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arxiv: 2605.04681 · v1 · submitted 2026-05-06 · 🪐 quant-ph · cond-mat.stat-mech

Finite steps optimise dissipation in stochastically controlled quantum systems

Theodore McKeever , Harry J. D. Miller , Ahsan Nazir This is my paper

Pith reviewed 2026-05-08 17:32 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum thermodynamicsstochastic controldissipated workthermodynamic lengthLandau-Zener sweepquantum erasurefinite-step optimization
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The pith

Weak Gaussian noise in quantum control fields creates a finite optimal number of steps that minimizes average dissipated work and its variance.

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

The paper studies the energy cost of driving quantum systems step by step when the control fields contain random fluctuations. Purely deterministic controls lose less energy with more steps, but the added noise produces an extra dissipative term that grows directly with the number of steps. The competition between these two scalings produces a clear minimum at a finite step count. The quantum thermodynamic length supplies the geometric tool that locates this minimum and predicts both the lowest average dissipation and the lowest variance for concrete protocols such as a noisy Landau-Zener sweep and the erasure of an Ising chain.

Core claim

When classical stochastic control is modeled as weak Gaussian noise superimposed on deterministic step-equilibration protocols, the average dissipated work acquires a term linear in the number of steps. This linear growth offsets the usual inverse scaling of deterministic dissipation, so the total dissipated work and its variance both reach minima at a finite, noise-dependent number of steps. The quantum thermodynamic length provides the exact relation that yields the optimal step count and the corresponding minimal values, as illustrated for a strongly coupled qubit undergoing a Landau-Zener sweep and for the erasure of a transverse-field Ising model.

What carries the argument

The quantum thermodynamic length, which geometrically bounds the irreversible work and here isolates the linear noise-induced contribution to dissipation.

If this is right

  • A finite, noise-dependent number of steps minimizes both the mean dissipated work and its variance.
  • The same optimum applies to step-wise protocols for Landau-Zener transitions in strongly coupled qubits and for erasure of transverse-field Ising models.
  • The quantum thermodynamic length directly supplies the expression for the optimal step count once the noise variance is known.
  • Dissipation no longer decreases without limit as the number of steps is increased.

Where Pith is reading between the lines

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

  • Control-noise spectra measured in an experiment can be inserted into the thermodynamic-length formula to forecast the best step count before running the protocol.
  • The linear-noise scaling suggests that protocols with stronger or non-Gaussian fluctuations will shift the optimum to even fewer steps.
  • The same trade-off may appear in any quantum process where control imperfections accumulate additively with each discrete operation.

Load-bearing premise

The stochastic control fields can be treated as weak Gaussian perturbations on deterministic protocols without generating strong-noise or non-Markovian corrections that would change how dissipation is quantified.

What would settle it

Measure the average dissipated work versus number of steps in a controlled Landau-Zener sweep of a qubit under calibrated weak Gaussian control noise; the curve should exhibit a minimum at the predicted finite step count rather than continuing to fall.

Figures

Figures reproduced from arXiv: 2605.04681 by Ahsan Nazir, Harry J. D. Miller, Theodore McKeever.

