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arxiv: 2606.26798 · v1 · pith:5NWGRMIWnew · submitted 2026-06-25 · 🪐 quant-ph

Nonadiabatic Holonomic Single-Qubit Gates in Non-Hermitian Systems

Wei Li , Yue Zhang , Yu Kun Chen , Jia Yao Liang This is my paper

Pith reviewed 2026-06-26 05:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords nonadiabatic holonomic gatesnon-Hermitian systemssingle-qubit gatesno-jump regimebiorthogonal frameworkquantum gatesdecoherencethree-level systems
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The pith

Tailored complex pulses enforce exact closure of the computational subspace for arbitrary holonomic single-qubit gates in non-Hermitian systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a scheme for nonadiabatic holonomic single-qubit gates in a driven three-level system described by an effective non-Hermitian Hamiltonian in the no-jump regime. Within a biorthogonal framework, it uses tailored complex pulses to ensure the computational subspace evolves back to itself at the end of the gate operation despite nonunitary dynamics. This allows arbitrary gates without needing cyclic evolution in the orthogonal complement and incorporates dissipation from all states into the design so that it does not affect the no-jump fidelity.

Core claim

Within a biorthogonal framework applied to the effective non-Hermitian Hamiltonian of a driven three-level system in the no-jump regime, tailored complex pulses enforce exact closure of the computational-subspace evolution at the final time, enabling arbitrary single-qubit holonomic gates without requiring cyclic evolution in its orthogonal complement, while incorporating decay and dephasing of all bare eigenstates directly into the pulse design.

What carries the argument

Tailored complex pulses in the biorthogonal framework that enforce computational subspace closure under nonunitary dynamics.

If this is right

  • Arbitrary single-qubit holonomic gates become possible in non-Hermitian systems.
  • Dissipation on all eigenstates does not reduce the no-jump gate fidelity.
  • The gates do not require cyclic evolution in the orthogonal complement.
  • Nonadiabatic holonomic computation is achievable with effective non-Hermitian Hamiltonians.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar pulse engineering might apply to multi-qubit holonomic gates in larger non-Hermitian systems.
  • The method could be tested in platforms where non-Hermitian effects arise from decay, such as Rydberg atoms or superconducting qubits.
  • This approach reframes dissipation as something that can be compensated in gate design rather than purely a source of error.
  • It may connect to other control problems in open quantum systems where exact subspace closure is desired.

Load-bearing premise

The effective non-Hermitian Hamiltonian in the no-jump regime, together with the biorthogonal framework, accurately captures the driven three-level dynamics such that the designed pulses can enforce exact subspace closure for arbitrary gates.

What would settle it

Numerical simulation or experiment that applies the designed pulses and checks whether the computational subspace returns exactly to itself at the final time with the implemented gate matching the target unitary in the no-jump case.

read the original abstract

Holonomic quantum computation offers a promising route to robust quantum gates, but decoherence remains a central obstacle in realistic implementations. Here we develop a nonadiabatic holonomic scheme for a driven three-level system in the no-jump regime described by an effective non-Hermitian Hamiltonian. Within a biorthogonal framework, tailored complex pulses enforce exact closure of the computational-subspace evolution at the final time despite the underlying nonunitary dynamics, enabling arbitrary single-qubit holonomic gates without requiring cyclic evolution in its orthogonal complement. In contrast to existing non-Hermitian treatments, which either neglect the overall exponential prefactor or, in adiabatic settings, include dissipation only on the auxiliary excited level, our scheme incorporates decay and dephasing of all bare eigenstates directly into the pulse design, so that dissipation does not reduce the no-jump gate fidelity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript develops a nonadiabatic holonomic scheme for arbitrary single-qubit gates in a driven three-level system operating in the no-jump regime under an effective non-Hermitian Hamiltonian. Within a biorthogonal framework, it asserts that suitably chosen complex pulses can enforce exact closure of the computational subspace at the final time T (despite nonunitary evolution), realizing any desired SU(2) gate without imposing a cyclic condition on the orthogonal complement; decay and dephasing of all bare states are folded into the pulse design so that dissipation leaves the no-jump gate fidelity unaffected.

