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Cited by 94Quantum Information and CryptographyQuantum Mechanics and ApplicationsAdvanced Thermodynamics and Statistical Mechanics

Mechanism for quantum speedup in open quantum systems

Open access
Liu, Hai-Bin, Yang, W. L., An, Jun-Hong, Xu, Zhen-Yu·2016·Physical Review A
DOI 10.1103/physreva.93.020105
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Abstract

The quantum speed limit (QSL) time for open system characterizes the most efficient response of the system to the environmental influences. Previous results showed that the non-Markovianity governs the quantum speedup. Via studying the dynamics of a dissipative two-level system, we reveal that the non-Markovian effect is only the dynamical way of the quantum speedup, while the formation of the system-environment bound states is the essential reason for the quantum speedup. Our attribution of the quantum speedup to the energy-spectrum character can supply another vital path for experiments when the quantum speedup shows up without any dynamical calculations. The potential experimental observation of our quantum speedup mechanism in the circuit QED system is discussed. Our results may be of both theoretical and experimental interest in exploring the ultimate QSL in realistic environments, and may open new perspectives for devising active quantum speedup devices.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper argues that in a dissipative two-level system the formation of system-environment bound states, rather than non-Markovianity itself, is the essential mechanism behind quantum speedup. It shows that the quantum speed limit time and non-Markovianity share thresholds matching bound-state formation, and proposes a circuit QED realization.

Key findings

  • For an Ohmic spectrum with \(s=1\), \(\omega_c\tau=800\), and \(\omega_0/\omega_c=0.1\), both the QSL time \(\tau_{\rm QSL}\) and non-Markovianity \(N\) show a threshold at \(\eta_c=0.1\): for \(\eta<\eta_c\), \(\tau_{\rm QSL}/\tau=1\) and \(N=0\); for \(\eta\ge\eta_c\), \(\tau_{\rm QSL}/\tau\) steeply decreases and \(N\) steeply increases.
  • The critical point for forming the bound state matches the threshold for quantum speedup and non-Markovianity. With a bound state, \(|c(t)|^2\) tends to a finite value and \(\Gamma(t)\) transiently becomes negative and asymptotically approaches zero; without it, \(\Gamma(t)\) tends to a positive constant and \(|c(t)|^2\) decays to zero.
  • In the proposed circuit QED setup with \(\xi=0.05\omega_c\), \(\omega_0=1.08\omega_c\), and \(N=800\): for \(\eta\lesssim0.015\) no bound state and no speedup; for \(0.015\lesssim\eta\lesssim0.07\) one bound state forms with remarkable speedup and non-Markovianity; for \(\eta\ge0.07\) two bound states form, \(\tau_{\rm QSL}/\tau\) almost reduces to zero, and \(N\) tends to divergence.
  • The mechanism is claimed to be generic, also applicable to a dissipative harmonic oscillator and the spin-boson model.

Methods

  • The model is a two-level system coupled dissipatively to a bosonic environment, governed by an exact master equation with time-dependent renormalized frequency \(\Omega(t)\) and decay rate \(\Gamma(t)\). The spectral density is Ohmic, \(J(\omega)=\eta\omega^s\omega_c^{1-s}e^{-\omega/\omega_c}\).
  • The QSL time is computed from the Bures angle and operator norm, and non-Markovianity from the trace-distance increase; the paper uses \(\tau_{\rm QSL}=\tau\frac{1-|c(\tau)|^2}{1-|c(\tau)|^2+2N}\). Energy-spectrum analysis identifies a bound state with isolated eigenvalue \(E_{\rm BS}\) in the band gap.
  • A possible experimental realization is discussed using a superconducting qubit interacting with an array of coupled microwave superconducting resonators.