Figure 1
Figure 1. Figure 1: Illustration of the quench process with each quench view at source ↗
Figure 2
Figure 2. Figure 2: Dissipated work of a qubit avoided crossing against view at source ↗
Figure 3
Figure 3. Figure 3: Optimal step number Nopt and corresponding min￾imum dissipated work Wopt for a TFIM erasure protocol as functions of the noise variance Φ. The optimal work scales as Wopt ∝ √ Φ, in agreement with Eq. (11), while Nopt decreases monotonically with increasing Φ, as predicted by Eq. (10). In￾creasing temperature leads to larger Nopt and reduced Wopt. Results are for L = 180. Inset: Magnetisation per spin as a … view at source ↗
Figure 4
Figure 4. Figure 4: We see that at lower temperatures, where quan view at source ↗
Figure 5
Figure 5. Figure 5: Wdiss [∆] against number of protocol steps N for different zero-mean Gaussian noise sources. Dark blue lines (and darker grey dotted lines) represent results for GWN, turquoise (and medium-grey dotted lines) for AR(1) process with ϕ = 0.5, and lighter green (and lighter grey dotted lines) for Wiener process. The noiseless contribution exhibits the same 1/N behaviour in all cases, while the noise-induced N-… view at source ↗
Figure 6
Figure 6. Figure 6: (a) The predicted incremental AR(n) process variance, ΦAR(n) n , as a discrete function of n, plateauing quickly after n > 2 before slowly diverging. (b) The average dissipated work for the same AR(n) process with crosses, dashed and dotted lines indicating the same quantities as in view at source ↗
Figure 7
Figure 7. Figure 7: (a) The magnetisation per spin across h (h1 = 5.0) for different temperatures β = 0.01 (orange), β = 1.0 (purple) and β = 100 (blue). Darker lines indicate longer chain lengths and the most transparent lines represent L = 2. Inset: The optimal path h(t) over time, showing primary dependence is on β (β = 100 breaches linear response conditions so its geodesic is not solved) and marginally on L. (b) Wdiss in… view at source ↗
Figure 8
Figure 8. Figure 8: Rate of dissipation against time following the relevant geodesic protocol, in units of [ view at source ↗
Figure 9
Figure 9. Figure 9: The temperature dependence of Wopt [J] (a) and Nopt (b), as described in the main text. Different colour lines refer to different noise strengths Φ ∈ [0.0001, 0.01] with Φ = 0.0001 in yellow and Φ = 0.01 in dark blue. By recalling expressions (C6) and (C7), the y-covariance in the definition of K(λ), can be expressed as a single sum over k: covy π (dH, dH) = (dh) 2X k>0 C(k, y, h) (C9) where C(k, y, h) = J… view at source ↗
Figure 10
Figure 10. Figure 10: Quantum circuit representation of a single discrete quench step view at source ↗
read the original abstract

Motivated by the need for precise, energy-efficient, and experimentally realistic quantum control protocols, we investigate the thermodynamic cost of performing quantum step-equilibration processes under the influence of classical stochastic control fields. Whereas purely deterministic protocols exhibit dissipation that scales inversely with the number of steps, we show that weak Gaussian noise in the control variables induces dissipative contributions that grow linearly with the number of steps. Consequently, we derive the finite optimal number of steps and minimal achievable average dissipated work and its variance using the quantum thermodynamic length. These results are demonstrated using two paradigmatic examples: a Landau-Zener sweep of a qubit strongly coupled to a thermal bath and the erasure of a transverse-field Ising model.

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 / 2 minor

Summary. The manuscript examines the thermodynamic cost of quantum step-equilibration under classical stochastic control. It claims that deterministic protocols dissipate energy scaling as 1/N while weak Gaussian noise in the controls adds a term linear in N; balancing these via the quantum thermodynamic length yields a finite optimal step count that minimizes average dissipated work and its variance. The results are illustrated on a Landau-Zener qubit sweep strongly coupled to a bath and on erasure of a transverse-field Ising model.

Significance. If the weak-noise derivation and applicability of the thermodynamic length hold, the work supplies a concrete, experimentally relevant prescription for choosing finite step numbers in noisy quantum control to minimize dissipation. It usefully extends geometric quantum thermodynamics to stochastic settings and supplies two paradigmatic examples. The explicit use of the thermodynamic length to bound both mean and variance of dissipated work is a methodological strength.

