in areas such as nuclear magnetic resonance (NMR), electron spin resonance (ESR), quantum computing and information processing, sampling-based learning control (SLC), quantum sensing and metrology, and quantum error correction.

DOI
10.1109/lcsys.2025.3573242

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Ensemble control of open quantum systems from one state to another is much more difficult than closed quantum systems due to their interactions with the environment. In this paper, a sampling-based learning control (SLC) method is applied to the optimal control design for inhomogeneous open quantum ensembles regarding the state-to-state transition task. A differential evolution (DE) algorithm is adopted for the training step of the SLC design to find the optimal control for the generalized system constructed by sampling members of the inhomogeneous quantum ensembles. Numerical results, including two-level, three-level and four-level open quantum ensembles, demonstrate the effectiveness of the proposed control approach for open quantum ensembles.
Quantum ensemble classification has significant applications in discrimination of atoms (or molecules), separation of isotopic molecules and quantum information extraction. In this paper, we recast quantum ensemble classification as a supervised quantum learning problem. A systematic classification methodology is presented by using a sampling-based learning control (SLC) approach for quantum discrimination. The classification task is accomplished via simultaneously steering members belonging to different classes to their corresponding target states (e.g., mutually orthogonal states). Numerical results demonstrate the effectiveness of the proposed approach for the discrimination of two quantum systems and the binary classification of two-level quantum ensembles.
Nuclear magnetic resonance (NMR) spectroscopy has proven to be a facile means of achieving small-scale demonstrations of quantum information processing. This was in large part made possible by the sophisticated methods of quantum control that have been developed by the NMR community over a span of more than 50 years. The traditional control methods, already perhaps the most complex examples of open-loop control available, were nevertheless designed primarily to assist in identifying the physical parameters of the underlying spin system, rather than to control the system with high precision. We have therefore extended the traditional methods in a variety of ways so as to achieve precise control, by taking advantage of prior knowledge of the physical parameters as determined by traditional methods. Due to the experimental challenges of real-time control, these developments relied upon our ability to simulate the evolution of the system under the action of radio-frequency control fields with essentially arbitrary precision, subject only to the available computing power. This talk presents an overview of our work in quantum control together with some of the on-going challenges still facing us
In this paper, a new method for controlling a quantum ensemble that its members have uncertainties in Hamiltonian parameters is designed. Based on combining the sampling-based learning control (SLC) and a new quantum genetic algorithm (QGA) method, the control of an ensemble of a two-level quantum system with Hamiltonian uncertainties is achieved. To simultaneously transfer the ensemble members to a desired state, an SLC algorithm is designed. For reducing the transfer error significantly, an optimization problem is defined. Considering the advantages of QGA and the nature of the problem, the optimization problem by using the QGA method is solved. For this purpose, N samples through sampling of the uncertainty parameters via uniform distribution are generated and an augmented system is also created. By using QGA in the training step, the best control signal is obtained. To test the performance and validation of the method, the obtained control is implemented for some random selected samples. A couple of examples are simulated for investigating the proposed model. The results of the simulations indicate the effectiveness and the advantages of the proposed method.
Compensation for parameter dispersion is a significant challenge for control of inhomogeneous quantum ensembles. In this paper, we present the systematic methodology of sampling-based learning control (SLC) for simultaneously steering the members of inhomogeneous quantum ensembles to the same desired state. The SLC method is employed for optimal control of the state-to-state transition probability for inhomogeneous quantum ensembles of spins as well as $\ensuremath{\Lambda}$-type atomic systems. The procedure involves the steps of (i) training and (ii) testing. In the training step, a generalized system is constructed by sampling members according to the distribution of inhomogeneous parameters drawn from the ensemble. A gradient flow based learning and optimization algorithm is adopted to find an optimal control for the generalized system. In the process of testing, a number of additional ensemble members are randomly selected to evaluate the control performance. Numerical results are presented, showing the effectiveness of the SLC method.