Trade-Off Between Speed and Cost in Shortcuts to Adiabaticity
Open accessAbstract
Achieving effectively adiabatic dynamics is a ubiquitous goal in almost all areas of quantum physics. Here, we study the speed with which a quantum system can be driven when employing transitionless quantum driving. As a main result, we establish a rigorous link between this speed, the quantum speed limit, and the (energetic) cost of implementing such a shortcut to adiabaticity. Interestingly, this link elucidates a trade-off between speed and cost, namely, that instantaneous manipulation is impossible as it requires an infinite cost. These findings are illustrated for two experimentally relevant systems-the parametric oscillator and the Landau-Zener model-which reveal that the spectral gap governs the quantum speed limit as well as the cost for realizing the shortcut.
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- Findingreasoning
The maximal speed with which a quantum state can undergo transitionless quantum driving is given by v_QSL = sqrt(ε_n^2(t) + (∂t C)^2) / (ħ cos(L_t) sin(L_t)).
“Substituting Eqs. ( 5) and ( 6) into Eq.( 3) we obtain the maximal speed with which a quantum state can undergo transitionless quantum driving v QSL ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε 2 n ðtÞ þ ð∂ t CÞ 2 p ℏ cosðL t Þ sinðL t Þ ;”
Quote in the paperSubstantial claim - Findingreasoning
For transitionless quantum driving, the quantity ε_t equals the square root of the sum of the squared instantaneous eigenenergy and the norm of the time derivative of the instantaneous eigenstate.
“By further noting that jψ t i ¼ jn t i and HðtÞ ¼ H 0 ðtÞ þ H 1 ðtÞ with H 1 ðtÞ as given in Eq. ( 1) ϵ t simply becomes ϵ t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε 2 n ðtÞ þ h∂ t n t j∂ t n t i q ;”
Quote in the paperSubstantial claim - Findingreasoning
The maximal speed v_QSL and the instantaneous cost ∂t C exhibit qualitatively opposite behaviors for the compression and expansion protocols.
“Remarkably, v QSL and ∂ t C exhibit qualitatively opposite behaviors for the two protocols.”
Quote in the paperSubstantial claim - Findingnone given
For the parametric harmonic oscillator, the instantaneous angle is L_t = arccos( sqrt(2 sqrt(ω0 ω_t)/(ω0+ω_t)) ).
“Finally, the instantaneous angle can be written as L t ¼ arccosð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffiffiffiffiffiffiffiffiffiffi ω 0 ω t p =ðω 0 þ ω t Þ q Þ:”
Quote in the paperSubstantial claim - Findingreasoning
For the parametric harmonic oscillator, the maximal quantum speed is v_QSL = sqrt((ħω_t/2)^2 + (∂t ω_t / sqrt(8) ω_t)^2) / (ħ cos(L_t) sin(L_t)).
“and the maximal quantum speed reads v QSL ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðℏω t =2Þ 2 þ ð∂ t ω t = ffiffi ffi 8 p ω t Þ 2 q ℏ cosðL t Þ sinðL t Þ :”
Quote in the paperSubstantial claim
- Findingown data
For the harmonic oscillator, the QSL time is generally significantly larger for an expansion than for a compression.
“Interestingly, we see that generally the QSL time is significantly larger for the expansion.”
Quote in the paperSubstantial claim - Findingown data
In the Landau-Zener model, the speed steadily decreases as the system approaches the avoided crossing and cusps at t=0.5τ.
“We observe that the speed steadily decreases as we approach the AC and cusps at t ¼ 0.5τ.”
Quote in the paperNot a substantial claim - Findingreasoning
The QSL time for transitionless quantum driving is τ_QSL = ħ τ [sin(L_τ)]^2 / (2 ∫0^τ dt sqrt(ε_n^2(t) + (∂t C)^2)).
“and the QSL time becomes τ QSL ¼ ℏτ½sinðL τ Þ 2 2 R τ 0 dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ε 2 n ðtÞ þ ð∂ t CÞ 2 p :”
Quote in the paperSubstantial claim - Findingown data
In the Landau-Zener model, as the system approaches the avoided crossing, the speed using the counterdiabatic field increases.
“In Fig. 2 we see as the system approaches the AC the speed using the counterdiabatic field increases.”
Quote in the paperSubstantial claim - Findingreasoning
For the parametric harmonic oscillator kept in its instantaneous ground state, the cost derivative is ∂t C = ∂t ω_t / (sqrt(8) ω_t).
