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

Chinese Science Bulletin (Chinese Version)

Nonadiabatic holonomic quantum computation with atom-cavity system

Tonghao Xing, D. M. Tong

Published Apr 10, 2020• 3 citations
Publisher

Abstract

Quantum computation is much more effective than classical computation in solving many problems such as factoring large integers and searching unsorted databases. However, such effectiveness relies on the ability to perform universal high-fidelity quantum gates. One main challenge in achieving such high-fidelity gates is to reduce control errors of a quantum system. To overcome this problem, various proposals of fault-tolerant quantum computation are proposed. A promising one of such proposals is nonadiabatic holonomic quantum computation. Nonadiabatic holonomic quantum computation is realized by using a quantum system with a subspace satisfying both the cyclic evolution and parallel transport conditions. For an N dimensional quantum system with Hamiltonian H(t) and evolution operator U(t)=Texp[−i∫0tH(t')dt'] , if there exists a time-dependent L dimensional subspace S(t) spanned by the orthonormal vectors {|ϕk(t)〉=U(t)|ϕk(0)〉}k=1L that satisfy the two conditions: (1) ∑k=1L|ϕk(τ)〉〈ϕk(τ)|= ∑k=1L|ϕk(0)〉〈ϕk(0)| , with τ being the evolution period, and (2) 〈ϕk(t)|H(t)|ϕl(t)〉=0, k,l=1,2,⋯,L, then the unitary transformation U(τ) is a holonomic gate on the L dimensional subspace S(0) spanned by {|ϕk(0)〉}k=1L . The work of realizing nonadiabatic holonomic gate is to build a physical system with a Hamiltonian satisfying both the cyclic evolution condition and the parallel transport condition. Nonadiabatic holonomic quantum computation is based on nonadiabatic non-Abelian geometric phases. Since nonadiabatic non-Abelian geometric phases are only dependent on evolution paths but independent of evolution details, nonadiabatic holonomic gates are robust against control errors. Besides, nonadiabatic holonomic gates do not require the long run-time evolution that is necessary for adiabatic holonomic gates. Due to the merits of both robustness against control errors and high-speed realization, nonadiabatic holonomic quantum computation has received increasing attention in theory and experiment. Various schemes of nonadiabatic holonomic quantum computation have been proposed based on different physical systems, and a number of these schemes have been experimentally demonstrated with nuclear magnetic resonance systems, nitrogen-vacancy center systems, and superconducting systems. The progress of nonadiabatic holonomic quantum computation is also reflected in the improvement of the methods of designing quantum gates. The single-shot scheme, single-loop multiple-pulse scheme and composite gate scheme of nonadiabatic quantum gate are based on the improved methods. In particular, the single-shot scheme simplifies the operation of the one-qubit gate, so that the operation of any one-qubit gate, which needs to be realized by combining two non-commuting π-rotation gate in the original scheme, can be simply realized by a single-shot operation. In this paper, we propose a scheme of nonadiabatic holonomic quantum computation with atom-cavity system. The universal set of quantum gates in our scheme consists of an arbitrary one-qubit nonadiabatic holonomic gate and a two-qubit nonadiabatic holonomic controlled-PHASE gate. To realize the arbitrary one-qubit gate, we use two resonant laser pulses and an off-resonant laser pulse to drive a single four-level atom. With the aid of Stark shifts, the one-qubit gate can be performed in one-step only by changing the Rabi frequencies and the phases of the laser pulses. To realize the two-qubit controlled-PHASE gate, we use two off-resonant laser pulses to drive the two four-level atoms in the cavity. The transitions between the two-atom ground state and the two-atom auxiliary state is facilitated by exchanging virtual photons through a common cavity mode. In this process, the effective two-atom transition is disentangled from the cavity mode and therefore the two-qubit gate is insensitive to the cavity decay.