Limitations

  • The paper states that the analysis is for a zero-temperature environment, so only the \(N=0\) and \(N=1\) subspaces are involved in the dynamics.
  • The paper discusses a possible experimental realization but does not report an experiment; the circuit QED scheme is proposed as accessible rather than demonstrated.
  • The text makes evident that the mechanism is supported by numerical and theoretical analysis and is described as a conjecture for the general case, with the authors stating “We thus conjecture” and that the mechanism is “expected” to be valid for other spectra and systems.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.93.020105

Cites

53 of 53 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 21 cited sources checked · 0 supported · 0 flagged

  • Findingnone given
    § Mechanism for quantum speedup in open quantum systems · p. 4

    In the proposed circuit QED system, two bound states at most could be formed.

    “The novel character of this system is that two bound states at most could be formed”
    Quote in the paperSubstantial claim
  • Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 2

    For the dissipative TLS with Ohmic spectrum s=1, both τ_QSL and N show an obvious threshold at the critical point η_c=0.1.

    “It is interesting to find that both τ QSL and N show an obvious threshold at the critical point η c = 0 . 1.”
    Quote in the paperSubstantial claim
  • Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 3

    For η<η_c (no bound state), Γ(t) tends to a positive constant after a short-time jolt.

    “We can see that Γ( t ) in the absence of the bound state when η &lt; η c tends to a positive constant after a short-time jolt.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § Mechanism for quantum speedup in open quantum systems · p. 3

    When the bound state is formed, after some short-time oscillations, |c(t)|^2 tends to a finite value matching Eq. (9).

    “Consequently, after some short-time oscillations, | c ( t ) | 2 tends to a finite value matching well with the result in Eq. (9).”
    Quote in the paperSubstantial claim
  • Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 4

    The threshold point from Markovian to non-Markovian and from no-speedup to speedup regimes matches well with the one for forming the bound state.

    “The results confirm that the threshold point from Markovian to non-Markovian and from no-speedup to speedup regimes matches well with the one for forming the bound state.”
    Quote in the paperSubstantial claim
  • May combine two claims. 13
  • Classified as probably not a substantial claim. 2

  • May combine two claims.

    Findingreasoning
    § Mechanism for quantum speedup in open quantum systems · p. 3

    For η<η_c, the environment has no backaction on the system and N is zero.

    “Here the environment has no backaction on the system and thus N is zero.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 4

    For η≲0.015, no bound state is formed and no speedup is present.

    “When η ≲ 0 . 015, no bound state is formed and no speedup is present.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 4

    For 0.015≲η≲0.07, one bound state is formed, and there are remarkable quantum speedup and non-Markovianity.

    “When 0 . 015 ≲ η ≲ 0 . 07, one bound state is formed, and there are remarkable quantum speedup and non-Markovianity.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 2

    When η≥η_c, τ_QSL/τ experiences a steep decrease and N experiences a steep increase.

    “However, when η ≥ η c , τ QSL /τ experiences a steep decrease and N experiences a steep increase.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Mechanism for quantum speedup in open quantum systems · p. 3

    The transient increase of |c(t)|^2 causes the increase of distinguishability and the decrease of τ_QSL.

    “The transient increase of | c ( t ) | 2 causes the increase of the distinguishability and the decrease of τ QSL .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § Mechanism for quantum speedup in open quantum systems · p. 4

    For η≥0.07, two bound states are formed, their effect is greatly enhanced and τ_QSL/τ almost reduces to zero and N tends to divergence.

    “When η ≥ 0 . 07, two bound states are formed, their effect is greatly enhanced and τ QSL /τ almost reduces to zero and N tends to divergence.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § Mechanism for quantum speedup in open quantum systems · p. 4

    Our mechanism is applicable to a dissipative harmonic oscillator, whose dynamics is governed by the same bound-state mechanism as our TLS, and the spin-boson model, where the bound state has been found playing a leading role in the quantum phase transition.

    “Our mechanism is applicable to a dissipative harmonic oscillator, whose dynamics is governed by the same bound-state mechanism as our TLS [45, 46], and the spinboson model, where the bound state has been found playing a leading role in the quantum phase transition of the model [39, 47].”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [45] Y.-Q. L¨ u, J.-H. An, X.-M. Chen, H.-G. Luo, and C. H. Oh, Phys. Rev. A 88 , 012129 (2013). source

      Not checked in the free run.