“A straightforward calculation then reveals that the cost of keeping the oscillator in its instantaneous ground state at all times is ∂ t C ¼ ∂ t ω t ffiffi ffi 8 p ω t ;”
Quote in the paperSubstantial claim - Findingown data
The cost of transitionless quantum driving and the quantum speed limit are rigorously linked.
“As a main result, we establish a rigorous link between this speed, the quantum speed limit, and the (energetic) cost of implementing such a shortcut to adiabaticity.”
- Also stated in: § Abstract · boosted
“In this Letter we show that the cost of TQD [34] and the QSL [51] are intimately connected.”
- Also stated in: § Y S I C A L R E V I E W L E T T E R S · boosted
“We have rigorously proven the relationship between the cost of TQD and the QSL.”
Quote in the paperSubstantial claim0 of 2 sources checked - Also stated in: § Abstract · boosted
- Interpretationreasoning
The spectral gap governs both the quantum speed limit and the cost of realizing the shortcut.
“These findings are illustrated for two experimentally relevant systems-the parametric oscillator and the Landau-Zener model-which reveal that the spectral gap governs the quantum speed limit as well as the cost for realizing the shortcut.”
- Also stated in: § Y S I C A L R E V I E W L E T T E R S · boosted
“Clearly, the gap between the driven state and the rest of the spectrum plays the most crucial role in determining both the QSL and the cost of achieving finite time adiabatic dynamics.”
- Also stated in: § Y S I C A L R E V I E W L E T T E R S · boosted
“we have highlighted the crucial role of the gap for the cost and the speed with which a shortcut can be facilitated.”
Quote in the paperSubstantial claim - Also stated in: § Y S I C A L R E V I E W L E T T E R S · boosted
- Interpretationown data
Counterintuitively, transitionless quantum driving encourages faster driving when the energy gap closes.
“Our results show that the shortcut comes from the increased allowed speed of evolution and, quite counterintuitively, TQD encourages faster driving when the energy gap closes.”
Quote in the paperSubstantial claim - Premisecitation
For the Landau-Zener model, the counterdiabatic correction term is H1 = - g'(t)Δ / (2(Δ^2 + [g(t)]^2)) σ_y.
“The corresponding correction term (1) is readily determined to be [18] H 1 ¼ - g 0 ðtÞΔ 2ðΔ 2 þ ½gðtÞ 2 Þ σ y ;”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [18] source
Not checked in the free run.
- Interpretationcitation
The shorter the time during which a quantum system is driven from the initial to the final energy eigenstate, the more thermodynamic resources must be expended.
“Equation (8) states that the shorter the time is during which a quantum system is driven from the initial to the final energy eigenstate, the more thermodynamic resources have to be expended [81].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [81] source
Not checked in the free run.
- Interpretationcitation
Achieving the same evolution in shorter times requires the use of the counterdiabatic field.
“Achieving the same evolution in shorter times requires the use of the counterdiabatic field (15).”
Quote in the paperSubstantial claim - Interpretationreasoning
If the gap is small, transitions can be suppressed by prohibiting the quantum system from lingering at the avoided crossing.
“Hence, if the relative magnitude of the perturbation is large, i.e., if the gap is small, transitions can be suppressed if the quantum system is prohibited from lingering at the AC.”
Quote in the paperSubstantial claim - Interpretationreasoning
Instantaneous manipulation is impossible because it would require an infinite cost.
“namely, that instantaneous manipulation is impossible as it requires an infinite cost.”
- Also stated in: § Y S I C A L R E V I E W L E T T E R S · plain
“In particular, we have found that effectively instantaneous yet adiabatic dynamics can be achieved at the expense of an infinite cost.”
Quote in the paperSubstantial claim - Also stated in: § Y S I C A L R E V I E W L E T T E R S · plain
- Premisecitation
The minimal time for a quantum system to evolve from its initial to its final state, the QSL time, is approximately ħ/(2E).
“Thus, the minimal time during which a quantum system can evolve from the initial to the final state, i.e., the QSL time τ QSL , is determined by τ QSL ≃ ℏ=2E [48,51].”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning
There is a trade-off between the speed of implementing a shortcut to adiabaticity and its thermodynamic cost: faster implementation requires higher cost.
“Interestingly, this link elucidates a trade-off between speed and cost, namely, that instantaneous manipulation is impossible as it requires an infinite cost.”