DOI
10.1360/tb-2020-0265

Related papers

Advances in nonadiabatic holonomic quantum computation

Peizi Zhao, Guofu Xu, D. M. Tong

Quantum computation has been shown to be superior to classical computation in solving some problems, and therefore can substantially change our life. However, the realization of quantum computation is still challenging, even if quantum technologies have been improved significantly. Two main obstacles to the realization of practical quantum computation are control errors and decoherence. Control errors are caused by inaccurate manipulations of quantum systems and decoherence is caused by the inevitable interaction between the system and its environment. Geometric phases are only dependent on evolution paths of quantum systems but independent of the evolution details and therefore quantum computation based on geometric phases, i.e., geometric quantum computation, is robust against control errors, benefiting the realization of practical quantum computation. The early proposals of geometric quantum computation are based on adiabatic geometric phases. These proposals require quantum systems to undergo adiabatic evolution, which makes quantum systems evolve for a long time. To circumvent this, nonadiabatic geometric quantum computation based on nonadiabatic Abelian geometric phases was proposed soon after. In 2012, nonadiabatic holonomic quantum computation based on nonadiabatic non-Abelian geometric phases was proposed, which also circumvents long-time evolutions. Moreover, compared with nonadiabatic geometric quantum computation that uses the geometric phase as one parameter of a quantum gate, nonadiabatic holonomic quantum computation uses the holonomic matrix itself as a quantum gate. This makes nonadiabatic holonomic quantum computation possess whole-geometric property. Due to the merits of both geometric robustness and high-speed implementation without the limit of adiabatic evolution, nonadiabatic holonomic quantum computation has been attracting much attention. Until now, much progress has been achieved in the field of nonadiabatic holonomic quantum computation. On one hand, various methods have been proposed to design more efficient nonadiabatic holonomic gates. Nonadiabatic holonomic gates were first realized by using resonant laser fields to drive a three-level system. After this, the single-shot proposal and the single-loop proposal were proposed, allowing us to realize one-qubit gates by a shorter path and thereby reducing the exposure time of nonadiabatic holonomic gates to the environment. To further shorten the exposure time, the path-shortening protocol was put forward, where nonadiabatic holonomic gates can be realized based on a class of extended evolution paths that are shorter than the former ones. Recently, a general approach of constructing Hamiltonians for nonadiabatic holonomic quantum computation was put forward, by using which one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. On the other hand, various methods have been proposed to combine nonadiabatic holonomic gates and various decoherence-resilient methods, making the resulting schemes robust against both control errors and decoherence. The first proposal in this aspect is combining nonadiabatic holonomic gates and decoherence-free subspaces. Gradually, proposals combining nonadiabatic holonomic gates with noiseless subsystems, dynamical decoupling and surface codes were put forward. Last but not least, various proposals suitable for specific physical systems have been proposed and particularly various experimental platforms have been used to demonstrate nonadiabatic holonomic gates. This also significantly improves the development of nonadiabatic holonomic quantum computation. In this paper, we review the above research advances on nonadiabatic holonomic quantum computation, aiming to help readers understand the main developments of nonadiabatic holonomic quantum computation.

Chinese Science Bulletin (Chinese Version)•Mar 2, 2021•10 citations

Single-loop multiple-pulse nonadiabatic holonomic quantum gates

Emmi Herterich, Erik Sjöqvist

Nonadiabatic holonomic quantum computation provides the means to perform fast and robust quantum gates by utilizing the resilience of non-Abelian geometric phases to fluctuations of the path in state space. While the original scheme [E. Sj\"oqvist et al., New J. Phys. 14, 103035 (2012)] needs two loops in the Grassmann manifold (i.e., the space of computational subspaces of the full state space) to generate an arbitrary holonomic one-qubit gate, we propose single-loop one-qubit gates that constitute an efficient universal set of holonomic gates when combined with an entangling holonomic two-qubit gate. Our one-qubit gate is realized by dividing the loop into path segments, each of which is generated by a $\mathrm{\ensuremath{\Lambda}}$-type Hamiltonian. We demonstrate that two path segments are sufficient to realize arbitrary single-loop holonomic one-qubit gates. We describe how our scheme can be implemented experimentally in a generic atomic system exhibiting a three-level $\mathrm{\ensuremath{\Lambda}}$-coupling structure by utilizing carefully chosen laser pulses.

