Erasing Classical Memory with Quantum Fluctuations: Shannon Information Entropy of Reverse Quantum Annealing

Cristiano Nisoli; Elijah Pelofske

REVIEW 3 major objections 5 minor 51 references

Erasing Classical Memory with Quantum Fluctuations: Shannon Information Entropy of Reverse Quantum Annealing

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Reverse annealing on odd antiferromagnetic rings shows a clear crossover from memory retention to memory loss as quantum fluctuations increase, tracked by the Shannon entropy of domain-wall positions.

desk verdict Domain-wall Shannon entropy is a genuinely new, useful observable for reverse-annealing diagnostics, and the measured crossover is real; but the title's 'quantum fluctuations' causal claim is softer than the data support, and the missing error analysis and thermal/noise separation need referee attention. read the letter →

arxiv 2509.10927 v1 pith:4TVKSUBF submitted 2025-09-13 quant-ph cond-mat.stat-mechcond-mat.str-elcs.ITmath.IT

Elijah Pelofske , Cristiano Nisoli This is my paper · ORCID

classification quant-phcond-mat.stat-mechcond-mat.str-elcs.ITmath.IT

keywords reversequantumannealingShannonentropydomainwalltransverseIsingmodelmemoryretentionfluctuationsD-Waveannealerantiferromagneticring

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 asks whether quantum fluctuations, not just thermal ones, can erase the memory of an initial classical spin configuration, and how to measure that erasure. It runs reverse quantum annealing on thousands of spins arranged in odd antiferromagnetic rings, initializing a single domain wall, then exposing the system to a transverse field of variable strength. The authors measure the Shannon entropy of the domain-wall position distribution and find a clear crossover: entropy near zero at low transverse field (the domain wall stays put, memory retained) rising to entropy one at high transverse field (the configuration is forgotten). The crossover position and shape depend on the device and on how long the system is held at the transverse field, and the entropy measure even detects failing qubits. If correct, this gives a general, device-agnostic probe of quantum-fluctuation-driven memory loss in programmable quantum annealers.

What carries the argument

The load-bearing object is the Shannon entropy of the domain-wall distribution on an odd antiferromagnetic ring, h[p] = −∑_e p_e log_N p_e, where N is the number of ring edges (spins) and p_e is the measured probability that a domain wall sits on edge e. Odd ring length plus antiferromagnetic couplings guarantees geometric frustration: at least one domain wall must always be present, so the initial pinned wall is a well-defined topological memory. The reverse-annealing schedule—rapid ramp to a pause at anneal fraction s, hold for τ, then ramp back—maps each s to a ratio Γ/J = A(s)/(B(s)J) of transverse to Ising energy, turning the annealer into a tunable source of quantum fluctuations. The e

What would settle it

Run the identical reverse-annealing protocol on an odd ring small enough to simulate exactly (or on the same device with the physical temperature lowered), and compare the measured domain-wall entropy against a closed-system Schrödinger evolution under the transverse Ising Hamiltonian. If the hardware crossover occurs at Γ/J values far from the closed-system prediction, or if a clean device at the same Γ/J shows an entropy floor above zero at low transverse field, then the observed memory erasure is thermal/noise-driven rather than quantum-fluctuation-driven.

Watch

Extended reading notes

Core claim

The central experimental claim is that the Shannon entropy h[p] = −∑_e p_e log_N p_e of magnetic domain-wall positions is a faithful order parameter for memory in the transverse Ising model. Prepared in a single-domain-wall eigenstate and exposed to quantum fluctuations via a pause in reverse annealing, an odd antiferromagnetic ring retains that memory at low Γ/J (h≈0), loses it partially as the domain wall diffuses (0<h<1), and forgets it completely at high Γ/J (h=1). The crossover is sigmoidal in the weakly coupled regime, broadens with exposure time—the partial-memory window spans four decades at 2 µs, five at 100 µs, and six at 2000 µs—and becomes non-monotonic at the longest time, a sig

Load-bearing premise

The crossover is attributed to quantum fluctuations tuned by Γ/J, but the annealer is an open, noisy quantum system; if thermalization, calibration drift, or accumulating hardware errors—not the transverse-field dynamics—drive the entropy rise, the central claim collapses.