- Also stated in: § Abstract · boosted
“As our main result we rigorously prove a trade-off between the speed and the thermodynamic cost: the faster one wants to implement a shortcut, the higher is the thermodynamic cost of realizing the quantum process.”
- Also stated in: § Abstract · boosted
“Second, Eqs. ( 7) and ( 8) express, in a transparent and immediate way, the trade-off between speed and cost of a shortcut to adiabaticity.”
Quote in the paperSubstantial claim - Also stated in: § Abstract · boosted
- Premisecitation
The energy-time Heisenberg uncertainty relation sets a fundamental upper bound on the speed of quantum evolution, known as the quantum speed limit.
“It has been established in virtually all areas of quantum physics [35][36][37][38] that the Heisenberg uncertainty relation for energy and time sets a quantum speed limit (QSL) [39][40][41][42][43], i.e., a fundamental upper bound on the speed of quantum evolution.”
- Also stated in: § Abstract · plain
“The QSL is a fundamental upper bound on the rate with which a quantum state can evolve.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [39][40][41][42][43]
Not checked in the free run.
- Also stated in: § Abstract · plain
- May combine two claims. 6
- The quote only partly states this claim. 1
- Classified as probably not a substantial claim. 1
May combine two claims.
Findingown dataFor a larger splitting Δ=0.01, the same qualitative behavior is observed but the cusp is smoothed out.
“Comparing to the same evolution time for a larger splitting, Δ ¼ 0.01, while the same qualitative behavior is observed we see that the cusp is smoothed out.”
Quote in the paperNot a substantial claimThe quote only partly states this claim.
Findingown dataA critical region emerges that depends delicately on both the energy splitting and the evolution time.
“A further interesting feature is the clear emergence of a”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataFor the Landau-Zener model with Δ=0.001 and τ=10^3, the evolution is close to the adiabatic limit and the instantaneous cost is approximately zero.
“In Fig. 2(a) we set Δ ¼ 0.001 and consider τ ¼ 10 3 (bottom curve), which is close to the adiabatic limit and therefore ∂ t C ≃ 0.”
Quote in the paperNot a substantial claimMay combine two claims.
Findingown dataWhen sufficiently far from the avoided crossing, v_QSL is largely independent of evolution time and the instantaneous cost is close to zero.
“It is clear in both panels that when sufficiently far from the AC, v QSL is largely independent of evolution time and in these regions the instantaneous cost is close to zero.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningApproaching the avoided crossing requires either slowing the evolution accordingly or using a counterdiabatic field.
“Approaching the AC requires that either the evolution is accordingly slowed down or a counterdiabatic field is used.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationThe maximal speed of quantum evolution is given by the time-averaged norm of the generator of the dynamics; for unitary dynamics and orthogonal states it reduces to the average energy E.
“It has been shown that the maximal speed of quantum evolution is given by the time averaged norm of the generator of the dynamics [51], which in the case of unitary dynamics and for orthogonal states reduces to the average energy E [48].”
Quote in the paperSubstantial claim0 of 2 sources checkedClassified as probably not a substantial claim.
PremisecitationFermi's golden rule for time-dependent perturbation theory states that the rate of quantum transitions is determined by the time-integrated magnitude of the perturbation.
“This rule states that the rate of quantum transitions is determined by the time-integrated magnitude of the perturbation.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [76] source
Not checked in the free run.
May combine two claims.
PremisecitationThe Ising model can be expressed as a series of independent Landau-Zener crossings, so results for the LZ model extrapolate to driving a critical many-body system.
“As shown in Ref. [71] the Ising model can be expressed as a series of independent LZ crossings, and therefore the following results extrapolate to driving a critical many-body system.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [71] source
Not checked in the free run.
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 transitionless quantum driving, the instantaneous cost, given by the trace norm of the counterdiabatic Hamiltonian, reduces to equation (122).
“It is easy to see [93], that in the case of transitionless quantum driving the instantaneous cost, i.e. the trace norm of the counterdiabatic Hamiltonian, H 1 (t), reduces to ∂ t C = H 1 (t) tr = ∂ t n t |∂ t n t ,(122)”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [93]
Campbell and Deffner formalised that arbitrarily fast shortcuts to adiabaticity are not possible and how the quantum speed limit enters the picture.
“That this is not the case and how the quantum speed limit enters the picture was formalised by Campbell and Deffner [93].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [93]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rescaling the Landau-Zener Hamiltonian gives a minimum gap of two
MinorEquation (14) has eigenvalues ±√(Δ2 + g2). Dividing the Hamiltonian by Δ therefore makes its minimum separation 2 at g = 0, not 1.