arXiv:1608.07418•Physical Review A•Nov 9, 2016•100 citations

Suppression of systematic errors in holonomic quantum gates via multipath correction

Shou-Yue Sun, Li‐Na Ji, Fei-Fan Cui +2 more

Nonadiabatic holonomic quantum computation (NHQC) based on non-Abelian geometric phases can possess potential robustness advantages due to its intrinsic error-resilient features. However, rigorous geometric conditions, including cyclic evolution and parallel transport conditions, typically demand complex and error-prone control and also involve additional auxiliary resources. Moreover, the limitations of optional evolution paths in previous NHQC schemes make it difficult to avoid path segments that are heavily affected by systematic errors, thereby weakening the error resilience of holonomic gates. To address this problem, we propose a multipath NHQC scheme, which can sufficiently improve the robustness of quantum gates by flexibly correcting geometric evolution paths. Numerical simulation results show that our scheme surpasses previous NHQC schemes in suppressing systematic errors for arbitrary gate types. In addition, we provide a physical implementation based on superconducting quantum circuits, with the decoherence-free-subspace encoding and an experiment-friendly two-body exchange interaction, demonstrating the excellent compatibility of our scheme. Therefore, our scheme offers a promising alternative for future fault-tolerant quantum computation.

Physical Review A•Dec 31, 2024•2 citations

Single-shot realization of nonadiabatic holonomic quantum gates in decoherence-free subspaces

P. Z. Zhao, Guofu Xu, Qi-Ming Ding +2 more

Nonadiabatic holonomic quantum computation in decoherence-free subspaces has attracted increasing attention recently, as it allows for high-speed implementation and combines both the robustness of holonomic gates and the coherence stabilization of decoherence-free subspaces. Since the first protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, a number of schemes for its physical implementation have been put forward. However, all previous schemes require two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to error sources as well as the resource expenditure. In this paper, we propose an alternative protocol for nonadiabatic holonomic quantum computation in decoherence-free subspaces, in which an arbitrary one-qubit gate in decoherence-free subspaces is realized by a single-shot implementation. The present protocol not only maintains the merits of the original protocol but also avoids the extra work of combining two gates to implement an arbitrary one-qubit gate and thereby reduces the exposure time to various error sources.

arXiv:1706.02967•Physical Review A•Jun 6, 2017•70 citations

Single-loop realization of universal nonadiabatic holonomic gates in decoherence-free subspaces

Chang Wang, Zhenping Guo

Control errors and decoherence are two main obstacles for realization of quantum computation. Nonadiabatic holonomic quantum computation in decoherence-free subspaces can protect quantum gates from both control errors and decoherence, and, therefore, it has received increasing attention. The original protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces needs two loops in the Grassmann manifold to generate an arbitrary nonadiabatic holonomic one-qubit gate. In this paper, we proposed a protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, where an arbitrary one-qubit gate is generated by only a single loop in the Grassmann manifold and an entangling two-qubit gate assisted by two single-qubit gates can be used to construct a two-qubit controlled unitary gate. In addition, the encoding of our protocol for nonadiabatic holonomic gates requires a lower number of physical qubits compared with previous schemes. Our scheme not only maintains the merits of the original protocol but it also avoids the extra work of combining two gates to realize an arbitrary one-qubit gate.

