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REVIEW 2 major objections 7 minor 44 references

A single AOD pair can rotate a surface-code atom array for a transversal Hadamard in logarithmic strokes by factoring the turn into three binary shears.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 16:19 UTC pith:TZ5VFCC3

load-bearing objection Clean, usable single-AOD schedules that cut surface-code H/S movement from O(d^{7/3}) to O(d^{1/3}); math checks out, novelty is real but incremental. the 2 major comments →

arxiv 2607.28149 v1 pith:TZ5VFCC3 submitted 2026-07-30 quant-ph

Efficient atom rearrangements for quantum error correction primitives with a single AOD

Tom Hartweg (1 , 2) , Asier Pi\~neiro Orioli (1) , Hugo Perrin (1) , Samuel Crew (1) ((1) QPerfect SAS , European Center for Quantum Sciences , 23 rue du Loess , Strasbourg
show 6 more authors
67200 France (2) University of Strasbourg CNRS CESQ ISIS (UMR 7006))
This is my paper
classification quant-ph
keywords neutral atomsacousto-optic deflectorsatom rearrangementsurface codetransversal gatesquantum error correctionPaeth decompositionbinary shears
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Neutral-atom machines can turn some logical gates into pure geometry: move the atoms instead of compiling long gate sequences. This paper shows how to shear, rotate, and reflect whole 2D arrays with only one crossed AOD pair, under the usual rectangular-grid and rigid-row/column constraints. The key is to build every shear from a short binary (or negabinary) list of strokes, then compose three such shears via the classical Paeth factorization to get a 90° rotation. For a rotated surface code of odd distance d that yields a transversal Hadamard in 3⌊log₂(d−1)⌋+4 strokes and constant-jerk time scaling as O(d^{1/3}), against quadratic stroke count and much worse time for atom-by-atom moves. The same primitives cover fold-transversal S, toric-code automorphisms, and Bacon-Shor gadgets for La-cross codes, so logical throughput is no longer limited by naive rearrangement cost.

Core claim

With a single dynamic crossed AOD pair addressing rectangular sub-arrays by rigid horizontal or vertical translations, any shear of a rectangular atom block can be realized in ⌊log₂(d−1)⌋+1 binary strokes, and a 90° rotation of a d×d block (d odd) therefore costs exactly 3⌊log₂(d−1)⌋+4 strokes and O(d^{1/3}) constant-jerk time when the rotation is factored into three Paeth shears. That is within a small constant of the information-theoretic lower bound on stroke count and a polynomial improvement over sequential atom-by-atom rearrangement.

What carries the argument

Paeth’s three-shear factorization of a 90° rotation, each shear executed by a binary (or, for signed displacements, negabinary) AOD stroke list that encodes every required row/column offset as a subset sum. The binary list saturates the logarithmic lower bound on the number of strokes while keeping total tweezer travel linear in d.

Load-bearing premise

The schedules assume collisions with both filled and empty static traps can always be avoided during every move, either by turning those traps off or by routing halfway between lattice sites, and that the extra half-spacing moves can be ignored in the reported time.

What would settle it

On a concrete static-lattice platform, attempt the binary Paeth rotation for several odd d and record whether every intermediate configuration is collision-free without extra costly routing; if the measured stroke count or constant-jerk duration exceeds the claimed 3⌊log₂(d−1)⌋+4 and O(d^{1/3}) scalings once realistic collision avoidance is included, the central efficiency claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Transversal Hadamard on a rotated surface code of distance d becomes an O(log d)-stroke geometric operation rather than an O(d²)-stroke rearrangement.
  • Fold-transversal S can be reduced to O(log d) strokes (space-permitting logarithmic variant) or to linear but still improved cost (compact variant).
  • Toric-code 90° automorphism and Dehn-twist shears, Iceberg reflections, and Bacon-Shor patch rotations for La-cross addressable Cliffords inherit the same logarithmic stroke counts.
  • Logical throughput on reconfigurable atom arrays is no longer dominated by naive atom-by-atom movement time for these primitives.

