Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Noncyclic geometric quantum computation

Axel Friedenauer, Erik Sjöqvist

Published Feb 28, 2003• 31 citations
Publisher

Abstract

Geometric quantum computation is extended to the noncyclic case. Adiabatic nuclear magnetic resonance implementations of geometric phase-shift gates at any precession angle and their concomitant fault tolerance with respect to the amplitude of the oscillating field are analyzed. Nonadiabatic one-qubit geometric quantum computation is considered from the noncyclic perspective, with particular emphasis on errors in precession angles.

DOI
10.1103/physreva.67.024303

Related papers

Nonadiabatic Conditional Geometric Phase Shift with NMR

Xiang‐Bin Wang, Matsumoto Keiji

A conditional geometric phase shift gate, which is fault tolerant to certain types of errors due to its geometric nature, was realized recently via nuclear magnetic resonance (NMR) under adiabatic conditions. However, in quantum computation, everything must be completed within the decoherence time. The adiabatic condition makes any fast conditional Berry phase (cyclic adiabatic geometric phase) shift gate impossible. Here we show that by using a newly designed sequence of simple operations with an additional vertical magnetic field, the conditional geometric phase shift gate can be run nonadiabatically. Therefore geometric quantum computation can be done at the same rate as usual quantum computation.

arXiv:quant-ph/0101038•Physical Review Letters•Aug 8, 2001•313 citations

Noncyclic geometric quantum computation in a nuclear-magnetic-resonance system

Z. S. Wang, G.Q. Liu, Y. H. Ji

A scheme is proposed to include both cyclic and noncyclic geometric quantum computations in nuclear-magnetic-resonance system by the invariant theory. By controlling magnetic field and arbitrary parameters in the invariant operator, the phases accumulated in the entangling quantum gates for single- and two-qubit systems are pure geometric phases. Thus, fault tolerance may occur in some critical magnetic field parameters for either cyclic or noncyclic evolution by differently choosing for gate time.

Physical Review A•May 1, 2009•88 citations

Noncyclic geometric quantum computation with shortcut to adiabaticity

Qing-Xian Lv, Zhen-Tao Liang, Hongzhi Liu +3 more

Fast and robust quantum gates are the cornerstones of fault-tolerance quantum computation. In this paper, as opposed to the previous proposals which use cyclic geometric phases, we propose to realize quantum computation based on noncyclic geometric evolution. The superiority of the noncyclic scheme is that the operation time is proportional to the rotation angle of the quantum state. Therefore, the noncyclic scheme becomes fairly fast in the case of quantum gates with small rotation angle, which will be more insensitive to the decoherence and the leakage to the states outside the computational basis. Similar to the cyclic version of geometric quantum computation, dynamical phases here during the evolution are canceled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabaticity. The proposed scheme is robust against random noise due to the geometric characteristic of projective Hilbert space. Since the proposed scheme is fast and robust, it is a particularly suitable way to manipulate the physical systems with weak nonlinearity, such as superconducting systems.

arXiv:1903.03397•Physical Review A•Feb 24, 2020•24 citations

Noncyclic and Nonadiabatic Geometric Quantum Gates with Smooth Paths

Li‐Na Ji, Cheng-Yun Ding, Tao Chen +1 more

Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quantum gates, which is the necessity of fault-tolerant quantum computation. However, it is limited by cyclic and mutative evolution path, which usually requires longer gate-time and abrupt pulse control, and thus weaken the gate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadiabatic evolutions, which effectively shorten the gate duration, and universal quantum gates can be realized in one step without path mutation. Our numerical simulation shows that, comparing with the conventional dynamical gates, the constructed geometric gates have stronger resistant not only to noise but also to decoherence effect. In addition, our scheme can also be implemented on a superconducting circuit platform, with the fidelities of single-qubit and two-qubit gates higher than 99.97$\%$ and 99.84$\%$, respectively. Therefore, our scheme provides a promising way to realize high-fidelity fault-tolerant quantum gates for scalable quantum computation.

•Feb 1, 2021•2 citations

Implementations of Nonadiabatic Geometric Quantum Computation using NMR

Jiangfeng Du, Mingjun Shi, Jihui Wu +2 more

Recently, geometric phases, which is fault tolerate to certain errors intrinsically due to its geometric property, are getting considerable attention in quantum computing theoretically. So far, only one experiment about adiabatic geometric gate with NMR through Berry phase has been reported. However, there are two drawbacks in it. First, the adiabatic condition of Berry phase makes such gate very slowly. Second, the extra operation to eliminate the dynamic phase. As we know, geometric phase can exist both adiabatic(Berry phase) and nonadiabatic(Aharonov-Anandan phase). In this letter, we reports the first experimental realization of nonadiabatic geometric gate with NMR through conditional-AA phase. In our experiment the gates can be made faster and more easily, and the two drawbacks mentioned above are removed.

arXiv:quant-ph/0207022•Jul 3, 2002•1 citation

One-step implementation of Rydberg nonadiabatic noncyclic geometric quantum computation in decoherence-free subspaces

Li-Na Sun, F.-Q. Guo, Zheng Shan +3 more

Quantum gates constructed by geometric phase are naturally robust to control errors due to the global nature of the geometric evolution path. Therefore, how to cope with the inevitable decoherence errors is worthy of serious attention for geometric quantum computation. Different from conventional nonadiabatic geometric quantum computation (NGQC), which needs the same evolution time for any geometric rotation angle, we have proposed and experimentally demonstrated nonadiabatic noncyclic geometric quantum computation (NNGQC) without the requirement of the cyclic evolution condition, which thus reduces decoherence errors due to shortened evolution time [J. W. Zhang et al., Phys. Rev. Lett. 127, 030502 (2021)]. In addition, the use of decoherence-free subspaces (DFS) is another effective strategy to protect quantum gates from dephasing error and has been studied in nonadiabatic holonomic quantum computation [G. F. Xu et al., Phys. Rev. Lett. 109, 170501 (2012)]. Motivated by previous studies, we propose a one-step scheme to implement single-qubit NNGQC operation in DFS constructed by two Rydberg atoms, and then generalize the scheme to the two-logical-qubit case in one step. The numerical results show that our scheme is robust against decoherence in contrast to the NGQC counterparts. The present scheme combines the features of NNGQC and DFS and can further reduce the impact of decoherence on gate infidelity, which may provide an option for the realization of Rydberg-atom-based quantum computation in the future.