Europhysics Letters (EPL)•Dec 10, 2018•10 citations

Robust holonomic quantum gates via cyclic evolution protection

Yan Liang, Tao Chen, Zheng‐Yuan Xue

Nonadiabatic holonomic quantum computation provides a promising approach toward fault-tolerant quantum control, due to its simple requirements for energy level structure and intrinsic robustness stemming from non-Abelian geometric phases. However, conventional nonadiabatic holonomic quantum computation relies on segmented evolution along a specific trajectory, which not only complicates experimental control but also exacerbates decoherence effects. Meanwhile, minor deviations in systematic parameters can directly disrupt the cyclic evolution process necessary to construct holonomic gates, leading to degraded gate robustness. To address these disadvantages, we here propose a general strategy to incorporate cyclic evolution protection into the holonomic gate construction. The aim is to design on-demand trajectories by modulating pulse shapes, thereby circumventing the detrimental impact of systematic errors on cyclic evolution. Consequently, universal holonomic gates implemented through a stable cyclic evolution process can maintain lower error sensitivity. Meanwhile, in our scheme, compressing the state population in the ancillary state ensures less energy consumption, resulting in higher gate fidelity. Therefore, our work serves as a practical solution for achieving high-fidelity and robust universal quantum gates, paving the way for large-scale quantum computation.

