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REVIEW 4 major objections 5 minor 35 references

Measuring Butterfly Velocity in the XY Model on Emerging Quantum Computers

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A teleportation-based OTOC protocol with Riemannian-trust-region circuit compilation estimates the XY model's butterfly velocity on a noisy five-qubit simulation, matching the analytic maximum group velocity.

desk verdict A plausible proof-of-concept for YKY+RTR butterfly velocity measurement on a simulator, with an overclaimed NISQ framing and some unfinished edges; the alleged OTOC mapping contradiction does not hold up. read the letter →

arxiv 2411.10206 v1 pith:D4IYASSO submitted 2024-11-15 quant-ph

Calum McCartney , Eric Chen , Subhayan Roy Moulik This is my paper · ORCID
classification quant-ph
keywords butterflyvelocityout-of-time-ordercorrelatorXYmodelquantumteleportationRiemanniantrust-regionoperatorgrowthNISQdevicesHamiltoniansimulation
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 establishes a practical path from a quantum many-body Hamiltonian to a measured scrambling speed: it runs a quantum-teleportation protocol that extracts the operator-averaged out-of-time-order correlator (OTOC), a correlation function that tracks how a local operator grows into a many-body operator, and it uses Riemannian trust-region optimization to compile the time-evolution unitary into a short gate circuit. From the OTOC at different lattice sites, the paper defines a spreading time at each site and fits a line whose inverse slope is the butterfly velocity, the speed at which a local perturbation travels through the system. On a noisy simulation of a five-qubit quantum device, the method yields 1.972 for the isotropic XY model and 3.745 for the anisotropic model, against analytic values of 2 and 3.75 obtained from the maximum group velocity. The paper argues that the protocol is robust to certain decoherence and coherent errors without any error-mitigation post-processing, and that the same machinery can be applied to lattice models that are not analytically solvable.

What carries the argument

Three pieces carry the argument. The YKY teleportation protocol is a doubled-Hilbert-space circuit in which a Bell-basis post-selection probability gives $F_{\mathrm{EPR}}=1/(4\langle\mathrm{OTOC}\rangle)$, converting operator growth into a measurable teleportation fidelity; the paper uses $F_{\mathrm{EPR}}$ to compute $C_j(t)=2-1/(2F_{\mathrm{EPR}})$. The Riemannian trust-region (RTR) optimizer on the product manifold $U(4)^m$ minimizes the Frobenius error $\|E(G)-e^{-iHt}\|_F^2$ over $m$-layer brick-wall circuits, producing a compiled unitary that is shallower than product-formula decompositions. The threshold rule $C_j(t)\ge 0.1$ defines a site-dependent spreading time $t_j$, and the inverse slope of the best-fit line through $t_j$ versus lattice position $j$ is the reported butterfly velocity.

What would settle it

Run the same YKY-RTR measurement on chains of length 7, 9, and 11 qubits, and re-extract the velocity with thresholds $C_j(t)\ge 0.05$, $0.1$, and $0.2$; if the fitted slope changes systematically with chain length or threshold by more than the few-percent scatter reported here, then the finite-chain linear-fit velocity is not the infinite-chain maximum-group-velocity butterfly velocity.

