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arXiv:2203.15727 · IEEE Conference on Decision and Control (CDC)

A Quantum Optimal Control Problem with State Constrained Preserving Coherence

Nahid Binandeh Dehaghani, Фернандо Лобо Перейра, A. Pedro Aguiar

Published Dec 6, 2022• 6 citations
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Abstract

In this work, we address the problem of maximizing fidelity in a quantum state transformation process controlled in such a way as to keep decoherence within given bounds. We consider a three-level Λ-type atom subjected to Markovian decoherence characterized by non-unital decoherence channels. We introduce fidelity as the performance index for the quantum state transformation process since the goal is to maximize the similarity of the final state density operator with the one of the desired target state. We formulate the quantum optimal control problem with state constraints where the latter reflect the fact that the decoherence level remains within a pre-defined bound. Optimality conditions of Pontryagin’s Maximum Principle in Gamkrelidze’s form are given. These provide a complete set of relations enabling the computation of the optimal control strategy. This is a novel approach in the context of quantum systems.

DOI
10.1109/cdc51059.2022.9993086

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In this work, we address the problem of maximizing fidelity in a quantum state transformation process controlled in such a way as to keep decoherence within given bounds. We consider a three-level Λ-type atom subjected to Markovian decoherence characterized by non-unital decoherence channels. We introduce fidelity as the performance index for the quantum state transformation process since the goal is to maximize the similarity of the final state density operator with the one of the desired target state. We formulate the quantum optimal control problem with state constraints where the latter reflect the fact that the decoherence level remains within a pre-defined bound. Optimality conditions of Pontryagin’s Maximum Principle in Gamkrelidze’s form are given. These provide a complete set of relations enabling the computation of the optimal control strategy. This is a novel approach in the context of quantum systems.
High fidelity quantum state transfer is an essential part of quantum information processing. In this regard, we address the problem of maximizing the fidelity in a quantum state transformation process satisfying the Liouville-von Neumann equation. By introducing fidelity as the performance index, we aim at maximizing the similarity of the final state density operator with the one of the desired target state. Optimality conditions in the form of a Maximum Principle of Pontryagin are given for the matrix-valued dynamic control systems propagating the probability density function. These provide a complete set of relations enabling the computation of the optimal control strategy.
This paper addresses the optimal control of quantum coherence in multi-level systems, modeled by the Lindblad master equation, which captures both unitary evolution and environmental dissipation. We develop an energy minimization framework to control the evolution of a qutrit (three-level) quantum system while preserving coherence between states. The control problem is formulated using Pontryagin’s Minimum Principle in the form of Gamkrelidze, incorporating state constraints to ensure coherence remains within desired bounds. Our approach accounts for Markovian decoherence, demonstrating that the Lindblad operator is non-unital, which reflects the irreversible decay processes inherent in the system. The results provide insights into effectively maintaining quantum coherence in the presence of dissipation.
This paper addresses the optimal control of quantum coherence in multi-level systems, modeled by the Lindblad master equation, which captures both unitary evolution and environmental dissipation. We develop an energy minimization framework to control the evolution of a qutrit (three-level) quantum system while preserving coherence between states. The control problem is formulated using Pontryagin's Minimum Principle in the form of Gamkrelidze, incorporating state constraints to ensure coherence remains within desired bounds. Our approach accounts for Markovian decoherence, demonstrating that the Lindblad operator is non-unital, which reflects the irreversible decay processes inherent in the system. The results provide insights into effectively maintaining quantum coherence in the presence of dissipation.
In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.
We address a control-constrained optimal control problem pertaining to the transformation of quantum states. Our objective is to steer a quantum system from an initial state to a desired target state while adhering to the principles of the Liouville-von Neumann equation. To achieve this, we introduce a cost functional that balances the dual goals of fidelity maximization and energy consumption minimization. We derive optimality conditions in the form of the Pontryagin Maximum Principle (PMP) for the matrix-valued dynamics associated with this problem. Subsequently, we present a time discretized computational scheme designed to solve the optimal control problem. This computational scheme is rooted in an indirect method grounded in the PMP, showcasing its versatility and efficacy. To illustrate the practicality and applicability of our methodology, we employ it to address the case of a spin ½ particle subjected to interaction with a magnetic field. Our findings shed light on the potential of this approach to tackle complex quantum control scenarios and contribute to the broader field of quantum state transformations.
Optimal control theory, also known as Pontryagin's maximum principle, is applied to the quantum parameter estimation in the presence of decoherence. An efficient procedure is devised to compute the gradient of quantum Fisher information with respect to the control parameters and is used to construct the optimal control protocol. The proposed procedure keeps the control problem in the time-invariant form so that both first-order and second-order optimality conditions derived from Pontryagin's maximum principle apply; the second-order condition turns out to be crucial when the optimal control contains singular arcs. Concretely we look for the optimal control that maximizes quantum Fisher information for the ``twist and turn'' problem. We find that the optimal control is singular without dissipation but can become unbounded once the quantum decoherence is introduced. An amplitude constraint is needed to guarantee a bounded solution. With quantum decoherence, the maximum quantum Fisher information happens at a finite time due to the decoherence, and the asymptotic value depends on the specific decoherence channel and the control of consideration.