APL Quantum•Feb 19, 2025•2 citations
Quantum computation is much more effective than classical computation in solving many problems such as factoring large integers and searching unsorted databases. However, such effectiveness relies on the ability to perform universal high-fidelity quantum gates. One main challenge in achieving such high-fidelity gates is to reduce control errors of a quantum system. To overcome this problem, various proposals of fault-tolerant quantum computation are proposed. A promising one of such proposals is nonadiabatic holonomic quantum computation. Nonadiabatic holonomic quantum computation is realized by using a quantum system with a subspace satisfying both the cyclic evolution and parallel transport conditions. For an N dimensional quantum system with Hamiltonian H(t) and evolution operator U(t)=Texp[−i∫0tH(t')dt'] , if there exists a time-dependent L dimensional subspace S(t) spanned by the orthonormal vectors {|ϕk(t)〉=U(t)|ϕk(0)〉}k=1L that satisfy the two conditions: (1) ∑k=1L|ϕk(τ)〉〈ϕk(τ)|= ∑k=1L|ϕk(0)〉〈ϕk(0)| , with τ being the evolution period, and (2) 〈ϕk(t)|H(t)|ϕl(t)〉=0, k,l=1,2,⋯,L, then the unitary transformation U(τ) is a holonomic gate on the L dimensional subspace S(0) spanned by {|ϕk(0)〉}k=1L . The work of realizing nonadiabatic holonomic gate is to build a physical system with a Hamiltonian satisfying both the cyclic evolution condition and the parallel transport condition. Nonadiabatic holonomic quantum computation is based on nonadiabatic non-Abelian geometric phases. Since nonadiabatic non-Abelian geometric phases are only dependent on evolution paths but independent of evolution details, nonadiabatic holonomic gates are robust against control errors. Besides, nonadiabatic holonomic gates do not require the long run-time evolution that is necessary for adiabatic holonomic gates. Due to the merits of both robustness against control errors and high-speed realization, nonadiabatic holonomic quantum computation has received increasing attention in theory and experiment. Various schemes of nonadiabatic holonomic quantum computation have been proposed based on different physical systems, and a number of these schemes have been experimentally demonstrated with nuclear magnetic resonance systems, nitrogen-vacancy center systems, and superconducting systems. The progress of nonadiabatic holonomic quantum computation is also reflected in the improvement of the methods of designing quantum gates. The single-shot scheme, single-loop multiple-pulse scheme and composite gate scheme of nonadiabatic quantum gate are based on the improved methods. In particular, the single-shot scheme simplifies the operation of the one-qubit gate, so that the operation of any one-qubit gate, which needs to be realized by combining two non-commuting π-rotation gate in the original scheme, can be simply realized by a single-shot operation. In this paper, we propose a scheme of nonadiabatic holonomic quantum computation with atom-cavity system. The universal set of quantum gates in our scheme consists of an arbitrary one-qubit nonadiabatic holonomic gate and a two-qubit nonadiabatic holonomic controlled-PHASE gate. To realize the arbitrary one-qubit gate, we use two resonant laser pulses and an off-resonant laser pulse to drive a single four-level atom. With the aid of Stark shifts, the one-qubit gate can be performed in one-step only by changing the Rabi frequencies and the phases of the laser pulses. To realize the two-qubit controlled-PHASE gate, we use two off-resonant laser pulses to drive the two four-level atoms in the cavity. The transitions between the two-atom ground state and the two-atom auxiliary state is facilitated by exchanging virtual photons through a common cavity mode. In this process, the effective two-atom transition is disentangled from the cavity mode and therefore the two-qubit gate is insensitive to the cavity decay.
Quantum computation has been shown to be superior to classical computation in solving some problems, and therefore can substantially change our life. However, the realization of quantum computation is still challenging, even if quantum technologies have been improved significantly. Two main obstacles to the realization of practical quantum computation are control errors and decoherence. Control errors are caused by inaccurate manipulations of quantum systems and decoherence is caused by the inevitable interaction between the system and its environment. Geometric phases are only dependent on evolution paths of quantum systems but independent of the evolution details and therefore quantum computation based on geometric phases, i.e., geometric quantum computation, is robust against control errors, benefiting the realization of practical quantum computation. The early proposals of geometric quantum computation are based on adiabatic geometric phases. These proposals require quantum systems to undergo adiabatic evolution, which makes quantum systems evolve for a long time. To circumvent this, nonadiabatic geometric quantum computation based on nonadiabatic Abelian geometric phases was proposed soon after. In 2012, nonadiabatic holonomic quantum computation based on nonadiabatic non-Abelian geometric phases was proposed, which also circumvents long-time evolutions. Moreover, compared with nonadiabatic geometric quantum computation that uses the geometric phase as one parameter of a quantum gate, nonadiabatic holonomic quantum computation uses the holonomic matrix itself as a quantum gate. This makes nonadiabatic holonomic quantum computation possess whole-geometric property. Due to the merits of both geometric robustness and high-speed implementation without the limit of adiabatic evolution, nonadiabatic holonomic quantum computation has been attracting much attention. Until now, much progress has been achieved in the field of nonadiabatic holonomic quantum computation. On one hand, various methods have been proposed to design more efficient nonadiabatic holonomic gates. Nonadiabatic holonomic gates were first realized by using resonant laser fields to drive a three-level system. After this, the single-shot proposal and the single-loop proposal were proposed, allowing us to realize one-qubit gates by a shorter path and thereby reducing the exposure time of nonadiabatic holonomic gates to the environment. To further shorten the exposure time, the path-shortening protocol was put forward, where nonadiabatic holonomic gates can be realized based on a class of extended evolution paths that are shorter than the former ones. Recently, a general approach of constructing Hamiltonians for nonadiabatic holonomic quantum computation was put forward, by using which one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. On the other hand, various methods have been proposed to combine nonadiabatic holonomic gates and various decoherence-resilient methods, making the resulting schemes robust against both control errors and decoherence. The first proposal in this aspect is combining nonadiabatic holonomic gates and decoherence-free subspaces. Gradually, proposals combining nonadiabatic holonomic gates with noiseless subsystems, dynamical decoupling and surface codes were put forward. Last but not least, various proposals suitable for specific physical systems have been proposed and particularly various experimental platforms have been used to demonstrate nonadiabatic holonomic gates. This also significantly improves the development of nonadiabatic holonomic quantum computation. In this paper, we review the above research advances on nonadiabatic holonomic quantum computation, aiming to help readers understand the main developments of nonadiabatic holonomic quantum computation.
