Experimental investigation of a quantum Otto heat engine with shortcuts to adiabaticity implemented using counter-adiabatic driving

Arvind; Kavita Dorai; Krishna Shende; Matreyee Kandpal

REVIEW 4 major objections 5 minor 42 references

Experimental investigation of a quantum Otto heat engine with shortcuts to adiabaticity implemented using counter-adiabatic driving

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read An NMR experiment demonstrates that driving a quantum Otto heat engine with counter-adiabatic shortcuts to adiabaticity yields more output power at shorter driving times than the non-adiabatic engine, even after subtracting the energy…

desk verdict A plausible and useful NMR demonstration of an STA quantum Otto engine, but the key quantitative claim rests on an unmeasured theoretical energy cost and the temperature values are internally inconsistent. read the letter →

arxiv 2412.20194 v1 pith:3EYG5SKA submitted 2024-12-28 quant-ph cond-mat.stat-mech

Krishna Shende , Matreyee Kandpal , Arvind , Kavita Dorai This is my paper · ORCID

classification quant-phcond-mat.stat-mech

keywords quantumOttoheatengineshortcutstoadiabaticitycounter-adiabaticdrivingLandau-ZenermodelNMRprocessorspin-1/2systemefficiencywithcostoutputpower

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to show that the usual finite-time penalty of a quantum Otto heat engine—the trade-off between efficiency and power caused by non-adiabatic transitions—can be sidestepped by counter-adiabatic driving, a shortcut-to-adiabaticity protocol that pins the working spin to its instantaneous energy eigenstates. Using spin-1/2 nuclei on an NMR quantum processor, the authors implement a Landau-Zener Otto cycle with shortcut strokes at several driving times and two hot-bath temperatures, and they compare the shortcut engine with the non-adiabatic engine after charging the shortcut's energy cost against the work output. They report that the shortcut engine delivers more output power at shorter driving times in both temperature settings. They also use the comparison to adjudicate between two published efficiency metrics that include the shortcut cost, concluding that the metric which subtracts the shortcut energy from the numerator is the appropriate one for the NMR platform.

What carries the argument

The load-bearing object is the counter-adiabatic (CD) Hamiltonian $H_{\text{CD}}(t) = -\frac{b_x \dot{b}_z(t)}{2[b_x^2 + b_z(t)^2]}\,\sigma_y$, which is added to the Landau-Zener Hamiltonian $H_0(t) = b_x\sigma_x + b_z(t)\sigma_z$ so that the spin follows the instantaneous eigenstates of $H_0$ in finite time. The profile $b_z(t)$ is a polynomial in $t/\tau$ that makes the CD term vanish at the start and end of each stroke. The shortcut cost is defined as $\langle \dot{H}_i\rangle_{\text{STA}} = \int_0^\tau \langle \dot{H}_{\text{CD}}(t)\rangle_{\text{STA}}\,dt$ per stroke, and the two candidate efficiency definitions place this cost either in the denominator (Eq. 9) or in the numerator (Eq. 10) of the efficiency expression; the experimental power data are used to decide which placement is physically correct for this platform.

What would settle it

Measure the actual radio-frequency power delivered by the GRAPE pulses that implement $H_0 + H_{\text{CD}}$ during a stroke and compare the time-integrated power with the theoretical $\langle \dot{H}_{\text{CD}}\rangle \tau$. If the measured integrated power disagrees with the theoretical cost by more than the experimental uncertainty, the reported STA efficiency and power curves would shift, and the conclusion that the second efficiency metric is the appropriate one would need to be re-examined.

Watch

Extended reading notes

Core claim

The paper claims that in a spin-1/2 Landau-Zener quantum Otto engine implemented on an NMR quantum processor, replacing the finite-time unitary strokes with counter-adiabatic shortcuts to adiabaticity improves the engine's performance relative to naively fast (non-adiabatic) driving, and that the improvement survives once the energy consumed by the shortcut is deducted from the work output. The paper further claims that, between the two published ways of incorporating the shortcut cost into efficiency, the metric that subtracts the shortcut energy from the work numerator (Eq. 10) is the one that correctly describes the NMR engine, because the other metric (Eq. 9) reports positive efficiency at driving times where the engine is actually consuming net energy and producing no power.

