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Russian Microelectronics

Lorentz Transformation and Its Generalizations in Problems of Precisely Controlling the States of Multilevel Quantum Systems

Yu. I. Bogdanov, Nadezhda A. Bogdanova, Yu. A. Kuznetsov, Vladimir F. Lukichev

Published Apr 1, 2022• 0 citations
Publisher

Abstract

Abstract A method of precision quantum measurements is proposed, which makes it possible to accurately track the states of multilevel quantum systems in Hilbert spaces of various dimensions. The developed quantum control algorithms are based on the use of the spinor representation of the Lorentz transformation group and its generalizations to the case of multilevel quantum systems. It is shown that feedback through weakly perturbing adaptive quantum measurements is capable of providing precise control of the quantum system, while introducing only weak perturbations in the initial quantum state.

DOI
10.1134/s1063739722020044

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Abstract A method of precision quantum measurements is proposed, which makes it possible to accurately track the states of multilevel quantum systems in Hilbert spaces of various dimensions. The developed quantum control algorithms are based on the use of the spinor representation of the Lorentz transformation group and its generalizations to the case of multilevel quantum systems. It is shown that feedback through weakly perturbing adaptive quantum measurements is capable of providing precise control of the quantum system, while introducing only weak perturbations in the initial quantum state.
The method of quantum tomography, which allows us to track with high accuracy the evolution of multilevel quantum systems (qudits) in Hilbert spaces of various dimensions is presented. The developed algorithms for quantum control are based on the use of the spinor representation of the Lorentz transformation group. In the simplest case of one-qubit states, it turns out that, in addition to three-dimensional rotations on the Bloch sphere, one can introduce four-dimensional Lorentz pseudorotations, similar to the transformations of the special theory of relativity. We show that feedback through weakly perturbing adaptive quantum measurements turns out to be capable of providing high-precision control of the quantum system, while introducing only weak perturbations into the initial quantum state. It turns out that, together with the control of a quantum system through its weak perturbation, the developed algorithms for controlling the evolution of the state of a quantum system can be super-efficient, providing a higher measurement accuracy than any standard POVM (Positive-Operator Valued Measure) protocols. The results of the study are important for the development of optimal adaptive methods for quantum states and operations controlling. The results obtained are essential for the development of high-precision control methods for quantum information technologies.
The method of quantum tomography, which allows us to track with high accuracy the evolution of multilevel quantum systems (qudits) in Hilbert spaces of various dimensions is presented. The developed algorithms for quantum control are based on the use of the spinor representation of the Lorentz transformation group. In the simplest case of one-qubit states, it turns out that, in addition to three-dimensional rotations on the Bloch sphere, one can introduce four-dimensional Lorentz pseudorotations, similar to the transformations of the special theory of relativity. We show that feedback through weakly perturbing adaptive quantum measurements turns out to be capable of providing high-precision control of the quantum system, while introducing only weak perturbations into the initial quantum state. It turns out that, together with the control of a quantum system through its weak perturbation, the developed algorithms for controlling the evolution of the state of a quantum system can be super-efficient, providing a higher measurement accuracy than any standard POVM (PositiveOperator Valued Measure) protocols. The results of the study are important for the development of optimal adaptive methods for quantum states and operations controlling.
Abstract Multilevel quantum states (qudits) represent a promising platform for scalable quantum computing. In this paper, we present a method for precisely controlling such systems using fuzzy quantum measurements. The developed method is used for a precise reconstruction of quantum states under conditions of a significant effect of decoherence and quantum noise. Protocols for quantum measurements based on mutually unbiased bases (MUBs) of various dimensions are considered. The accuracy characteristics of sets of random states uniformly distributed with respect to the Haar measure are studied.
This paper proposes and analyzes an adaptive quantum feedback control system designed to optimize the performance of quantum devices. The core of the system lies in a real-time monitoring and adjustment mechanism utilizing a quantum sensor to measure the state of the quantum system. This measurement data is then fed back into a control algorithm, which dynamically adjusts the system parameters to achieve precise control. Unlike traditional control systems that are sensitive to parameter uncertainties, this adaptive approach leverages the inherent properties of quantum systems to enhance stability and performance. The theoretical framework presented here establishes a foundation for future research and development in adaptive quantum control, addressing key challenges in quantum device operation and manipulation. The system aims to provide a robust and flexible solution for controlling quantum systems with varying characteristics. The primary focus is on the control loop design and the feedback algorithm implementation, with an emphasis on achieving optimal performance under conditions of parameter uncertainty. This work contributes to the growing field of quantum control by introducing a novel adaptive strategy that can significantly improve the operational capabilities of quantum devices.
Abstract This work is a continuation of our research [1], aimed at developing methods for high-precision control of multilevel quantum systems (qudits). This study presents a method for precision quantum measurements of such systems using fuzzy quantum measurements. The developed method is used for precision reconstruction of quantum states under conditions of significant influence of amplitude and phase relaxation. The protocols of quantum measurements based on mutually unbiased bases of various dimensions are considered. The accuracy characteristics of sets of random states uniformly distributed with respect to the Haar measure are studied. A study was made of the dependence of the accuracy loss function on the parameters of phase and amplitude relaxation. It is shown that the results of the performed numerical experiments are in good agreement with the analytical results obtained on the basis of the universal distribution of accuracy losses using fuzzy measurement protocols.
This paper presents a novel adaptive quantum control algorithm incorporating a feedback mechanism to enhance the robustness and precision of quantum system manipulation. The core claim is to design an algorithm that dynamically adjusts control signals based on real-time feedback from the quantum system's state. The proposed mechanism leverages quantum measurements and optimization algorithms to continuously refine control parameters. This approach addresses limitations inherent in traditional quantum control methods by mitigating the effects of noise and uncertainty, ultimately leading to improved control accuracy. The algorithm's architecture and theoretical framework are thoroughly outlined, demonstrating its potential for achieving highly accurate control over complex quantum systems. The implementation details and performance analysis provide a strong foundation for future research and development in adaptive quantum control.