Editorial extensions

If this is right

The Shannon entropy of domain-wall positions provides a quantitative, device-agnostic measure of memory retention in reverse quantum annealing, valid across weakly and strongly coupled regimes.
The partial-memory window (0<h<1) widens with exposure time, from four orders of magnitude in Γ/J at 2 µs to six at 2000 µs, so longer exposure erases memory at progressively weaker transverse fields.
At τ=2000 µs the entropy curves become non-monotonic and machine-specific, showing that error accumulation and coupling to the environment dominate the long-time dynamics.
Domain-wall motion, not multi-domain-wall proliferation, accounts for most of the partial-memory window in the strongly coupled case, with the maximum motion probability tracking h≈1/2.
The entropy measure is sensitive enough to detect hardware malfunctions, as shown when two qubits failing on one device left a residual entropy floor of h≈0.2.

Reading between the lines

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

The paper does not separate quantum-fluctuation-driven erasure from thermalization and hardware noise; if the same protocol were run at different physical temperatures on the same chip, the crossover shift would reveal how much of the entropy rise is genuinely quantum.
The reported Γ_init ∼ 1/√τ scaling, based on only three times, is consistent with diffusive domain-wall motion under quantum fluctuations; testing at more exposure times could turn this into a quantitative signature of the dynamics.
Because the 2000 µs curves lose monotonicity, the shorter exposure times (2 and 100 µs) are likely the regime where the annealer behaves closest to a coherent transverse-field simulator; extending the method to smaller systems closed to the environment would let one compare against exact simulation.
The entropy probe should transfer to other frustrated topological features, such as 2D domain walls pinned by geometric frustration, giving a route to studying memory and scrambling in higher-dimensional quantum annealer simulators.

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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

Colleague,

The useful novelty here is the domain-wall Shannon entropy (Eq. 3) as a memory metric for reverse quantum annealing, mapped across a crossover from h=0 to h=1 on three D-Wave QPUs at three exposure times and two couplings. The crossover is directly measured, not fitted, and the sigmoid fits are descriptive. The hardware diagnostic angle is real: the entropy caught two failing qubits on Advantage2. If you work on annealer benchmarking or frustration physics, this is a useful observable.

Where it holds up: the measured curves in Figs. 2-4 are internally consistent, the weakly coupled J=0.001 case is a clean single-qubit benchmark, and the paper is unusually honest about its own limitations — open quantum system, thermalization, hardware malfunction, non-monotonic long-time behavior, only three exposure times. The reader's conditional verdict is about right.

The soft spots, in order of weight:

First, the causal attribution to quantum fluctuations is not established. The paper never separates the quantum contribution from thermal relaxation to the Gibbs state at device temperature (15/16/20 mK) or from hardware noise. The stress-test flag lands: without a classical-thermal control or simulation comparison, the title's "Erasing Classical Memory with Quantum Fluctuations" overclaims. The measured crossover is real, but the "by quantum fluctuations" part is an interpretation.

Second, no error bars on the entropy and proportion estimates despite 8000 samples per point. Since p_e is estimated from a finite sample, the plug-in Shannon entropy is biased low, with sample-size-dependent bias. A bootstrap or analytical correction would make the crossover claims quantitative. Addressable.

Third, the Gamma_init ~ 1/sqrt(tau) scaling is fit to three points, and one slope (0.841) is a clear outlier; the paper acknowledges that exception, but three points is three points. Suggestive, not a law.

Fourth, no raw data or code released. Less critical for a hardware experiment, but it blocks independent verification.

Overall, the central memory-crossover claim holds up as a measured phenomenon. The paper deserves a serious referee; the referee should push on the quantum-vs-thermal separation and the error analysis. I'd accept, conditionally.