Load-bearing premise

The cost of the shortcut is taken from the ideal counter-adiabatic Hamiltonian's time-integral (Eq. 12), not from a measurement of the power actually supplied by the GRAPE-optimized radio-frequency pulses.

Editorial extensions

If this is right

At short driving times the shortcut engine produces positive output power while the non-adiabatic engine still consumes net energy; the maximum-power stroke time is 1550 µs and 1210 µs for the shortcut engine, versus 2000 µs and 1500 µs for the non-adiabatic engine, at the two hot temperatures studied.
The efficiency of the shortcut engine lies above that of the non-adiabatic engine at all finite driving times, and both merge at the Otto limit (≈0.629) for long cycle times.
The second efficiency definition (subtracting shortcut energy from the numerator) is the appropriate descriptor for NMR engines, since the first definition gives positive efficiency in a regime where no net work is produced.
The shortcut cost for the expansion stroke is higher than for compression, and the compression cost decreases as the hot spin temperature increases; hence the performance of the STA engine depends on stroke direction and bath temperatures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The quantitative efficiency and power values rest on the theoretical shortcut cost; measuring the actual power delivered by the GRAPE pulses would provide a direct check of Eq. (12), and any discrepancy would shift the numerical comparisons without necessarily overturning the qualitative STA-versus-NA ordering.
The choice between the two cost-accounting metrics is likely to be platform-specific: in platforms where the shortcut is implemented by different physical drives (e.g., a second microwave tone rather than the same RF channel), the 'appropriate' efficiency definition may differ from the one chosen here.
For working media larger than a single spin-1/2, the counter-adiabatic term becomes multiqubit and its experimental cost may scale differently with system size, so the demonstrated benefit of shortcut driving may not transfer directly to many-body quantum engines.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real experimental implementation of a counter-adiabatic STA quantum Otto engine on NMR, and the qualitative story — STA gives useful power at shorter driving times than the non-adiabatic engine — is plausible and supported by the data *if* the energy-cost accounting is right. But the cost in Eqs. (9)-(11) comes from the theoretical H_CD, not from the energy the GRAPE pulses actually deliver, and no measurement of the real pulse power is reported. That gap leaves the central quantitative claim unproven. The manuscript also has internal temperature inconsistencies (cold value given as 11.94 peV and 1.9 peV; hot values differ by the same factor) and a typo in the efficiency-definition discussion.

What's new: putting STA via counter-adiabatic driving into a two-qubit NMR Otto engine and comparing two cost-accounted efficiency metrics on the same platform. Earlier work had theoretical STA-engine analysis and separate NMR engine experiments; this combined them. The experimental work is careful: state tomography, error bars, GRAPE pulses with high fidelity, and the theoretical curves track the data reasonably well.

Soft spots, in order: 1. Resource-accounting gap. Eq. (12) computes the STA cost from the ideal counter-adiabatic Hamiltonian, but the implemented unitaries are GRAPE pulses optimized for fidelity, not for matching that energy. The efficiency and power curves in Fig. 4 therefore rely on an assumption. The claim "even after spending an extra amount of energy" is not validated for the real pulses. Fixable by measuring or bounding the RF power, but as written it is load-bearing. 2. Temperature inconsistencies. Sec. III B and Sec. IV give different cold temperatures (11.94 vs 1.9 peV), and the Fig. 4 caption quotes hot temperatures roughly six times larger than those in the text. The reader cannot tell which is correct. 3. The discussion of the two efficiency definitions is muddled, with Eq. 9 called the "second definition" when it is the first. The argument that one metric is more appropriate rests on a consistency condition, which is fine, but the presentation needs cleaning.