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Extended reading notes

Core claim

The central claim is that the butterfly velocity of the one-dimensional anisotropic XY model, defined as the maximum group velocity of its quasiparticles, can be estimated on noisy near-term quantum hardware by combining a teleportation-based OTOC measurement with Riemannian-trust-region (RTR) circuit compilation. Analytically, the paper diagonalizes $H=J\sum_j(\frac{1+r}{2}X_jX_{j+1}+\frac{1-r}{2}Y_jY_{j+1}+hZ_j)$ via Jordan-Wigner, Fourier, and Bogoliubov transformations, obtaining the dispersion $\varepsilon(k)=-2J\sqrt{(h-\cos k)^2+r^2\sin^2 k}$ and group velocity $v_g(k)=-2J[\sin k(h-\cos k)+r^2\sin k\cos k]/\sqrt{(h-\cos k)^2+r^2\sin^2 k}$, so $v_B=\max_k v_g(k)$. Numerically, the YKY teleportation circuit gives the squared commutator $C_j(t)=2-1/(2F_{\mathrm{EPR}})$ from the teleportation fidelity $F_{\mathrm{EPR}}$, the spreading time at site $j$ is $t_j=\min\{t:C_j(t)\ge 0.1\}$, and a least-squares fit to $t_j$ for $j=2,\ldots,5$ has inverse slope equal to the butterfly velocity. The paper reports that these noisy-simulation estimates agree with the analytic $v_B$ to within a few percent, and that RTR compilation uses significantly fewer circuit layers than Lie-Trotter-Suzuki product formulas.

Load-bearing premise

The load-bearing premise is that the inverse slope of a best-fit line through the spreading times at four lattice sites, fixed by an arbitrary $C_j(t)\ge 0.1$ threshold on a five-qubit chain, equals the infinite-chain butterfly velocity obtained from the maximum group velocity.

Editorial extensions

If this is right

  • On a noisily simulated five-qubit device, the protocol recovers the isotropic XY butterfly velocity within about 1.4 percent (1.972 versus the analytic 2) and the anisotropic value within about 0.1 percent (3.745 versus 3.75).
  • Because the estimator obeys $1/(4F_{\mathrm{EPR}}) \ge \langle\mathrm{OTOC}\rangle$, the protocol inherits a one-sided robustness to decoherence and small coherent errors, so it does not require explicit error mitigation to remain usable near the noise threshold of current devices.
  • RTR circuit compilation requires substantially fewer layers than Lie-Trotter-Suzuki splitting at the same target error, which is the step that makes the OTOC circuit short enough to run on noisy hardware.
  • The analytic diagonalization provides a closed-form benchmark, $v_B=\max_k v_g(k)$, that future hardware measurements on longer chains can be tested against as quantum devices improve.

Reading between the lines

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

  • One could turn the fixed threshold into a diagnostic: repeat the $t_j$ extraction at several thresholds and check that the fitted velocity is stable; if it is not, the linear-fit estimator should be replaced by a light-cone-edge extrapolation.
  • A natural next test would be to apply the same YKY-RTR measurement to a non-integrable or disordered spin chain, where no analytic $v_B$ exists, and compare the output against exact-diagonalization or tensor-network light-cone speeds.
  • Because the protocol measures $C_j(t)$ separately at each site, it could be adapted to measure asymmetric or directional butterfly velocities, and to extract the full light cone rather than a single speed, in models with anisotropic interactions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper combines the Yoshida-Kitaev-Yao (YKY) teleportation protocol with Riemannian trust-region (RTR) Hamiltonian-to-circuit compilation to estimate the operator-averaged out-of-time-order correlation function and, from it, the butterfly velocity of the one-dimensional XY model. The authors derive the maximum group velocity of the XY model analytically via Jordan-Wigner, Fourier, and Bogoliubov transformations, and then compare these values with velocities extracted from 5-qubit simulations of the YKY-RTR protocol. The numerical results are obtained with the IBM FakeTorino noisy simulator and are reported to agree with the analytical butterfly velocities to within a few percent.

Significance. If the measurement pipeline is correct, the paper offers a useful proof-of-concept for estimating butterfly velocities on near-term devices in models that are not analytically solvable. The analytical derivation in Section II and Appendix A is standard and internally consistent, and the comparison with the independently computed maximum group velocity is a genuine consistency check rather than a fit to the target. The RTR-based circuit compilation is a promising tool for reducing circuit depth relative to product formulas. However, the current manuscript has a load-bearing inconsistency in the relation between the measured teleportation probability and the OTOC, and the numerical extraction lacks the error analysis and hardware context needed to support the claimed few-percent agreement.