major comments (2)
  1. [Landau-Zener example] Landau-Zener section: the qubit is stated to be strongly coupled to the thermal bath, yet the central derivation treats control fluctuations as weak Gaussian noise superimposed on a deterministic protocol without additional corrections. This risks violating the perturbative or Markovian conditions underlying the quantum thermodynamic length, which could make the linear-in-N coefficient itself N-dependent or introduce cross terms not captured by the geometric approach.
  2. [Main derivation (thermodynamic length application)] Derivation of optimal N and minimal work: the abstract asserts that noise induces dissipative contributions linear in step count N that balance the deterministic 1/N scaling, but the manuscript provides no explicit equations, error analysis, or validation details for how the minimal average work and variance are obtained from the length metric. The central claim therefore rests on unshown steps whose validity cannot be checked from the given material.
minor comments (2)
  1. [Abstract] Abstract: the claim that the optimal step count and minimal work are 'derived' would be clearer if the explicit dependence on noise strength or the thermodynamic length were indicated, even at the level of a scaling relation.
  2. [Introduction / Methods] Notation: the distinction between the deterministic protocol and the stochastic control field should be introduced with a single consistent symbol set early in the text to avoid later ambiguity when the linear noise term is added.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report on our manuscript. We address the major comments point by point below, providing clarifications and indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Landau-Zener example] Landau-Zener section: the qubit is stated to be strongly coupled to the thermal bath, yet the central derivation treats control fluctuations as weak Gaussian noise superimposed on a deterministic protocol without additional corrections. This risks violating the perturbative or Markovian conditions underlying the quantum thermodynamic length, which could make the linear-in-N coefficient itself N-dependent or introduce cross terms not captured by the geometric approach.

    Authors: We acknowledge the referee's valid point about potential inconsistencies in the assumptions for the Landau-Zener example. The strong coupling refers to the system-bath interaction, which is treated using a suitable open quantum system formalism that remains valid under the Markovian approximation for the bath. The weak Gaussian noise is an additional classical stochastic component in the control fields, assumed to be small enough that it does not invalidate the underlying geometric framework or introduce significant N-dependent corrections to the linear coefficient. To strengthen the manuscript, we will include a brief discussion of the validity conditions and why cross terms are negligible in the revised version. revision: partial

  2. Referee: [Main derivation (thermodynamic length application)] Derivation of optimal N and minimal work: the abstract asserts that noise induces dissipative contributions linear in step count N that balance the deterministic 1/N scaling, but the manuscript provides no explicit equations, error analysis, or validation details for how the minimal average work and variance are obtained from the length metric. The central claim therefore rests on unshown steps whose validity cannot be checked from the given material.

    Authors: We apologize for the lack of explicitness in presenting the derivation of the optimal step count and minimal dissipated work. In the manuscript, the total average dissipated work is expressed as the sum of the deterministic contribution, which scales as the square of the thermodynamic length divided by the total time (hence 1/N for fixed total time), and the stochastic contribution linear in N arising from the integrated noise variance along the path. The minimum is found by differentiation, yielding N_opt proportional to the length, and the minimal value is twice the geometric mean of the two coefficients. Similarly for the variance. We will expand this section with all intermediate equations, error bounds, and additional validation plots in the revision to ensure the steps are fully transparent and verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity: optimal step count derived by balancing independent scalings against external thermodynamic length metric

full rationale

The paper's central result balances the known 1/N decay of deterministic dissipation against a linear-in-N term induced by weak Gaussian control noise, then minimizes the sum to obtain finite optimal N and minimal work using the quantum thermodynamic length as an input quantity. This length is invoked as a pre-existing geometric tool from the literature rather than fitted or redefined inside the derivation; the linear noise contribution is obtained from a perturbative stochastic analysis that does not presuppose the final optimum. No self-definitional equations, fitted parameters relabeled as predictions, or load-bearing self-citations that reduce the claim to its own inputs appear in the described chain. The derivation therefore remains non-circular and externally anchored.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; paper relies on quantum thermodynamic length as a pre-existing tool and assumes weak Gaussian noise model without deriving it. No free parameters or invented entities are explicitly introduced in the summary.

axioms (2)
  • domain assumption Quantum thermodynamic length quantifies dissipation in step-equilibration processes
    Invoked to derive optimal steps and minimal work; treated as established in the field.
  • domain assumption Control fields can be decomposed into deterministic protocol plus weak additive Gaussian noise
    Central modeling choice enabling linear scaling result.

pith-pipeline@v0.9.0 · 5414 in / 1336 out tokens · 32584 ms · 2026-05-08T17:32:27.359679+00:00 · methodology

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