Physical Review A•Jun 10, 2022•14 citations
Geometric quantum computation is extended to the noncyclic case. Adiabatic nuclear magnetic resonance implementations of geometric phase-shift gates at any precession angle and their concomitant fault tolerance with respect to the amplitude of the oscillating field are analyzed. Nonadiabatic one-qubit geometric quantum computation is considered from the noncyclic perspective, with particular emphasis on errors in precession angles.
A conditional geometric phase shift gate, which is fault tolerant to certain types of errors due to its geometric nature, was realized recently via nuclear magnetic resonance (NMR) under adiabatic conditions. However, in quantum computation, everything must be completed within the decoherence time. The adiabatic condition makes any fast conditional Berry phase (cyclic adiabatic geometric phase) shift gate impossible. Here we show that by using a newly designed sequence of simple operations with an additional vertical magnetic field, the conditional geometric phase shift gate can be run nonadiabatically. Therefore geometric quantum computation can be done at the same rate as usual quantum computation.
A scheme is proposed to include both cyclic and noncyclic geometric quantum computations in nuclear-magnetic-resonance system by the invariant theory. By controlling magnetic field and arbitrary parameters in the invariant operator, the phases accumulated in the entangling quantum gates for single- and two-qubit systems are pure geometric phases. Thus, fault tolerance may occur in some critical magnetic field parameters for either cyclic or noncyclic evolution by differently choosing for gate time.
Fast and robust quantum gates are the cornerstones of fault-tolerance quantum computation. In this paper, as opposed to the previous proposals which use cyclic geometric phases, we propose to realize quantum computation based on noncyclic geometric evolution. The superiority of the noncyclic scheme is that the operation time is proportional to the rotation angle of the quantum state. Therefore, the noncyclic scheme becomes fairly fast in the case of quantum gates with small rotation angle, which will be more insensitive to the decoherence and the leakage to the states outside the computational basis. Similar to the cyclic version of geometric quantum computation, dynamical phases here during the evolution are canceled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabaticity. The proposed scheme is robust against random noise due to the geometric characteristic of projective Hilbert space. Since the proposed scheme is fast and robust, it is a particularly suitable way to manipulate the physical systems with weak nonlinearity, such as superconducting systems.
Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quantum gates, which is the necessity of fault-tolerant quantum computation. However, it is limited by cyclic and mutative evolution path, which usually requires longer gate-time and abrupt pulse control, and thus weaken the gate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadiabatic evolutions, which effectively shorten the gate duration, and universal quantum gates can be realized in one step without path mutation. Our numerical simulation shows that, comparing with the conventional dynamical gates, the constructed geometric gates have stronger resistant not only to noise but also to decoherence effect. In addition, our scheme can also be implemented on a superconducting circuit platform, with the fidelities of single-qubit and two-qubit gates higher than 99.97%\%% and 99.84%\%%, respectively. Therefore, our scheme provides a promising way to realize high-fidelity fault-tolerant quantum gates for scalable quantum computation.
Recently, geometric phases, which is fault tolerate to certain errors intrinsically due to its geometric property, are getting considerable attention in quantum computing theoretically. So far, only one experiment about adiabatic geometric gate with NMR through Berry phase has been reported. However, there are two drawbacks in it. First, the adiabatic condition of Berry phase makes such gate very slowly. Second, the extra operation to eliminate the dynamic phase. As we know, geometric phase can exist both adiabatic(Berry phase) and nonadiabatic(Aharonov-Anandan phase). In this letter, we reports the first experimental realization of nonadiabatic geometric gate with NMR through conditional-AA phase. In our experiment the gates can be made faster and more easily, and the two drawbacks mentioned above are removed.
Quantum gates constructed by geometric phase are naturally robust to control errors due to the global nature of the geometric evolution path. Therefore, how to cope with the inevitable decoherence errors is worthy of serious attention for geometric quantum computation. Different from conventional nonadiabatic geometric quantum computation (NGQC), which needs the same evolution time for any geometric rotation angle, we have proposed and experimentally demonstrated nonadiabatic noncyclic geometric quantum computation (NNGQC) without the requirement of the cyclic evolution condition, which thus reduces decoherence errors due to shortened evolution time [J. W. Zhang et al., Phys. Rev. Lett. 127, 030502 (2021)]. In addition, the use of decoherence-free subspaces (DFS) is another effective strategy to protect quantum gates from dephasing error and has been studied in nonadiabatic holonomic quantum computation [G. F. Xu et al., Phys. Rev. Lett. 109, 170501 (2012)]. Motivated by previous studies, we propose a one-step scheme to implement single-qubit NNGQC operation in DFS constructed by two Rydberg atoms, and then generalize the scheme to the two-logical-qubit case in one step. The numerical results show that our scheme is robust against decoherence in contrast to the NGQC counterparts. The present scheme combines the features of NNGQC and DFS and can further reduce the impact of decoherence on gate infidelity, which may provide an option for the realization of Rydberg-atom-based quantum computation in the future.