Where Pith is reading between the lines

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

  • If the remaining factor-3/2 gap to the stroke lower bound can be closed, large-distance surface-code Hadamards would sit at the absolute information limit of single-AOD control.
  • The same binary-shear vocabulary likely extends to other stabilizer automorphisms and to non-square patches once rectangular addressing is retained.
  • Hardware that cannot cheaply blank static traps or route at half-spacing would force a redesign of the collision model and could erase the O(d^{1/3}) timing advantage.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 7 minor

Summary. The manuscript develops AOD movement schedules that implement shears, 90° rotations, axis reflections, and related 2D rearrangements of atom arrays using only a single crossed AOD pair and rigid row/column translations. Under an explicit hardware model (§2.1–2.2), binary and negabinary stroke lists realize a shear (resp. axis reflection) in ⌊log₂(d−1)⌋+1 (resp. ⌈log₂ ℓ⌉) strokes, saturating or nearly saturating the subset-sum lower bound of Lemma 1. Composing three shears via Paeth’s decomposition yields a 90° rotation of a d×d block in exactly 3⌊log₂(d−1)⌋+4 strokes and O(d^{1/3}) constant-jerk time (eq. (5), Table 1), versus O(d²) strokes and O(d^{7/3}) time for atom-by-atom moves. The same primitives are applied to transversal H and fold-transversal S on the surface code, selected code automorphisms (Iceberg, toric Dehn twist), and the Bacon–Shor gadget for La-cross codes.

Significance. Movement time is a first-order bottleneck for logical throughput on reconfigurable neutral-atom processors. Showing that several standard QEC geometric primitives can be done with a single AOD pair, logarithmic stroke count, and O(d^{1/3}) jerk time—without non-rigid row/column crossings—is a concrete and useful systems result. Strengths include a clean lower bound (Lemma 1), matching binary/negabinary constructions, an elementary collision-freedom argument for axis negation (Prop. 1), explicit resource tables against sequential baselines, and direct mapping onto surface-code H/S, toric automorphisms, and La-cross addressable Cliffords. The work is self-contained combinatorial algorithmics with falsifiable stroke-count claims; if the hardware assumptions hold, the schedules are immediately actionable.

major comments (2)
  1. [§2.1 Assumptions 5–9; Table 1; §2.5] §2.1 Assumptions 5 and 9, together with Table 1 and eq. (5): the reported D_tweezer and T_jerk omit the λ/2 offset strokes used to avoid collisions with static sites. For pure shears the paper argues offsets may be unnecessary if static traps on the moved line can be switched off; for axis reflection (§2.5) offsets “before and after every move” are stated as required. Because the main selling point is O(d^{1/3}) time and the comparison in Table 1, the manuscript should either (i) fold a worst-case O(1) offset per stroke into the Table 1 / eq. (5) bounds (asymptotics are unchanged for binary schedules, since N_moves = O(log d)), or (ii) clearly separate “stroke count (exact)” from “wall-clock time under technology X” and state for which static-lattice technologies the no-offset shear timing applies. This is a modeling completeness issue, not a flaw in the stroke-count theorem.
  2. [§3.1; Table 1] §3.1 fold-transversal S, logarithmic variant 3.a): the protocol needs enough free space to translate the upper-triangular block away, reflect it, and return. Unlike the compact variant, the spatial footprint is not quantified (in lattice spacings as a function of d). For dense or near-packed arrays this can dominate the architectural cost and may force the compact O(d)-stroke schedule. A short bound on the auxiliary workspace (or a statement that the log-S schedule assumes a sparse region of size Θ(d²)) would make Table 1’s “logarithmic” row usable for architecture sizing.
minor comments (7)
  1. [§2.4 eq. (5); Figure 2] Figure 2 caption and §2.4: state explicitly that the middle shear runs over 2d−1 columns and that ⌊log₂(2(d−1))⌋+1 = ⌊log₂(d−1)⌋+2 is what produces the “+4” in eq. (5). A one-line derivation would help readers verify the constant.
  2. [§2.4 Remark 2] Remark 2 (sub-block parallel shears) and Figures 3–4: the improved move counts are useful (including the d=3 saturation of the lower bound) but sit outside the main cost formulas. Either promote a single “best known N_moves(d)” expression into Table 1 or mark Remark 2 as optional and keep Table 1 strictly Paeth+binary for clarity.
  3. [§2.2 Lemma 1] Lemma 1: the proof relaxes to arbitrary (non-rectangular) selections and multi-occupancy. It would help to add one sentence that the binary shear constructions meet the bound under the actual rectangular+rigid constraints, so the relaxation is only for the converse lower bound.
  4. [§2.6; §3.1] §2.6 / Figure 6: the “stretched 45° rotation” snaps to the lattice and splits even/odd sublattices. Clarify whether subsequent CZ alignment for fold-transversal S undoes the 0.5-lattice stagger automatically or needs an extra correction stroke.
  5. [Table 1; Abstract] Table 1 header “Dtweezer” and several ⪅ entries: define ⪅ once in the table caption (as in the main text) so the table is self-contained. Also fix the inconsistent spacing in “F old-transversal” and the broken math in the abstract line “3⌊log 2(d− 1)⌋+ 4”.
  6. [§3.2; Figure 8] §3.2 Dehn twist: the extra ℓ−1 strokes of length ℓ after the binary shear dominate both N_moves and T_jerk and prevent a pure O(log ℓ) claim. State the leading term up front (N = ℓ + ⌊log₂(ℓ−1)⌋) so it is not misread as logarithmic.
  7. [Abstract; §1; §2.1] Minor typos: “Those translate directly” → “These”; “adistance” spacing artifacts in the abstract PDF text; “intermodulations” is fine but a pointer to Ref. [14] at first use in §2.1 would help.