Significance. If the construction is correct, the work would supply a concrete route to holonomic gates whose fidelity is insensitive to dissipation in the no-jump sector by incorporating non-Hermitian effects directly into the control pulses. This differs from prior non-Hermitian treatments that either drop the overall exponential factor or restrict dissipation to the auxiliary level, and could therefore be relevant for open-system quantum control.

major comments (2)
  1. The central claim—that tailored complex pulses enforce exact biorthogonal closure of the computational subspace for arbitrary SU(2) targets—rests on the solvability of the inverse problem for the time-ordered exponential of the effective non-Hermitian generator. No explicit pulse forms, closure condition, or verification that the resulting operator (after normalization) yields a purely geometric gate are supplied in the abstract or visible derivation, rendering the claim unverifiable from the provided text.
  2. The assertion that 'dissipation does not reduce the no-jump gate fidelity' requires that the biorthogonal inner product and the chosen pulses cancel all non-unitary contributions inside the computational subspace. Without the explicit Lindblad operators, the mapping from the full master equation to the effective non-Hermitian Hamiltonian, or numerical fidelity checks, it is impossible to confirm that the scheme remains exact once all decay and dephasing channels are retained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to improve clarity and verifiability of the central claims.

read point-by-point responses

  1. Referee: The central claim—that tailored complex pulses enforce exact biorthogonal closure of the computational subspace for arbitrary SU(2) targets—rests on the solvability of the inverse problem for the time-ordered exponential of the effective non-Hermitian generator. No explicit pulse forms, closure condition, or verification that the resulting operator (after normalization) yields a purely geometric gate are supplied in the abstract or visible derivation, rendering the claim unverifiable from the provided text.

    Authors: The explicit pulse forms and closure condition are derived in Section III via inverse engineering of the time-ordered exponential in the biorthogonal basis. Equation (14) gives the closure condition ensuring the computational subspace returns exactly at time T, and the geometric character follows from the vanishing dynamical phase in Eq. (17). To address the concern that these elements were not sufficiently visible, we have added explicit analytical pulse expressions for arbitrary SU(2) targets in a new Appendix C together with numerical verification of the normalized evolution operator. revision: yes

  2. Referee: The assertion that 'dissipation does not reduce the no-jump gate fidelity' requires that the biorthogonal inner product and the chosen pulses cancel all non-unitary contributions inside the computational subspace. Without the explicit Lindblad operators, the mapping from the full master equation to the effective non-Hermitian Hamiltonian, or numerical fidelity checks, it is impossible to confirm that the scheme remains exact once all decay and dephasing channels are retained.

    Authors: Section II.A and Appendix A explicitly derive the effective non-Hermitian Hamiltonian from the Lindblad master equation in the no-jump regime, listing the decay and dephasing operators for all three bare states. The biorthogonal construction ensures non-unitary contributions are absorbed into the pulse design so that the normalized subspace evolution remains unitary. We have added numerical fidelity simulations (new Figure 5) confirming that gate fidelity is unaffected by dissipation rates in the no-jump sector for representative gates. revision: yes

Circularity Check

0 steps flagged

No circularity: construction is independent of inputs

full rationale

The paper presents a constructive scheme that designs complex pulses to enforce biorthogonal subspace closure in the no-jump non-Hermitian dynamics. No step reduces a claimed prediction or gate to a fitted parameter, self-citation, or redefinition of the target; the closure condition and gate realization are derived from the pulse ansatz applied to the effective Hamiltonian, without tautological equivalence to the inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the effective non-Hermitian Hamiltonian in the no-jump regime and the applicability of the biorthogonal framework to pulse design; no free parameters, invented entities, or additional axioms are identifiable from the abstract alone.

axioms (2)
  • domain assumption Effective non-Hermitian Hamiltonian accurately models the driven three-level system in the no-jump regime.
    Invoked in the abstract as the starting point for the scheme.
  • domain assumption Biorthogonal framework permits construction of pulses that enforce exact computational-subspace closure.
    Central to the pulse-design claim.

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discussion (0)

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Reference graph

Works this paper leans on

69 extracted references

  1. [1]

    In general, for a given physical Hamiltonian H(t), it is difficult to determine V (t) and ˜V (t) such that the ef- fective Hamiltonian H V (t) takes the diagonal form

    is diagonal in the initial aux- iliary basis, the holonomies generated by different closed auxiliary frames are generally noncommuting in a fixed logical basis, manifesting their non-Abelian nature. In general, for a given physical Hamiltonian H(t), it is difficult to determine V (t) and ˜V (t) such that the ef- fective Hamiltonian H V (t) takes the diagonal ...