Referee Report

3 major / 5 minor

Summary. The paper reports reverse quantum annealing experiments on three D-Wave QPUs using odd antiferromagnetic rings of 4059-4905 spins with periodic boundary conditions. The system is initialized in a one-domain-wall state, exposed to a transverse field by pausing at anneal fractions s corresponding to different Gamma/J = A(s)/(B(s)J) for 2, 100, or 2000 microseconds, and then read out. The authors extract the Shannon entropy h of the normalized domain-wall position distribution (Eq. 3). They observe a crossover from h approximately 0 at low Gamma/J to h approximately 1 at high Gamma/J, with a broad window of partial memory that grows with exposure time and differs between J=1 and J=0.001 and across the three devices. They also report domain-wall spatial densities and inflection points from sigmoid fits, and demonstrate that the entropy can flag hardware malfunctions. The paper interprets the crossover as evidence of memory erasure by quantum fluctuations.

Significance. The strengths of the paper are its scale and reproducibility: thousands of spins, three hardware generations, 8000 readout cycles per setting, and a simple, directly measured entropy metric. If the interpretation is accepted, the protocol offers a device-agnostic diagnostic of reverse annealing dynamics and a sensitive probe of hardware noise. The observation that h=0.2 at low Gamma/J for a malfunctioning chip is a nice validation of the method's sensitivity. However, the central interpretive claim - that the memory erasure is caused by quantum fluctuations rather than by thermalization or hardware noise - is not tested, and the paper's own statements about open-system effects and non-monotonic long-time behavior weaken the causal reading. The result is better characterized as a clear empirical crossover in domain-wall entropy vs Gamma/J, whose microscopic origin remains to be established.

major comments (3)

[Abstract and 'At these annealing times' paragraph] The paper states that the QPU 'is an open quantum system [14,15,18], with both thermalization effects and various sources of error,' yet the title and abstract attribute the observed memory erasure to 'quantum fluctuations.' Because no classical finite-temperature control (e.g., path-integral Monte Carlo at the device temperature, or a classical master equation) is provided, the sigmoidal h vs Gamma/J curves could in principle be produced by thermal relaxation to the Gibbs state of Eq. (2) or by noise. This is a load-bearing issue: the title's causal claim is not established by the experiments. The authors should either add such a control or explicitly reframe the conclusion as 'transverse-field-driven' memory erasure, with the quantum/thermal mixture left open.
[Fig. 4 and 2000 microseconds paragraph] At tau=2000 microseconds, the entropy curves 'lose their smoothness and in fact are not even monotonic,' which the authors attribute to 'accumulation and propagation of hardware errors and of coupling to the environment.' This directly undercuts the 'clear crossover' statement in the abstract when all data are included. The paper should either analyze the 2000 microsecond data separately, exclude it from the main crossover claim, or discuss how the non-monotonicity affects the extracted onset field Gamma_init and window-of-partial-memory widths. As written, the 'dependence on simulation time' claim conflates a clean sigmoidal shift (2 and 100 microseconds) with a qualitatively different regime (2000 microseconds).
[S.M. B and Fig. 7] The sigmoid fits are performed on h vs s (the hardware anneal fraction), but the main results are presented as functions of Gamma/J = A(s)/(B(s)J). Since the mapping between s and Gamma/J is nonlinear and device-specific, a sigmoid in s does not imply a sigmoid in Gamma/J, and the inflection points reported in Fig. 7 should be extracted from fits in Gamma/J (or the conversion should be shown and validated). In addition, the text acknowledges that some data 'does not have a single inflection point'; for the non-monotonic 2000 microsecond data, a single global sigmoid fit is not a meaningful descriptor. The absence of fit parameters (L, x0, k) and goodness-of-fit statistics further weakens the quantitative claims based on these fits.

minor comments (5)