major comments (4)
  1. [Section III A, Eq. (15), and Section IV] Eq. (15) states F_EPR = (1/4)⟨OTOC⟩, while the sentence following it and Section IV use ⟨OTOC⟩ = 1/(4F_EPR), with C_j(t) = 2 - 1/(2F_EPR). These two relations are reciprocal and cannot both hold. A direct two-qubit check (e.g., U = CNOT, for which the operator-averaged OTOC in Eq. (10) equals 1/2 and the conditional Bell probability F_EPR equals 1/2) confirms the reciprocal form used in Section IV, indicating that Eq. (15) and the diagrammatic derivation around Eqs. (16)-(18) contain an inverted factor. Because every numerical C_j(t) value in Section IV is derived from F_EPR, the mapping from the measured probability to the OTOC must be corrected and re-derived before the reported velocities are fully supported; the numerical implementation appears to use the correct form, but the paper as written does not establish the mapping.
  2. [Section IV, Table I, Fig. 4] The butterfly-velocity estimates rest entirely on the arbitrary spreading threshold C_j(t) >= 0.1, a linear fit to spreading times t_j over only j = 2, ..., 5 on a 5-qubit chain, and the identification of the fitted slope with the infinite-chain maximum group velocity of Eq. (9). No error bars are reported for the FakeTorino estimates, no sensitivity of t_j or the fitted slope to the threshold is given, and finite-size effects, including the boundary term dropped in Eq. (4), are not assessed. These omissions are load-bearing because the claimed few-percent agreement in Table I could be coincidental for a single threshold choice.
  3. [Abstract and Section IV] The abstract claims a "proof-of-concept demonstration of this method to estimate the butterfly velocity on NISQ-devices," but all quantum results in Section IV are obtained with the FakeTorino noisy simulator rather than a physical quantum device. This overstates the experimental content of the paper; the claim should be narrowed to a noisy-simulation demonstration or supported by data from actual hardware.
  4. [Sections III B and IV] The RTR mapping produces a brick-wall circuit of preselected depth m, but the manuscript never states the value of m, the optimization tolerance, the achieved Frobenius error ||E(G) - U||_F, or the number of shots used in the FakeTorino runs. Without these details the numerical results are not reproducible, and the possible bias of the circuit approximation on C_j(t) cannot be assessed.
minor comments (5)
  1. [Eq. (9)] The numerator of the group-velocity expression lacks parentheses; as written, it is ambiguous and should be checked against the derivative of Eq. (8).
  2. [Section III A] The unparenthesized expression "1/4F_EPR" should be written as "1/(4F_EPR)" to avoid ambiguity; this ambiguity is directly related to the inconsistency in Eq. (15).
  3. [Figure 2 and Section IV] The text alternates between referring to the final Bell measurement on A1,B1 and on A0,B0; the notation should be made consistent across Eq. (15), the circuit description, and the results section.
  4. [References [24], [25]] The text contains placeholder citations "[ ? ]" for "standard libraries" and for the comparison with product formulas; complete references should be supplied.
  5. [Table I] The column labels "Numerical 5 qubits" and "Quantum 5 qubits" are not defined in the text; the paper should state whether these are noiseless and FakeTorino results, respectively, and should report the statistical uncertainty of each entry.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical butterfly-velocity estimates are benchmarked against an independently computed analytical maximum group velocity, and the YKY-to-OTOC relation is used consistently.