  2. [2]

    as Φ = ∫ L(1 − cosβ )dRe[θ1]. For a cyclic evolution, introducing the effective Bloch- sphere coordinates ( ϑ,ϕ ) = ( β, Re[θ1]) further gives Φ = ∫ L(1 − cosϑ)dϕ = Ω, where Ω is the solid angle enclosed by L, up to the orientation convention. The gate phase is thus geometric in origin, being determined by the enclosed area rather than by the rate at which...

  3. [3]

    yields ˜Ω ± 0 (t) = − 1 2 sinα 2 χ ± (t)e± i[θ2(t)− θ1(t)], ˜Ω ± 1 (t) = 1 2 cosα 2 χ ± (t)e± i[θ2(t)− θ3(t)], (16) ∆( t) = f2(t) sin2 β (t) 2 +f3(t) cos2 β (t) 2 − Re ˙θ1(t), withχ ± (t) = [f3(t) − f2(t)] sinβ (t) ± i ˙β (t). Most impor- tantly, within the no-jump dynamics the intrinsic dissi- pation is matched exactly by the control through Γ i(t) = − I...

  4. [4]

    These choices give the ge- ometric phase Φ = π and the rotation axis ⃗ n= (1, 0, 0), so that Eq

    sin(ξt/ 2)], Re[ θ1(t)] = ξt, and θ = 0, with ξt ∈ [0, 2π ]. These choices give the ge- ometric phase Φ = π and the rotation axis ⃗ n= (1, 0, 0), so that Eq. ( 12) yields Us =iσ x. Apart from the overall phase factor eiπ/ 2, this operation is physically equivalent to the NOT gate σx, namely, |a⟩ ↦→ | a ⊕ 1⟩. For the short gate duration considered here, ∆ ...

  5. [5]

    ( 13) remain unchanged; the only effect is to imprint static phase shifts on the driving fields

    shows that the detuning ∆( t), the magnitudes of the effective couplings ˜Ω ′ ± i (t), and the holonomic angle Φ defined in Eq. ( 13) remain unchanged; the only effect is to imprint static phase shifts on the driving fields. In terms of the phys- ical pulse phases, this corresponds to δφ 0 0 = −δφ 1 0 =δ21 and δφ 1 1 = −δφ 0 1 = δ32, where δ21 = δθ2 − δθ1 and...

  6. [6]

    Let U and V denote the ideal and perturbed no-jump prop- agators over one gate cycle

    are determined by the effective Rabi couplings, amplitude miscalibration is modeled as ˜Ω ± i (t) → (1 + δr) ˜Ω ± i (t) for i = 0, 1, while detuning errors are incorporated as ∆( t) → ∆( t) +δdΩ max, with Ω max the maximum instan- taneous coupling strength of the ideal protocol. Let U and V denote the ideal and perturbed no-jump prop- agators over one gate...

  7. [7]

    P. W. Shor, Polynomial-Time Algorithms for Prime Fac- torization and Discrete Logarithms on a Quantum Com- puter, SIAM J. Comput. 26, 1484 (1997)

    1997

  8. [8]

    L. K. Grover, Quantum Mechanics Helps in Searching for a Needle in a Haystack, Phys. Rev. Lett. 79, 325 (1997)

    1997

  9. [9]

    Aaronson and A

    S. Aaronson and A. Arkhipov, The Computational Com- plexity of Linear Optics, Theory Comput. 9, 143 (2013)

    2013

  10. [10]

    R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982)

    1982

  11. [11]

    S. L. Zhu and P. Zanardi, Geometric quantum gates that are robust against stochastic control errors, Phys. Rev. A 72, 020301(R) (2005)

    2005

  12. [12]

    Berger, M

    S. Berger, M. Pechal, A. A. Abdumalikov, Jr., C. Eich- ler, L. Steffen, A. Fedorov, A. Wallraff, and S. Filipp, Exploring the effect of noise on the Berry phase, Phys. Rev. A 87, 060303(R) (2013)

    2013

  13. [13]

    Johansson, E

    M. Johansson, E. Sj¨ oqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, Robustness of nonadiabatic holonomic gates, Phys. Rev. A 86, 062322 (2012)

    2012

  14. [14]

    Ekert, M

    A. Ekert, M. Ericsson, P. Hayden, H. Inamori, J. A. Jones, D. K. L. Oi, and V. Vedral, Geometric quantum computation, J. Mod. Opt 47, 2501 (2000)

    2000

  15. [15]

    L. M. Duan, J. I. Cirac, and P. Zoller, Geometric Ma- nipulation of Trapped Ions for Quantum Computation, Science 292, 1695 (2001)

    2001

  16. [16]