[Eq. (3) and following text] Because h is based only on the edge probabilities, a global spin flip (or an orientation reversal of the domain wall at the same edge) leaves h=0. The abstract's 'memory of the initial configuration' is broader than positional memory. Please clarify that h measures positional memory and state explicitly why orientation flips are not counted (or alternatively, define a combined entropy).
[Figs. 7-9 and Table I] Device name inconsistency: Figs. 7, 8, and 9 use 'Advantage2_system1.5' while the text and Table I use 'Advantage2_system1.3'.
[S.M. A, paragraph after Table I] 'Advantage system1.3' should be 'Advantage2 system1.3'.
[Fig. 5, lower left] The claim of Gamma_init ~ 1/sqrt(tau) is based on three points; the listed slopes include 0.841, which is far from 0.5. The text's exception clause is vague. Please specify which device/regime gives the deviating slope and provide confidence intervals for the fitted exponents.
[Figs. 2-4 and methodology] No uncertainty estimates are provided for the entropy values. While finite-sampling error from 8,000 samples is likely small, showing error bars or confidence intervals would strengthen the platform-comparison claims, especially because device-to-device differences are discussed.

Circularity Check

1 steps flagged · score 1.0 of 10

Directly measured crossover; only a minor by-construction sigmoid-inflection statement.

fitted input called prediction [Supplementary Material B: Sigmoid Curve Fitting and Domain Wall Entropy Inflection Points]
"The sigmoid model function we use is y = L 1+exp(−k·(x−x0)) . ... We see that the entropy inflection point, as expected, consistently occurs very close to 0.5."

The fitted sigmoid has its inflection point at y = L/2 by calculus, and the fit fixes L = 1.0, so y = 0.5 at the inflection point is an analytic property of the chosen fitting function, not an empirical finding. The data-dependent quantity is the x-coordinate x0 (the Γ/J value), which does shift with annealing time and hardware; the y-coordinate 0.5 is imposed by construction. This is a minor by-construction artifact and does not bear on the main measured crossover, which is computed directly from the sampled domain-wall distribution via Eq. (3).

full rationale

The paper's central result—the Shannon entropy h of the domain-wall distribution crossing from ~0 to ~1 as Γ/J increases—is a direct measurement: h is computed from 8,000 anneal-readout samples per setting using Eq. (3), with no parameter fitted to the predicted quantity. The sigmoid fits are used only descriptively to locate the crossover; the reported shift of the inflection x-position with annealing time is data-driven. The only by-construction statement is that the entropy at the fitted sigmoid inflection point is 0.5, which follows from the sigmoid form with L=1.0 and is not load-bearing. The paper explicitly acknowledges that the QPU is an open quantum system with thermalization and hardware errors, so the attribution of the crossover to quantum fluctuations is an interpretation subject to external validation, not a circular derivation. No load-bearing self-citation chain appears: the cited prior work by the authors is contextual or comparative, not the source of the measured crossover. Overall circularity is minimal.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the hardware realizing the transverse Ising Hamiltonian, on the reverse-annealing pause acting as controlled exposure to Gamma, on the entropy metric being a valid memory measure, and on the sampling being sufficient. No new physical entities are introduced. The fitted quantities (sigmoid parameters, entropy thresholds, scaling slopes) are descriptive of the data rather than inputs that generate the claimed crossover.

free parameters (3)

Sigmoid fit parameters (L, x0, k) and extracted inflection points = L near 1, k initialized at -40, x0 per device/condition (Fig. 7)
Used to locate the entropy inflection point and to define the transition region. Descriptive fitting of measured data, not a predictive model, and the claim that the inflection occurs near h=0.5 is partly inherited from the sigmoid form.
Entropy thresholds 0.05 and 0.95 for onset/saturation (Gamma_init and transition region) = 0.05 and 0.95 entropy change
The paper defines the onset field Gamma_init 'arbitrarily as the value where h = 0.05' (Fig. 5 lower-left), and the transition region shading in Fig. 6 uses the 0.05 to 0.95 window. These hand-chosen thresholds set where the crossover is said to begin and end.
Log-log slope of Gamma_init vs annealing time = 0.465, 0.841, 0.509, 0.484, 0.505, 0.535
Linear fits to three data points per device support the Gamma_init ~ 1/sqrt(tau) claim. The 0.841 outlier for Advantage system4.1 at J=1.0 contradicts the claimed exponent; the paper acknowledges the three-point limitation.