    • cites [46] C.-J. Yang, J.-H. An, H.-G. Luo, Y. Li, and C. H. Oh, Phys. Rev. E 90 , 022122 (2014). source

      Not checked in the free run.

    • cites [39] H.-B. Liu, J.-H. An, C. Chen, Q.-J. Tong, H.-G. Luo, and C. H. Oh, Phys. Rev. A 87 , 052139 (2013). source

      Not checked in the free run.

    • cites [47] Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. B 84 , 174301 (2011). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § Mechanism for quantum speedup in open quantum systems · p. 4

    It is expected that N cannot converge with respect to the final time τ and τ_QSL tends to zero in the proposed circuit QED system.

    “It is expected that N cannot converge with respect to the final time τ and τ QSL tends to zero.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § Mechanism for quantum speedup in open quantum systems · p. 1

    The non-Markovian effect is only the dynamical way of quantum speedup, while the formation of system-environment bound states is the essential reason for quantum speedup.

    “Via studying the dynamics of a dissipative two-level system, we reveal that the non-Markovian effect is only the dynamical way of the quantum speedup, while the formation of the system-environment bound states is the essential reason for the quantum speedup.”
    • Also stated in: § Mechanism for quantum speedup in open quantum systems · p. 3 · plain
      “Hence the bound state supplies a latent capability, while the non-Markovianity only supplies the dynamical way to the system for quantum speedup.”
    • Also stated in: § Mechanism for quantum speedup in open quantum systems · p. 4 · boosted
      “It is revealed that the bound state of the whole system, as an isolated eigenstate with eigenvalue residing in the bandgap of the eigenspectrum, significantly changes the decoherence dynamics and plays a decisive role in quantum speedup.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § Mechanism for quantum speedup in open quantum systems · p. 2

    When η≥η_c, the system presents a dramatic speedup and non-Markovian effect.

    “It indicates that the system presents a dramatic speedup and nonMarkovian effect.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § Mechanism for quantum speedup in open quantum systems · p. 4

    Our mechanism cannot be captured by the previous treatment with Lorentzian spectrum, where the lower limit of the frequency integration is artificially extended from 0 to -∞, and this approximation loses the availability of the bound state.

    “Note that our mechanism cannot be captured by the previous treatment with Lorentzian spectrum [25, 26, 28, 33, 34], where the lower limit of the frequency integration in f ( t -τ ) is artificially extended from 0 to -∞ . This approximation is mathematically convenient but loses the availability of the bound state [40].”
    Quote in the paperSubstantial claim0 of 6 sources checked
    • cites [25] S. Deffner and E. Lutz, Phys. Rev. Lett. 111 , 010402 (2013). source

      Not checked in the free run.

    • cites [26] Y.-J. Zhang, W. Han, Y.-J. Xia, J.-P. Cao, and H. Fan, Sci. Rep. 4 , 4890 (2014). source

      Not checked in the free run.

    • cites [28] Z. Sun, J. Liu, J. Ma, and X. Wang, Sci. Rep. 5 , 8444 (2015). source

      Not checked in the free run.

    • cites [33] Z.-Y. Xu, S. Luo, W. L. Yang, C. Liu, and S. Zhu, Phys. Rev. A 89 , 012307 (2014). source

      Not checked in the free run.

    • cites [34] C. Liu, Z.-Y. Xu, and S. Zhu, Phys. Rev. A 91 , 022102 (2015). source

      Not checked in the free run.

    • cites [40] Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys. Rev. A 81 , 052330 (2010). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Mechanism for quantum speedup in open quantum systems · p. 1

    Non-Markovian effect characterized by non-Markovianity can speed up quantum evolution.