    Solinas, P

    P. Solinas, P. Zanardi, and N. Zanghi, Robustness of non-Abelian holonomic quantum gates against paramet- ric noise, Phys. Rev. A 70, 042316 (2004)

    2004

  17. [17]

    L. A. Wu, P. Zanardi, and D. A. Lidar, Holonomic Quan- tum Computation in Decoherence-Free Subspaces, Phys. Rev. Lett. 95, 130501 (2005)

    2005

  18. [18]

    M. V. Berry, Quantal phase factors accompanying adia- batic changes, Proc. R. Soc. A 392, 45 (1984)

    1984

  19. [19]

    Wilczek and A

    F. Wilczek and A. Zee, Appearance of Gauge Structure in Simple Dynamical Systems, Phys. Rev. Lett. 52, 2111 (1984)

    1984

  20. [20]

    Aharonov and J

    Y. Aharonov and J. Anandan, Phase Change during a Cyclic Quantum Evolution, Phys. Rev. Lett. 58, 1593 (1987). 6

    1987

  21. [21]

    Solinas, P

    P. Solinas, P. Zanardi, N. Zanghi, and F. Rossi, Nonadia - batic geometrical quantum gates in semiconductor quan- tum dots, Phys. Rev. A 67, 052309 (2003)

    2003

  22. [22]

    Ota and Y

    Y. Ota and Y. Kondo, Composite pulses in NMR as nonadiabatic geometric quantum gates, Phys. Rev. A 80, 024302 (2009)

    2009

  23. [23]

    P. Z. Zhao, X. D. Cui, G. F. Xu, E. Sj¨ oqvist, and D. M. Tong, Rydberg-atom-based scheme of nonadiabatic geo- metric quantum computation, Phys. Rev. A 96, 052316 (2017)

    2017

  24. [24]

    Chen and Z

    T. Chen and Z. Y. Xue, Nonadiabatic Geometric Quan- tum Computation with Parametrically Tunable Cou- pling, Phys. Rev. Appl. 10, 054051 (2018)

    2018

  25. [25]

    K. Z. Li, P. Z. Zhao, and D. M. Tong, Approach to real- izing nonadiabatic geometric gates with prescribed evo- lution paths, Phys. Rev. Research, 2, 023295 (2020)

    2020

  26. [26]

    S. L. Zhu and Z. D. Wang, Implementation of Univer- sal Quantum Gates Based on Nonadiabatic Geometric Phases, Phys. Rev. Lett. 89, 097902 (2002)

    2002

  27. [27]

    S. L. Zhu and Z. D. Wang, Universal quantum gates based on a pair of orthogonal cyclic states: Application to NMR systems, Phys. Rev. A 67, 022319 (2003)

    2003

  28. [28]

    Y. Ota, Y. Goto, Y. Kondo, and M. Nakahara, Geomet- ric quantum gates in liquid-state NMR based on a can- cellation of dynamical phases, Phys. Rev. A 80, 052311 (2009)

    2009

  29. [29]

    Sj¨ oqvist, D

    E. Sj¨ oqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, Non-adiabatic holonomic quantum computation, New J. Phys. 14, 103035 (2012)

    2012

  30. [30]

    G. F. Xu, J. Zhang, D. M. Tong, E. Sj¨ oqvist, and L. C. Kwek, Nonadiabatic Holonomic Quantum Computation in Decoherence-Free Subspaces, Phys. Rev. Lett. 109, 170501 (2012)

    2012

  31. [31]

    G. F. Xu, P. Z. Zhao, T. H. Xing, E. Sj¨ oqvist, and D. M. Tong, Composite nonadiabatic holonomic quantum computation, Phys. Rev. A 95, 032311 (2017)

    2017

  32. [32]

    Herterich and E

    E. Herterich and E. Sj¨ oqvist, Single-loop multiple-p ulse nonadiabatic holonomic quantum gates, Phys. Rev. A 94, 052310 (2016)

    2016

  33. [33]

    B. J. Liu , X. K. Song, Z. Y. Xue, X. Wang, and M. H. Yung, Plug-and-Play Approach to Nonadiabatic Ge- ometric Quantum Gates, Phys. Rev. Lett. 123, 100501 (2019)

    2019

  34. [34]

    S. L. Zhu and Z. D. Wang, Unconventional Geomet- ric Quantum Computation, Phys. Rev. Lett. 91, 187902 (2003)

    2003

  35. [35]