The physics is standard and the qualitative conclusion probably survives a corrected cost accounting. As written, the numbers are not trustworthy. Send it to peer review — it deserves referee time — but referees should demand the actual pulse-energy accounting and consistent temperatures.

Referee Report

4 major / 5 minor

Summary. The manuscript reports an experimental study of a finite-time quantum Otto heat engine implemented on a two-qubit NMR processor, using counter-adiabatic (shortcuts-to-adiabaticity, STA) driving in a spin-1/2 Landau-Zener model. The authors measure efficiency and output power as functions of the driving time for two hot-reservoir spin temperatures, incorporate the cost of the STA drive through two efficiency definitions (Eqs. 9 and 10), and compare the STA engine with a non-adiabatic (NA) engine. They conclude that the STA engine outperforms the NA engine at shorter driving times and that the second efficiency definition (Eq. 10) is the appropriate descriptor for their platform.

Significance. If the claims are correct, the paper provides a useful experimental demonstration of STA in a quantum heat engine and a concrete comparison of cost-inclusive efficiency metrics, which is of interest for quantum thermodynamics and quantum control communities. The experimental data are independent measurements, the theoretical curves are based on a standard Hamiltonian model, and the state tomography plus pulse fidelities are reported. The main value is experimental rather than conceptual, and the conclusions depend critically on the correct accounting of the STA resource cost and on the consistency of the reported reservoir temperatures.

major comments (4)

[§III B, §IV, Fig. 4 caption, Fig. 5 caption] The reservoir spin temperatures are internally inconsistent across the text and figure captions. Section III B and Fig. 4 state kBTC = 11.94 peV and hot temperatures kBT_H1 = 40.54 peV and kBT_H2 = 53.11 peV, while Section IV and Fig. 5 state kBTC = 1.9 peV and hot temperatures 6.45 peV and 8.45 peV; the two sets are related by a common factor of about 6.28. More seriously, the Section III B combination (cold 11.94 peV with hot 6.45/8.45 peV) violates the engine working condition of Eq. (19), since TH/TC is about 0.54, below νf/νi ≈ 2.69, whereas the Section IV combination satisfies the condition. Because the efficiency and power curves, and the maximum-power times (1550/1210 μs), all depend on the actual temperatures, this inconsistency must be resolved before the experimental claims can be evaluated.
[Eqs. (9)–(12)] There is a dimensional inconsistency in the definition and use of the STA cost. Equation (12) defines ⟨Ḣ_i⟩_{STA} as an integral of ⟨Ḣ_CD(t)⟩ over time, which has units of energy. However, Eqs. (9) and (10) add ⟨Ḣ_i⟩_{STA} τ to energy quantities, giving energy×time in the denominator and numerator, and Eq. (11) subtracts the same product. If ⟨Ḣ_i⟩ is intended as a time-averaged power, the right-hand side of Eq. (12) needs an explicit 1/τ prefactor; if it is intended as an energy, the extra τ factors in Eqs. (9)–(11) should be removed. This is not a typographical nuance: the numerical values of η_STA and P_STA, and therefore the comparison with the NA engine, change with the correct accounting.
[§III B, §IV, Fig. 5] The STA cost in Eq. (12) and in the efficiency/power formulas is computed theoretically from the ideal counter-adiabatic Hamiltonian H_CD(t), but the actual expansion and compression unitaries are GRAPE-optimized RF pulses whose energy delivery is not measured or calibrated. The paper reports pulse fidelities but no estimate of the actual RF power consumed by these pulses. Since the central claim is that the STA engine performs better 'even after spending an extra amount of energy,' the resource accounting should include the experimentally delivered STA cost, or at least a quantitative justification that the GRAPE pulse energy matches the ideal ∫⟨H_CD⟩dt. As it stands, the claimed advantage and the stated maximum-power times rest on an unverified equivalence between the theoretical control Hamiltonian and the physical pulse power.
[§IV] The discussion of which efficiency definition is appropriate contains a contradictory citation and a potential swap. The text says 'if we use the second definition of efficiency (Eq.9)' when Eq. (9) is the first definition, and it also says the second definition makes both efficiency and power zero at low driving times, after first saying the first definition gives positive efficiency at low times. This confusion directly affects the paper's conclusion that Eq. (10) is the appropriate metric, so the statements should be carefully corrected and aligned with the curves in Fig. 4.