Nonadiabatic holonomic quantum computation provides the means to perform fast and robust quantum gates by utilizing the resilience of non-Abelian geometric phases to fluctuations of the path in state space. While the original scheme [E. Sj\"oqvist et al., New J. Phys. 14, 103035 (2012)] needs two loops in the Grassmann manifold (i.e., the space of computational subspaces of the full state space) to generate an arbitrary holonomic one-qubit gate, we propose single-loop one-qubit gates that constitute an efficient universal set of holonomic gates when combined with an entangling holonomic two-qubit gate. Our one-qubit gate is realized by dividing the loop into path segments, each of which is generated by a $\mathrm{\ensuremath{\Lambda}}$-type Hamiltonian. We demonstrate that two path segments are sufficient to realize arbitrary single-loop holonomic one-qubit gates. We describe how our scheme can be implemented experimentally in a generic atomic system exhibiting a three-level $\mathrm{\ensuremath{\Lambda}}$-coupling structure by utilizing carefully chosen laser pulses.
Nonadiabatic holonomic quantum computation (NHQC) based on non-Abelian geometric phases can possess potential robustness advantages due to its intrinsic error-resilient features. However, rigorous geometric conditions, including cyclic evolution and parallel transport conditions, typically demand complex and error-prone control and also involve additional auxiliary resources. Moreover, the limitations of optional evolution paths in previous NHQC schemes make it difficult to avoid path segments that are heavily affected by systematic errors, thereby weakening the error resilience of holonomic gates. To address this problem, we propose a multipath NHQC scheme, which can sufficiently improve the robustness of quantum gates by flexibly correcting geometric evolution paths. Numerical simulation results show that our scheme surpasses previous NHQC schemes in suppressing systematic errors for arbitrary gate types. In addition, we provide a physical implementation based on superconducting quantum circuits, with the decoherence-free-subspace encoding and an experiment-friendly two-body exchange interaction, demonstrating the excellent compatibility of our scheme. Therefore, our scheme offers a promising alternative for future fault-tolerant quantum computation.
Nonadiabatic holonomic quantum computation in decoherence-free subspaces has attracted increasing attention recently, as it allows for high-speed implementation and combines both the robustness of holonomic gates and the coherence stabilization of decoherence-free subspaces. Since the first protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, a number of schemes for its physical implementation have been put forward. However, all previous schemes require two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to error sources as well as the resource expenditure. In this paper, we propose an alternative protocol for nonadiabatic holonomic quantum computation in decoherence-free subspaces, in which an arbitrary one-qubit gate in decoherence-free subspaces is realized by a single-shot implementation. The present protocol not only maintains the merits of the original protocol but also avoids the extra work of combining two gates to implement an arbitrary one-qubit gate and thereby reduces the exposure time to various error sources.
Control errors and decoherence are two main obstacles for realization of quantum computation. Nonadiabatic holonomic quantum computation in decoherence-free subspaces can protect quantum gates from both control errors and decoherence, and, therefore, it has received increasing attention. The original protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces needs two loops in the Grassmann manifold to generate an arbitrary nonadiabatic holonomic one-qubit gate. In this paper, we proposed a protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, where an arbitrary one-qubit gate is generated by only a single loop in the Grassmann manifold and an entangling two-qubit gate assisted by two single-qubit gates can be used to construct a two-qubit controlled unitary gate. In addition, the encoding of our protocol for nonadiabatic holonomic gates requires a lower number of physical qubits compared with previous schemes. Our scheme not only maintains the merits of the original protocol but it also avoids the extra work of combining two gates to realize an arbitrary one-qubit gate.
Nonadiabatic holonomic quantum computation provides a promising approach toward fault-tolerant quantum control, due to its simple requirements for energy level structure and intrinsic robustness stemming from non-Abelian geometric phases. However, conventional nonadiabatic holonomic quantum computation relies on segmented evolution along a specific trajectory, which not only complicates experimental control but also exacerbates decoherence effects. Meanwhile, minor deviations in systematic parameters can directly disrupt the cyclic evolution process necessary to construct holonomic gates, leading to degraded gate robustness. To address these disadvantages, we here propose a general strategy to incorporate cyclic evolution protection into the holonomic gate construction. The aim is to design on-demand trajectories by modulating pulse shapes, thereby circumventing the detrimental impact of systematic errors on cyclic evolution. Consequently, universal holonomic gates implemented through a stable cyclic evolution process can maintain lower error sensitivity. Meanwhile, in our scheme, compressing the state population in the ancillary state ensures less energy consumption, resulting in higher gate fidelity. Therefore, our work serves as a practical solution for achieving high-fidelity and robust universal quantum gates, paving the way for large-scale quantum computation.