“For the sake of clarity we further rescale H LZ by Δ, H 0 ¼ H LZ =Δ, and hence set the minimal energy gap to 1.”
“H LZ ¼ Δσ x þ gðtÞσ z ;ð14Þ”
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Masuda, S., Nakamura, K., & del Campo, A. (2014) · Physical Review Letters · doi:10.1103/physrevlett.113.063003 · Cited by 77 · see citing works
Optimal control of a qubit in an optical cavity
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Self-citationQuantum work statistics of linear and nonlinear parametric oscillators
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Self-citationAnalytically solvable two-level quantum systems and Landau-Zener interferometry
Barnes, E. (2013) · Physical Review A · doi:10.1103/physreva.88.013818 · Cited by 72 · see citing works
Masuda, S., & Rice, S. A. (2015) · The Journal of Physical Chemistry A · doi:10.1021/acs.jpca.5b00525 · Cited by 71 · see citing works
Quantum speed limit and optimal evolution time in a two-level system
Poggi, P. M., Lombardo, F. C., & Wisniacki, D. A. (2013) · EPL (Europhysics Letters) · doi:10.1209/0295-5075/104/40005 · Cited by 70 · see citing works
Criticality revealed through quench dynamics in the Lipkin–Meshkov–Glick model
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Self-citationThe speed limit of quantum unitary evolution
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Self-citationThermodynamic length for far-from-equilibrium quantum systems
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Self-citationQuantum speed limits and optimal Hamiltonians for driven systems in mixed states
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Energetic Cost of Superadiabatic Quantum Computation
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Non-classical role of potential energy in adiabatic quantum annealing
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Flagged first, then by citations
Research works printed but not linked: 2
- L. Mandelstam, 1945, J. Phys., 9, 249
- A. Messiah, 1966, Quantum Mechanics
Cited by
Sorted by citations
Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control
Deffner, S., & Campbell, S. (2017) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8121/aa86c6 · Cited by 585 · see citing works
Self-citationTightening Quantum Speed Limits for Almost All States
Campaioli, F., Pollock, F. A., Binder, F. C., & Modi, K. (2018) · Physical Review Letters · doi:10.1103/physrevlett.120.060409 · Cited by 148 · see citing works
Energy efficient quantum machines
Abah, O., & Lutz, E. (2017) · EPL (Europhysics Letters) · doi:10.1209/0295-5075/118/40005 · Cited by 101 · see citing works
Tight, robust, and feasible quantum speed limits for open dynamics
Campaioli, F., Pollock, F. A., & Modi, K. (2019) · Quantum · doi:10.22331/q-2019-08-05-168 · Cited by 92 · see citing works
Geometric quantum speed limits: a case for Wigner phase space
Deffner, S. (2017) · New Journal of Physics · doi:10.1088/1367-2630/aa83dc · Cited by 78 · see citing works
Self-citationCampaioli, F., Pollock, F. A., & Vinjanampathy, S. (2018) · Fundamental Theories of Physics · doi:10.1007/978-3-319-99046-0_8 · Cited by 71 · see citing works
- Authors
- Campbell, S., & Deffner, S.
- Affiliations
- Queen's University Belfast (GB)
- University of Maryland, Baltimore County (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2017-03-08
- Volume
- 118
- Issue
- 10
- Pages
- 100601-100601
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2521175289
- Cited by
- 235
- References
- 80
- Citations by year
- 2026: 18 · 2025: 20 · 2024: 19 · 2023: 16 · 2022: 32 · 2021: 28 · 2020: 32 · 2019: 27 · 2018: 23 · 2017: 20
- FWCI
- 13.69
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- Maryland Shared Open Access Repository (USMAI Consortium) (submittedVersion)
- arXiv (Cornell University)
- Digital Repository at the University of Maryland (University of Maryland College Park)
- Bulletin of the American Physical Society
- Funders
- Seventh Framework Programme (618074)
- European Cooperation in Science and Technology (MP1209)
- National Science Foundation (CHE-1648973)
- Julian Schwinger Foundation (JSF-14-7-0000)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Cold Atom Physics and Bose-Einstein Condensates
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- shortcuts to adiabaticity, transitionless quantum driving, quantum speed limit, parametric oscillator, Landau-Zener model, spectral gap, quantum control
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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