Circularity Check

0 steps flagged

No circularity: combinatorial AOD schedules derived from binary place-value and Paeth, not from fitted inputs or self-justifying citations.

full rationale

The paper’s load-bearing claims are explicit movement schedules and resource counts under a stated AOD model (§2.1–2.2). Lemma 1 is a subset-sum lower bound on stroke count and distance; the binary/negabinary shear and reflection schemes saturate or approach that bound by construction of place-value representations; the 90° rotation cost 3⌊log₂(d−1)⌋+4 follows from composing three such shears via the classical Paeth factorization (cited as external graphics literature [31]), with the middle shear width 2d−1 accounted for in eq. (5). Constant-jerk timing is the standard α∑|m_i|^{1/3} model, not a fit. Applications to transversal H, fold-transversal S, automorphisms, and Bacon-Shor gadgets are uses of these primitives, not predictions forced by data or by the authors’ prior uniqueness theorems. Citations supply background (surface-code gates, AOD hardware, Paeth) and do not close a definitional loop. No fitted-parameter-as-prediction, self-definitional identity, or load-bearing self-citation chain appears.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 3 invented entities

The central claims rest on a short list of hardware modeling choices and standard mathematical facts. No free parameters are fitted to data. The only invented objects are the named movement primitives themselves, which are fully constructive.

axioms (6)
  • domain assumption A single crossed AOD pair addresses only axis-aligned rectangular sub-arrays and applies a common rigid horizontal or vertical displacement to all selected atoms (assumptions 1–3, §2.1).
    Defines the legal stroke set; every construction is built inside this model.
  • domain assumption Collisions with static traps can always be avoided (by extinguishing traps or routing at λ/2); extra λ/2 cost is ignored in timing (assumptions 5, 9).
    Required for the collision-free claim and for the reported T_jerk figures.
  • domain assumption Stroke time follows constant-jerk scaling T = α ℓ^{1/3} with α = (12/jλ)^{1/3}; pickup/release time is negligible (assumptions 7–8).
    Converts stroke lists into the O(d^{1/3}) time claim.
  • standard math Any integer displacement in {0…n} (resp. signed) is a subset sum of the binary (resp. negabinary) place values (standard place-value theorem).
    Used for the logarithmic shear and reflection constructions and for Lemma 1 tightness.
  • standard math Paeth’s identity: a 90° rotation factors into three shears (Paeth 1990, eq. (4)).
    Reduces rotation to three calls of the shear primitive.
  • domain assumption Transversal H on the rotated surface code requires a 90° data-block rotation; fold-transversal S requires 45° diagonal CZ pairing (Horsman, Chen, Moussa, Breuckmann).
    Links the geometric primitives to the claimed QEC gate costs.
invented entities (3)
  • Binary / block / line-by-line shear primitives under single-AOD rigid constraints independent evidence
    purpose: Realize affine shears with O(log d) or O(d) strokes while obeying rectangular selection and no row/column crossing.
    Constructive schedules; fully specified by stroke lists; no extra physical postulate.
  • Paeth + binary 90° rotation schedule (3⌊log₂(d−1)⌋+4 strokes) independent evidence
    purpose: Concrete single-AOD realization of the surface-code transversal Hadamard geometry.
    Composition of the shear primitive with Paeth; counts derived, not postulated.
  • Negabinary axis-reflection schedule independent evidence
    purpose: Realize x→−x (and diagonal reflections via composition) with ⌈log₂ ℓ⌉ strokes.
    Standard negabinary encoding plus a short collision-freedom proof (Prop. 1).