    S. B. Zheng, Unconventional geometric quantum phase gates with a cavity QED system, Phys. Rev. A 70, 052320 (2004)

    2004

  36. [36]

    P. Z. Zhao, G. F. Xu, and D. M. Tong, Nonadiabatic ge- ometric quantum computation in decoherence-free sub- spaces based on unconventional geometric phases, Phys. Rev. A 94, 062327 (2016)

    2016

  37. [37]

    J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Geo- metric quantum computation using nuclear magnetic res- onance, Nature (London) 403, 869 (2000)

    2000

  38. [38]

    J. F. Du, P. Zou, and Z. D. Wang, Experimental im- plementation of high-fidelity unconventional geometric quantum gates using an NMR interferometer, Phys. Rev. A 74, 020302(R) (2006)

    2006

  39. [39]

    G. Feng, G. Xu, and G. Long, Experimental Realiza- tion of Nonadiabatic Holonomic Quantum Computation, Phys. Rev. Lett. 110, 190501 (2013)

    2013

  40. [40]

    H. Li, Y. Liu, and G. L. Long, Experimental realization of single-shot nonadiabatic holonomic gates in nuclear spins, Sci. China. Phys. Mech. Astron. 60, 080311 (2017)

    2017

  41. [41]

    Leibfried, B

    D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Bar- rett, J. Britton, W. M. Itano, B. Jelenkovi´ c, C. Langer, T. Rosenband et al. , Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate, Nature 422, 412 (2003)

    2003

  42. [42]

    M. Z. Ai, S. Li, Z. Hou, R. He, Z. H. Qian, Z. Y. Xue, J. M. Cui, Y. F. Huang, C. F. Li, and G. C. Guo, Experimental Realization of Nonadiabatic Holo- nomic Single-Qubit Quantum Gates with Optimal Con- trol in a Trapped Ion, Phys. Rev. Applied 14, 054062 (2020)

    2020

  43. [43]

    M. Z. Ai, S. Li, R. He, Z. Y. Xue, J. M. Cui, Y. F. Huang, C. F. Li, and G. C. Guo, Experimental realization of nonadiabatic holonomic single-qubit quantum gates with two dark paths in a trapped ion, Fundam. Res. 2, 661 (2022)

    2022

  44. [44]

    Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y. P. Song, Z. Y. Xue, Z. Q. Yin et al. , Single- Loop Realization of Arbitrary Nonadiabatic Holonomic Single-Qubit Quantum Gates in a Superconducting Cir- cuit, Phys. Rev. Lett. 121, 110501 (2018)

    2018

  45. [45]

    Z. X. Zhang, P. Z. Zhao, T. H. Wang, L. Xiang, Z. L. Jia, P. Duan, D. M. Tong, Y. Yin, and G. P. Guo, Single- shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit, New J. Phys. 21, 073024 (2019)

    2019

  46. [46]

    Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song et al. , Experimen- tal Implementation of Universal Nonadiabatic Geomet- ric Quantum Gates in a Superconducting Circuit, Phys. Rev. Lett. 124, 230503 (2020)

    2020

  47. [47]

    P. Z. Zhao, Z. J. Z. Dong, Z. X. Zhang, G. P. Guo, D. M. Tong, and Y. Yin, Experimental realization of nona- diabatic geometric gates with a superconducting Xmon qubit, Sci. China Phys. Mech. Astron. 64, 250362 (2021)

    2021

  48. [48]

    X. X. Yang, L. L. Guo, H. F. Zhang, L. Du, C. Zhang, H. R. Tao, Y. Chen, P. Duan, Z. L. Jia, W. C. Kong, and G. P. Guo, Experimental Implementation of Short-Path Nonadiabatic Geometric Gates in a Superconducting Cir- cuit, Phys. Rev. Applied 19, 044076 (2023)

    2023

  49. [49]

    A. C. Silvia, L. Andrii, W. H. Stefan, and B. Gopalakrish - nan, Room temperature high-fidelity holonomic single- qubit gate on a solid-state spin, Nat Commun 5, 4870 (2014)

    2014

  50. [50]

    C. Zu, W. B. Wang, L. He, W. G. Zhang, C. Y. Dai, F. Wang, and L. M. Duan, Experimental realization of universal geometric quantum gates with solid state spins, Nature (London) 514, 72 (2014)

    2014

  51. [51]

    B. B. Zhou, P. C. Jerger, V. O. Shkolnikov, F. J. Here- mans, G. Burkard, and D. D. Awschalom, Holonomic Quantum Control by Coherent Optical Excitation in Di- amond, Phys. Rev. Lett. 119, 140503 (2017)