    “Deffner and Lutz [25] showed that non-Markovian effect characterized by non-Markovianity [29-32] can speed up quantum evolution, which has been verified in different settings [26, 28, 33, 34].”
    • Also stated in: § Mechanism for quantum speedup in open quantum systems · p. 2 · plain
      “It has been found that the non-Markovian effect can speed up quantum evolution [25, 33].”
    Quote in the paperSubstantial claim0 of 5 sources checked
    • cites [25] S. Deffner and E. Lutz, Phys. Rev. Lett. 111 , 010402 (2013). source

      Not checked in the free run.

    • cites [26] Y.-J. Zhang, W. Han, Y.-J. Xia, J.-P. Cao, and H. Fan, Sci. Rep. 4 , 4890 (2014). source

      Not checked in the free run.

    • cites [28] Z. Sun, J. Liu, J. Ma, and X. Wang, Sci. Rep. 5 , 8444 (2015). source

      Not checked in the free run.

    • cites [33] Z.-Y. Xu, S. Luo, W. L. Yang, C. Liu, and S. Zhu, Phys. Rev. A 89 , 012307 (2014). source

      Not checked in the free run.

    • cites [34] C. Liu, Z.-Y. Xu, and S. Zhu, Phys. Rev. A 91 , 022102 (2015). source

      Not checked in the free run.

  • May combine two claims.

    Premisereasoning
    § Mechanism for quantum speedup in open quantum systems · p. 2

    The decay rate Γ(t) is not positively definite; negative Γ(t) causes energy backflow from the environment to the TLS.

    “Note that the decay rate Γ( t ) is by no means positively definite. Actually it is just the negative Γ( t ) which causes the energy backflow from the environment to the TLS.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § Mechanism for quantum speedup in open quantum systems · p. 2

    When η<η_c, there is no quantum speedup and no non-Markovian effect.

    “Thus there does not exist any quantum speedup and non-Markovian effect.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning
    § Mechanism for quantum speedup in open quantum systems · p. 4

    This mechanism can supply experimenters with a useful guideline to judge when the quantum speedup shows up without any calculations to its explicit dynamics.

    “This mechanism can supply experimenters with a useful guideline to judge when the quantum speedup shows up without any calculations to its explicit dynamics.”
    Quote in the paperNot a substantial claim
15 of 15

  • The paper relies on this without support.

    The non-Markovianity is not an experimental observable.

    “such attribution of the quantum speedup to the non-Markovianity is experimentally meaningless because the non-Markovianity is not an experimental observable.”
1 of 1

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • In addition to non-Markovianity, initial-state preparation, driving protocol, and energy-spectrum characteristics govern the maximal speed of quantum evolution.

    “Liu et al [155] showed that in addition to non-Markovianity, the preparation of the initial state, and the choice of the driving protocol, the characteristics of the energy spectrum govern the maximal speed of quantum evolution.”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [155]

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. L. Mandelstam, 1945, J. Phys. (USSR), 9, 249

Cited by

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Related

Authors
Liu, H.-B., Yang, W. L., An, J.-H., & Xu, Z.-Y.
Affiliations
  • Chinese Academy of Sciences (CN)
  • Lanzhou University (CN)
ORCID
An, Jun-Hong
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2016-02-25
Volume
93
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 2469-9926
  • 2469-9934
DOI
10.1103/physreva.93.020105
OpenAlex
W2281606705
Cited by
94
References
53
Citations by year
2026: 3 · 2025: 9 · 2024: 15 · 2023: 6 · 2022: 10 · 2021: 10 · 2020: 12 · 2019: 8 · 2018: 10 · 2017: 8 · 2016: 3
FWCI
7.96
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Lanzhou University Institutional Repository (submittedVersion)
  • arXiv (Cornell University)
Funders
National Natural Science Foundation of China (11175072, 11274351, 11574353, 11204196, 11474139)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
Keywords
quantum speed limit, open quantum systems, non-Markovianity, quantum advantage, energy spectrum, circuit quantum electrodynamics

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