pith-pipeline@v1.2.0-daily-grok45 · 20926 in / 3500 out tokens · 69874 ms · 2026-07-31T16:19:09.224888+00:00 · methodology

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read the original abstract

Neutral-atom quantum computers offer arbitrary connectivity enabled by atom transport. Some logical operations can then be simplified or reduced entirely to geometric rearrangements of the atoms. Minimizing the duration of these movements is therefore essential for high logical throughput. We introduce new primitives to shear, rotate and reflect 2D arrays of atoms in a static lattice using sweeps of a single dynamic crossed acousto-optic deflector (AOD) pair. Using (nega-)binary and geometric decompositions, we achieve an AOD stroke count scaling logarithmically in the linear size of the array. In one example, we use the Paeth decomposition to implement a $90^{\circ}$ rotation for a transversal Hadamard gate in a rotated surface code of distance $d$ in $3\lfloor\log_2(d-1)\rfloor + 4$ AOD strokes and $O(d^{1/3})$ constant-jerk time, against $O(d^2)$ strokes and $O(d^{7/3})$ time for atom-by-atom rearrangement.

Figures

Figures reproduced from arXiv: 2607.28149 by 2), 23 rue du Loess, (2) University of Strasbourg, 67200, Asier Pi\~neiro Orioli (1), CESQ, CNRS, European Center for Quantum Sciences, France, Hugo Perrin (1), ISIS (UMR 7006)), Samuel Crew (1) ((1) QPerfect SAS, Strasbourg, Tom Hartweg (1.

Figure 1
Figure 1. Figure 1: The three shear schemes applied to a 5 × 5 atom array with the top row as reference. Each panel shows the array after a single AOD stroke. (a) Line-by-line moves one row per stroke (d − 1 = 4 strokes). (b) Block moves all rows whose target displacement still exceeds the current cumulative shift, four strokes of displacement 1 each. (c) Binary decomposition encodes each row’s target displacement in binary: … view at source ↗
Figure 2
Figure 2. Figure 2: Paeth decomposition of the 90◦ rotation of a 5 × 5 array into three shears. Panels 1–4 show the array after each of the three shears. Here, for each shear, the reference line is chosen to be the middle row/column. 0) 1) 2) 3) 4) 5) 6) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimal 6-move 90◦ rotation for d = 3. pM(i) = i + P k∈M bk(i)(−2)k is the position of the atom that started in i after the strokes in M are ap￾plied. Subtracting the two positions, i − i ′ = X k∈M ck(−2)k , where ck := bk(i ′ ) − bk(i) ∈ {−1, 0, 1}. Subtracting the negabinary expansions of −2i and −2i ′ instead gives 2(i − i ′ ) = P k≥0 ck(−2)k . Eliminating i − i ′ between the two, X k≥0 ck(−2)k  1k∈M −… view at source ↗
Figure 4
Figure 4. Figure 4: Sub block moves for d = 9. Twelve moves beats the Paeth+binary protocol by one move. mirror 0) 1) 2) 3) −2 +4 −8 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reflection of a line of 5 atoms across a mirror axis, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Approximate rotation by 45◦ decomposed as two shears. The dashed red line represents the diagonal of the original configuration (0). The final configuration (2) shows how the diagonal gets stretched and staggered to snap back to the static lattice. needs only 2 strokes (no intermediate stops on the return). To implement the fold-transversal S gate on the rotated surface code, one can morph it into the regu… view at source ↗
Figure 7
Figure 7. Figure 7: The compact fold-transversal S-gate protocol for a distance- [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A Dehn twist realised by binary decomposed shears. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

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