assumptions (5)

domain assumption The D-Wave QPU during reverse annealing evolves under H = (B(s)J/2) sum sigmaz_i sigmaz_{i+1} - (A(s)/2) sum sigmax_i with machine-specific A(s) and B(s).
The entire Gamma/J axis and the crossover interpretation rest on this hardware Hamiltonian (Eq. 2, Fig. 1d). The paper qualifies the device as an open quantum system with thermalization and error sources.
ad hoc to paper The symmetric reverse-anneal schedule with a pause at s of duration tau realizes exposure to Gamma/J for time tau.
The 0.5 microsecond ramps before and after the pause also contribute dynamics; the pause is assumed to dominate the exposure. This is a protocol choice stated in the 'We utilize reverse quantum annealing' paragraph.
ad hoc to paper Shannon entropy of the normalized domain-wall distribution is a valid operational measure of memory.
Eq. 3 defines h; the paper argues it captures memory via information gain, but the metric is proposed, not derived from the model, and it ignores the total proportion of domain walls sampled.
domain assumption 8000 anneal-readout cycles per condition give sufficient statistics for pe and h.
The paper states the goal of 'mitigating finite sampling effects' but provides no error bars, confidence intervals, or convergence checks on the entropy estimates.
domain assumption The Glasgow subgraph isomorphism embedding faithfully maps the 1D ring onto the hardware graph.
Direct qubit-to-spin mapping is used; the paper notes hardware defects and a mid-run hardware graph change (two malfunctioning qubits) that altered the embedding, so the mapped ring can differ from the ideal 1D chain.

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Pith. "Pith review of Erasing Classical Memory with Quantum Fluctuations: Shannon Information Entropy of Reverse Quantum Annealing." pith.science (2026). https://pith.science/paper/4TVKSUBF

@misc{pith2026250910927,
  author       = {Pith},
  title        = {Pith review of: Erasing Classical Memory with Quantum Fluctuations: Shannon Information Entropy of Reverse Quantum Annealing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TVKSUBF}},
  note         = {Machine review of arXiv:2509.10927}
}

read the original abstract

Quantum annealers can provide non-local optimization by tunneling between states in a process that ideally eliminates memory of the initial configuration. We study the crossover between memory loss and retention due to quantum fluctuations, in a transverse Ising model on odd numbered antiferromagnetic rings of thousands of spins with periodic boundary conditions, by performing reverse quantum annealing experiments on three programmable superconducting flux qubit quantum annealers. After initializing the spins to contain a single domain wall, we then expose it to quantum fluctuations by turning on the transverse Zeeman energy. We characterize the crossover between memory retention at low transverse field, and memory loss at high transverse field by extracting the Shannon information entropy of magnetic domain wall distributions. We demonstrate a clear crossover in memory retention, and its dependence on hardware platform and simulation time. Our approach establishes a general probe of the interplay between quantum fluctuations and memory.

Figures

Figures reproduced from arXiv: 2509.10927 by the authors.

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

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: , top-row, shows three representative pe distributions for Advantage2 system1.3, strong J = 1 coupling, at τ = 100 µs, corresponding to the beginning, middle, and end of the WPM, illustrating the regime of domain-wall almost stationary (red), domain wall motion (blue…

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Middle: Total single-domain wall sample proportion (either both spin up ↑↑ or both spin down ↓↓). Bottom: Total single- domain wall proportion (abbreviated as SDWP) which moved from the initial pinned position specified by the reverse anneal initial spin configuration (or has the opposite domain wall orientation to that initial domain wall). Notice that i...
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