minor comments (5)

[§III B] The relation between the driving time τ (varied from 200 to 2250 μs) and the stated GRAPE pulse duration (600 to 6000 μs) is unclear; the unitary strokes should have duration τ, so the discrepancy should be explained.
[Fig. 4 caption and §IV] The maximum-power times for the NA engine are given in the text (2000 and 1500 μs) but are not marked in Fig. 4; adding these lines or explaining their omission would make the comparison easier to assess.
[§IV] The sentence 'This could be due to the fact that TH2 has lower population difference as compared to TH2' contains a typo; the second TH2 should likely be TH1.
[§III A] The sentence reporting relaxation times is grammatically ambiguous: 'The measured T1 and T2 relaxation times the for 13C1 and 13C2 spins are 28.57 s and 3.28 s and are 1.84 s and 1.25 s, respectively.' It should clearly state which value is T1 and which is T2 for each spin.
[Figs. 4 and 5 captions] The stray block of text beginning 'Hz 15N 13C 1H ...' appears in both figure captions and should be removed, as it looks like an unintended insertion from the figure-generation software.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the experimental measurements are independent, the theoretical curves are derived from the stated Hamiltonian with no fitting to the outputs, and the STA-cost concern is a resource-accounting gap rather than a circular step.

full rationale

The paper's central comparison between the STA and non-adiabatic (NA) Otto engines is based on independent NMR measurements of the working qubit's density matrices at each driving time, combined with theoretical curves obtained directly from the Landau-Zener model, the counter-adiabatic Hamiltonian of Eq. (6), and the stated driving schedule of Eq. (7). No parameter is fitted to the measured efficiencies or powers, and the experimental data points are not derived from the claimed conclusion. The two STA efficiency definitions (Eqs. (9) and (10)) and the power definition (Eq. (11)) are adopted from the cited literature and evaluated using the measured work values, so the comparison between them is not self-referential. The most substantive concern raised by a skeptic is that the STA cost in Eq. (12) is computed from the ideal counter-adiabatic Hamiltonian rather than from the actual GRAPE-optimized pulse energy; however, this is an experimental resource-accounting limitation or potential modeling error, not a circularity, because it does not make the predicted efficiency or power equivalent to the paper's inputs by construction. There are also no load-bearing self-citations: the cited earlier work on counter-adiabatic driving, cost definitions, and driving schedules is external to the present authors, and no uniqueness theorem or ansatz is smuggled in through self-citation. The conclusion that the second efficiency metric is more appropriate for the NMR platform is a consistency argument based on the observed behavior of efficiency and power, not a prediction that reduces to a fitted parameter. Overall, the derivation chain is self-contained against external benchmarks, and no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard quantum thermodynamics, spin temperature initializations, and the counter-adiabatic driving framework. No new physical entities are introduced. The main ledger items are the chosen spin temperatures and field parameters, which are experimental controls rather than fitted parameters, and the modeling assumptions about thermalization and STA cost accounting. The largest unverified assumption is that the theoretical H_CD cost equals the real pulse energy expenditure.

free parameters (4)

Cold spin temperature kBTC = 1.9 peV (Sec. IV); 11.94 peV (Sec. III B and Fig. 4 caption)
Chosen experimental setting for the working qubit's initial thermal state. The two inconsistent values in the text are likely a typo, but the value directly affects all heat and work calculations.
Hot spin temperatures kBTH1 and kBTH2 = 6.45 peV and 8.45 peV (Sec. IV); 40.54 peV and 53.11 peV (Fig. 4 caption)
Chosen experimental settings for the auxiliary qubit that plays the role of the hot bath. The inconsistency between figure and body affects the reported efficiency and power curves.
Energy gap endpoints νi and νf = νi=1000 Hz, νf=2692.6 Hz
Choosen to satisfy the engine condition TH/TC > νf/νi and to set the Otto limit η=1-νi/νf≈0.629. These are design parameters, not fitted to data.
Driving time τ = 200-2250 µs
Independent variable swept experimentally to study the power-efficiency trade-off. Not fitted; it is the control parameter of the experiment.