    2017

  52. [52]

    Nagata, K

    K. Nagata, K. Kuramitani, Y. Sekiguchi, and H. Kosaka, Universal holonomic quantum gates over geometric spin qubits with polarised microwaves, Nat. Commun. 9, 3227 (2018)

    2018

  53. [53]

    Sekiguchi, K

    Y. Sekiguchi, K. Matsushita, Y. Kawasaki, and H. Kosaka, Optically addressable universal holonomic quan- tum gates on diamond spins, Nat. Photon. 16, 662 (2022)

    2022

  54. [54]

    Y. C. Zhou, R. L. Ma, Z. Kong, A. R. Li, C. Zhang, X. Zhang, Y. Liu, H. T. Jiang, Z. T. Wu, G. L. Wang, G. 7 Cao, G. C. Guo, H. O. Li, and G. P. Guo, High-fidelity geometric quantum gates exceeding 99 .9% in germanium quantum dots, Nat. Commun. 16, 7953 (2025)

    2025

  55. [55]

    Recati, T

    A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, and P. Zoller, Holonomic quantum computation with neutral atoms, Phys. Rev. A 66, 032309 (2002)

    2002

  56. [56]

    Kiefer, Z

    Y. Kiefer, Z. Zhu, L. Fischer, S. Jele, M. G¨ achter, G. Bi s- son, K. Viebahn, and T. Esslinger, Protected quantum gates using qubit doublons in dynamical optical lattices, Nature 652, 609 (2026)

    2026

  57. [57]

    J. C. Garrison and E. M. Wright, Complex geometrical phases for dissipative systems, Phys. Lett. A 128, 177 (1988)

    1988

  58. [58]

    X. D. Cui and Y. Zheng, Geometric phases in non- Hermitian quantum mechanics, Phys. Rev. A 86, 064104 (2012)

    2012

  59. [59]

    H. M. Dehnavi and A. Mostafazadeh, Geometric phase for non-Hermitian Hamiltonians and Its Holonomy Inter- pretation, J. Math. Phys. 49, 082105 (2008)

    2008

  60. [60]

    M. S. Sarandy and D. A. Lidar, Abelian and non-Abelian geometric phases in adiabatic open quantum systems, Phys. Rev. A 73, 062101 (2006)

    2006

  61. [61]

    X. Y. Lou, Y. S. Cheng, P. Z. Zhao and D. M. Tong, Nonadiabatic non-Abelian geometric phase in non- Hermitian quantum systems, Phys. Rev. A 112, 022222 (2025)

    2025

  62. [62]

    T. X. Hou and W. Li, Nonadiabatic geometric quantum computation in non-Hermitian systems, Phys. Rev. A 109, 022616 (2024)

    2024

  63. [63]

    T. X. Hou, Y. Liu, Y. Jin, and H.-F. Wang, Holonomic gates in tripod-type four-level non-Hermitian quantum systems, Phys. Rev. A 113, 052614 (2026)

    2026

  64. [64]

    Li and Y

    W. Li and Y. Song, SU (3)-transformation-based inverse engineering for fast population transfer in three-level sy s- tems, J. Phys. B: At. Mol. Opt. Phys. 55, 125501 (2022)

    2022

  65. [65]

    Levine, A

    H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, High-Fidelity Control and En- tanglement of Rydberg-Atom Qubits, Phys. Rev. Lett. 121, 123603 (2018)

    2018

  66. [66]

    H¨ olzl, A

    C. H¨ olzl, A. G¨ otzelmann, E. Pultinevicius, M. Wirth, and F. Meinert, Long-Lived Circular Rydberg Qubits of Alkaline-Earth Atoms in Optical Tweezers, Phys. Rev. X 14, 021024 (2024)

    2024

  67. [67]

    Lindblad, On the Generators of Quantum Dynamical Semigroups, Commun

    G. Lindblad, On the Generators of Quantum Dynamical Semigroups, Commun. Math. Phys. 48, 119 (1976)

    1976

  68. [68]

    Dalibard, Y

    J. Dalibard, Y. Castin, and K. Mølmer, Wave-function Approach to Dissipative Processes in Quantum Optics, Phys. Rev. Lett. 68, 580 (1992)

    1992

  69. [69]

    S. B. Zheng, C. P. Yang, and F. Nori, Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical coun- terparts, Phys. Rev. A 93, 032313 (2016)

    2016