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arXiv:2203.04361 · IFAC-PapersOnLine

High Fidelity Quantum State Transfer by Pontryagin Maximum Principle

Nahid Binandeh Dehaghani, Фернандо Лобо Перейра

Published Jan 1, 2022• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1016/j.ifacol.2022.09.026

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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.
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.
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.
Reliable high-fidelity quantum state transformation has always been considered as an inseparable part of quantum information processing. In this regard, Pontryagin Minimum (or maximum) Principle (PMP) has proved to play an important role to achieve the maximum fidelity in an optimum time or energy. Motivated by this, in this work, we formulate a control constrained optimal control problem where we aim to minimize time and also energy subjected to a quantum system satisfying the bilinear Schrödinger equation. We derive the first order optimality conditions through the application of PMP resulting in a boundary value problem. Next, in order to obtain efficient numerical results, we exploit a particular family of physics-informed neural networks that are specifically designed to tackle the indirect method based on the PMP. We show that this method can significantly speed up the process by first obtaining a set of relations which finally let us compute the optimal control strategy to determine the time- and energy-optimal protocol driving a general initial state to a target state by a quantum Hamiltonian with bounded control. We make use of the so-called “qutip” package in python, and the newly developed “tfc” python package.
Optimal control provides powerful tools and concepts that can be applied to control quantum systems. It has been used extensively to improve the performance of quantum processes in a variety of active areas in quantum technologies. This paper reviews the necessary conditions of optimality in the form of a Pontryagin Maximum Principle (PMP), and introduces its formulations for quantum systems. After formulating an optimal control problem for a two-level quantum system, we then study the application of PMP to obtain an optimal control strategy.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.
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.