full rationale

The paper's central comparison is between (i) a numerical measurement of the squared commutator C_j(t) obtained by simulating the YKY teleportation circuit with an RTR-optimized Hamiltonian circuit, and (ii) an analytical butterfly velocity v_B = max_k v_g(k; J, r, h) computed from the dispersion relation in Eq. (8). The analytical value is not used to set the spreading threshold, the linear-fit slope, or any parameter of the simulation; it is an independent benchmark. The RTR optimization fits the circuit to U = e^{-iHt} via the cost function f(G) = ||E(G) - U||_F, not to the OTOC or to the butterfly velocity, so the subsequent OTOC measurement is not forced by the fit. The relation between the teleportation probability and the OTOC, F_EPR = 1/(4<OTOC>), is consistently used in Section IV through C_j(t) = 2 - 1/(2 F_EPR) = 2 - 2<OTOC>; the apparent formatting ambiguity in Eqs. (15) and (18) is resolved by the surrounding sentence 'the estimated <OTOC> as 1/(4 F_EPR)' and by the fact that the Section IV formula requires the reciprocal relation. The only imported physical input is the standard identification of the butterfly velocity with the maximum group velocity, cited to [26]; that citation is external, not self-referential, and is used as a benchmark rather than as a constraint on the measured data. No fitted parameter is renamed as a prediction, and no load-bearing step reduces to its own input.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central numerical claim rests on an arbitrary spreading threshold, an unstated circuit depth, the large-n boundary approximation applied at n=5, and the identification of the butterfly velocity with the maximum group velocity. No new physical entities are introduced.

free parameters (2)
  • spreading threshold = C_j >= 0.1
    Defines the spreading time t_j in Section IV; chosen by hand, no sensitivity analysis provided.
  • RTR circuit depth m = not reported
    The depth of the brick-wall circuit in the RTR approximation is not stated; it controls the approximation error and is a free choice that affects the OTOC values.
assumptions (4)
  • domain assumption The boundary term in the Jordan-Wigner transform is negligible for n=5
    Section II says the boundary term is irrelevant in the large n limit, but the simulations use n=5 where finite-size effects may be significant.
  • domain assumption Butterfly velocity equals the maximum group velocity of Eq. (9)
    Section II states 'Following work done in [26], the butterfly velocity is expected to match the maximum of (9) over k.' This is not derived for the XY model in this paper.
  • domain assumption YKY protocol is robust to decoherence and small coherent errors
    Section III.A relies on the robustness analysis of [19,20] to drop explicit error mitigation, without verifying it for the specific circuit and noise model used.
  • domain assumption FakeTorino simulator represents a current NISQ device
    Section IV uses the FakeTorino simulator in place of a real device; no real hardware data are presented, yet the abstract claims a demonstration on NISQ-devices.

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Cite this review

Pith. "Pith review of Measuring Butterfly Velocity in the XY Model on Emerging Quantum Computers." pith.science (2026). https://pith.science/paper/D4IYASSO

@misc{pith2026241110206,
  author       = {Pith},
  title        = {Pith review of: Measuring Butterfly Velocity in the XY Model on Emerging Quantum Computers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D4IYASSO}},
  note         = {Machine review of arXiv:2411.10206}
}
read the original abstract

The butterfly velocity is commonly used to understand information transport properties in quantum dynamical systems and is related to growth of operators. Here we utilise a quantum teleportation based protocol and Riemannian Trust-Region method to estimate the butterfly velocity via the operator averaged out-of-time-order correlation function. We particularly study the XY model and analytically find the maximum group velocity. We then report a proof-of-concept demonstration of this method to estimate the butterfly velocity on NISQ-devices. The numerical simulation results obtained here are compared with our analytical calculations and found to be in agreement. The quantum algorithmic methods presented here can be more generally utilised to study information transport properties in more complicated lattice models.

Figures

Figures reproduced from arXiv: 2411.10206 by the authors.

Figure 1
Figure 1. shows a colour map of butterfly velocities vB(r, h) (taking J = 1 to match energy and time units) over a range of the two parameters. It is clear from this plot that vB has symmetry under parity transforms r → −r and h → −h. Furthermore, the butterfly velocity in the XY-model is state independent and subsequently it suffices to study infinite temperature states containing a uniform mixture of all eigenstates [PITH_… view at source ↗
Figure 2
Figure 2. FIG. 2: Quantum circuit encoding [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Plot of error as a function of the number of layers [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Plots demonstrating proof-of-concept for the YKY-RTR algorithm on parameter sets in Table I. Figures (a) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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