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Geometric quantum computation

SL Zhu, Wang, ZD

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

實現可集成的量子計算的關鍵步驟是實現保真度足夠高的一組普適量子邏輯門 .最近幾年發展的幾何量子計算使用幾何位相來實現量子邏輯門 ,其特點是利用幾何位相的整體幾何性質來避免某些局域的無規噪聲的影響 ,從而實現較高保真度的量子門 .文章先簡要介紹常規幾何量子邏輯門的概念 ,然后重點介紹最近提出的非常規幾何量子計算 :量子計算中使用的邏輯門的總位相既包含有幾何位相 ,又包含有動力學位相 ,但它僅依賴于一些幾何特征 .而且 ,對于任意的量子位輸入態 ,在量子門操作過程中積累的位相要么是零 ,要么是僅依賴幾何特征的位相 . \nThe physical implementation of a universal set of high-fidelity quantum gates is a key step in quantum computation. Geometric quantum computation is a promising scenario to achieve high-fidelity quantum gates. In this scheme, quantum gates are realized based on geometric phases, which are dependent only on some global geometric features. In this Communication, we present an overview of geometric quantum computation, especially the unconventional geometric quantum computation recently proposed by us. In the latter scheme, the total phase accumulated in a gate operation consists of both geometric and dynamic components, but for any input state it still only depends on the global geometric features.

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實現可集成的量子計算的關鍵步驟是實現保真度足夠高的一組普適量子邏輯門 .最近幾年發展的幾何量子計算使用幾何位相來實現量子邏輯門 ,其特點是利用幾何位相的整體幾何性質來避免某些局域的無規噪聲的影響 ,從而實現較高保真度的量子門 .文章先簡要介紹常規幾何量子邏輯門的概念 ,然后重點介紹最近提出的非常規幾何量子計算 :量子計算中使用的邏輯門的總位相既包含有幾何位相 ,又包含有動力學位相 ,但它僅依賴于一些幾何特征 .而且 ,對于任意的量子位輸入態 ,在量子門操作過程中積累的位相要么是零 ,要么是僅依賴幾何特征的位相 . \nThe physical implementation of a universal set of high-fidelity quantum gates is a key step in quantum computation. Geometric quantum computation is a promising scenario to achieve high-fidelity quantum gates. In this scheme, quantum gates are realized based on geometric phases, which are dependent only on some global geometric features. In this Communication, we present an overview of geometric quantum computation, especially the unconventional geometric quantum computation recently proposed by us. In the latter scheme, the total phase accumulated in a gate operation consists of both geometric and dynamic components, but for any input state it still only depends on the global geometric features.
Quantum computers may possess much greater computing power than classical computers.But it is one of key step for implementation of quantum computer to use geometric quantum phase to realize high-fidelity universal quantum gates,and its characteristic is to utilize global geometric features to avoid noise influence in some regions,then high-fidelity quantum gates are realized.In this paper,first we briefly introduce both fundamental and existing difficulties of quantum computer,finally discuss geometric quantum phase having relation with scheme such as nuclear magnetic resonance(NMS)、cavity-QED and superconducting Josephson-junction.
From the prinecple of quanturn mechanics,we explain that quantuare computation is a theoretical basis of quantum computer.Quantum computers may provide much greater computing power than classical computers.Using geometric quantum phase to realize high-fidelity universal quantum gates is the key to realize quantum computer and its characteristic is to utilize global geometric features to avoid noise influence in some regions,thus realizing high-fidelity quantum gates.First,we briefly introduce the fundamental principle of quantum computation and existing difficulties of quantum computer implementation;finally,we discuss geometric quantum phase related to scheme of quantum computer implementation such as nuclear magnetic resonance(NMS)、cavity-QED and superconducting Josephson-junction.
Following a key idea of unconventional geometric quantum computation developed earlier [S. L. Zhu and Z. D. Wang, Phys. Rev. Lett. 91, 187902 (2003)], here we propose a more general scheme in such an intriguing way: ${\ensuremath{\gamma}}_{d}={\ensuremath{\alpha}}_{g}+\ensuremath{\eta}{\ensuremath{\gamma}}_{g}$, where ${\ensuremath{\gamma}}_{d}$ and ${\ensuremath{\gamma}}_{g}$ are respectively the dynamic and geometric phases accumulated in the quantum gate operation, with $\ensuremath{\eta}$ as a constant and ${\ensuremath{\alpha}}_{g}$ being dependent only on the geometric feature of the operation. More interestingly, we demonstrate an experiment to implement a universal set of such kind of generalized unconventional geometric quantum gates with high fidelity in an NMR system.
A different way to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic resonance, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore the phase accumulated in the geometric gate is a pure geometric phase for any input state. In comparison with the conventional geometric gates by rotating operations, our approach simplifies experimental implementations making them robust to certain experimental errors. In contrast to the unconventional geometric gates, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases.
The geometric phase stands as a foundational concept in quantum physics, revealing deep connections between geometric structures and quantum dynamical evolution. Unlike dynamical phases, geometric phases exhibit intrinsic resilience to certain types of perturbation, making them particularly valuable for quantum information processing, where maintaining coherent quantum operations is essential. This article provides a review of geometric phases in the context of universal quantum gate construction, i.e., the geometric quantum computation (GQC), with special attention to recent progress in nonadiabatic implementations that enhance gate fidelity and/or operational robustness. We first review a unified theoretical framework that can encompass all existing nonadiabatic GQC approaches, then systematically examine the design principles of nonadiabatic geometric gates with a particular focus on how optimal control techniques can be leveraged to improve the accuracy and noise resistance. In addition, we conducted detailed numerical comparisons of various nonadiabatic GQC protocols, offering a quantitative assessment of their respective performance characteristics and practical limitations. Through this focused investigation, our aim is to provide researchers with both fundamental insights and practical guidance for advancing geometric approaches in quantum computing.
We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault-tolerant geometric features can be implemented in real physical systems.