assumptions (5)

domain assumption After each thermalization stroke, the working medium is exactly a Gibbs state at the designated spin temperature.
Required for the heat and work formulas (Eqs. 1-2) and for the engine cycle analysis. The experimental implementation redistributes populations and applies a pulsed field gradient, and the states are verified by tomography, but exact Gibbs form is assumed.
domain assumption The SWAP gate with the auxiliary pseudothermal spin constitutes a proper heat bath for the working qubit.
The heating stroke is realized by swapping the working qubit with the 13C1 spin prepared at inverse temperature βH (Sec. III B). The paper assumes the auxiliary spin behaves as a memoryless bath at the intended temperature.
domain assumption The driving strokes implement exactly H0(t) + HCD(t) with the counter-adiabatic term from Eq. 6.
GRAPE pulses approximate the target unitary with fidelity ≥0.999 (Sec. III A), and the ideal CD term is assumed to fully suppress non-adiabatic transitions. The paper does not provide the actual pulse shapes or verify suppression directly.
domain assumption The STA cost defined in Eq. 12, the time integral of the expectation value of H_CD, correctly quantifies the extra energy spent on shortcut driving.
This accounting convention is one of several proposed in the literature (Refs. 33-35). The choice between Eqs. 9 and 10 depends on whether the cost is treated as heat-like or work-like input, and the paper does not derive the correct accounting from first principles.
standard math The polynomial driving field b_j(t) in Eq. 7 satisfies the boundary conditions and represents the physical field sweep.
The form is constructed to enforce b_CD(0)=b_CD(τ)=0 and its derivative conditions, which is a standard method for STA pulses. This is a mathematical construction, not an empirical assumption.

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Cite this review

Pith. "Pith review of Experimental investigation of a quantum Otto heat engine with shortcuts to adiabaticity implemented using counter-adiabatic driving." pith.science (2026). https://pith.science/paper/3EYG5SKA

@misc{pith2026241220194,
  author       = {Pith},
  title        = {Pith review of: Experimental investigation of a quantum Otto heat engine with shortcuts to adiabaticity implemented using counter-adiabatic driving},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3EYG5SKA}},
  note         = {Machine review of arXiv:2412.20194}
}

read the original abstract

The finite time operation of a quantum Otto heat engine leads to a trade-off between efficiency and output power, which is due to the deviation of the system from the adiabatic path. This trade-off caveat can be bypassed by using the shortcut-to-adiabaticity protocol. We experimentally implemented a quantum Otto heat engine using spin-1/2 nuclei on a nuclear magnetic resonance (NMR) quantum processor. We investigated its performance using the shortcut-to-adiabaticity technique via counter-adiabatic driving with the inclusion of the cost to perform the shortcut. We use two different metrics that incorporate the cost of shortcut-to-adiabaticity to define engine efficiency and experimentally analyze which one is more appropriate for the NMR platform. We found a significant improvement in the performance of the quantum Otto heat engine driven by shortcut-to-adiabaticity, as compared to the non-adiabatic heat engine.

Figures

Figures reproduced from arXiv: 2412.20194 by the authors.

**Figure 2.** FIG. 2. Molecular structure of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a)Schematic circuit diagram to realize QOHE using a two-qubit system in NMR. (b) NMR pulse sequence to implement [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Quantum Otto heat engine dynamics: Efficiency((a)-(b)) and output power((c)-(d)) as a function of driving time( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Quantum Otto Engine STA cost dynamics: The the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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