{"id":"6ac134fd-f806-4b0b-be19-0cd1bc2cdcc2","arxiv_id":"2607.28149","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A single crossed AOD pair can rotate a distance-d surface-code data block in 3⌊log₂(d−1)⌋+4 strokes and O(d^{1/3}) constant-jerk time via Paeth shears and binary decompositions.","lead":"Neutral-atom quantum computers can run some logical gates by physically rearranging atoms instead of applying many two-qubit gates. This paper shows how to shear, rotate, and reflect large atom arrays with a single AOD using only a logarithmic number of moves, cutting rearrangement time from polynomial in code distance to roughly its cube root.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper is a clean algorithms contribution whose strongest claim is purely combinatorial under an explicitly idealized AOD model. The lower-bound argument, binary shear primitive, Paeth factorization, and resource table are all derived rather than fitted and contain no internal inconsistency. The hardware idealizations (collision avoidance by trap shut-off or λ/2 routing, ignored transfer time, rigid-only moves) are the real external risk, exactly as the reader identified; they affect whether the O(d^{1/3}) wall-clock bound is attainable on a given platform, not whether the stroke schedule is correct inside the model. Because that risk is already flagged and does not undermine the combinatorial core, no verdict adjustment is warranted. The suggested concrete test simply re-executes the claimed schedule on small instances and would catch any off-by-one or collision error if one existed.","tokens_in":16871,"tokens_out":594,"duration_ms":11061,"concrete_test":"For a concrete odd d (e.g. d=5 and d=9), enumerate the exact sequence of rectangular selections and displacements given by the three binary shears of §2.4 (central reference lines) and verify that (i) every atom ends at its Paeth-rotated lattice site, (ii) no two atoms occupy the same site after any stroke, and (iii) the stroke count equals 3⌊log₂(d−1)⌋+4. This can be done by a short deterministic simulator of the integer-lattice model; any mismatch would falsify the headline schedule.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central combinatorial claim (Paeth + binary shears giving exactly 3⌊log₂(d−1)⌋+4 strokes and the stated D_tweezer/T_jerk bounds for a 90° rotation under the AOD model of §2.1–2.2) is internally consistent. Lemma 1, the binary/negabinary constructions, the collision-freeness argument for pure shears (atoms stay in-row and move rigidly), and the Paeth composition all check out; the residual-translation handling and the middle-shear width 2d−1 are correctly accounted for in eq. (5). The reader’s weakest assumption (Assumption 5 on collision routing / ignored λ/2 cost) is a genuine modeling idealization, but it is stated explicitly, does not create an internal contradiction, and does not falsify the stroke-count claim that is the paper’s strongest result. No tighter load-bearing flaw in the argument itself was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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.","tokens_in":17057,"tokens_out":1614,"duration_ms":48455,"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":[{"comment":"§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.","section":"§2.1 Assumptions 5–9; Table 1; §2.5"},{"comment":"§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.","section":"§3.1; Table 1"}],"minor_comments":[{"comment":"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.","section":"§2.4 eq. (5); Figure 2"},{"comment":"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.","section":"§2.4 Remark 2"},{"comment":"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.","section":"§2.2 Lemma 1"},{"comment":"§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.","section":"§2.6; §3.1"},{"comment":"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”.","section":"Table 1; Abstract"},{"comment":"§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.","section":"§3.2; Figure 8"},{"comment":"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.","section":"Abstract; §1; §2.1"}],"recommendation":"minor_revision","confidential_remarks":"I agree with the reader’s high-confidence accept lean: the combinatorial core is correct and the contribution is real. I am recommending minor_revision only to force explicit resource accounting for λ/2 offsets and for the workspace of log-S, which are the two places an architect could otherwise mis-apply Table 1. No novelty or citation concerns; fit for a quant-ph / QEC systems venue is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: under a strict single crossed-AOD, rigid-translation-only model they give concrete logarithmic schedules for the rearrangements that actually show up in FTQC—especially a 90° data-block rotation for transversal H in 3⌊log₂(d−1)⌋+4 strokes and O(d^{1/3}) constant-jerk time. That is a direct clock-rate win for neutral-atom surface-code and La-cross stacks.\n\nWhat is new is not Paeth or binary subset-sum (both classical), nor logarithmic rearrangement in weaker AOD models (Xu, Constantinides). It is the combination under the hardware constraints people actually face with one AOD pair and no non-rigid moves, plus the worked-out QEC costs (Table 1), the negabinary reflection, the 45° stretch-rotate for fold-transversal S, and the automorphism/Dehn-twist/Bacon-Shor applications. Lemma 1’s lower bounds are correct; the binary/negabinary constructions nearly saturate them; Prop. 1’s collision argument for axis negation is elementary and complete; the middle-shear width 2d−1 is handled properly. The resource table is derived, not fitted.\n\nSoft spots are modeling, not internal cracks. Assumption 5 (collisions always avoidable by turning off traps or λ/2 routing, with those extras ignored in the timing) is stated openly; if your static lattice cannot do that cheaply the O(d^{1/3}) claim softens, though the stroke count still stands. Transfer time and intermodulation are waved away. No code, no hardware demo—reproducibility is “methods-clear.” Novelty is solid inside the subfield but not paradigm-shifting. Citation pattern is appropriate.\n\nThis is for people building or scheduling neutral-atom FTQC movement layers. Bring it to reading group if that is your stack; skip if you only care about thresholds or new codes. I would cite the rotation and shear primitives when discussing logical clock rate. It deserves a serious referee—send it out.","headline":"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.","tokens_in":17834,"tokens_out":541,"would_cite":true,"duration_ms":14847,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"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.","keywords":["neutral atoms","acousto-optic deflectors","atom rearrangement","surface code","transversal gates","quantum error correction","Paeth decomposition","binary shears"],"falsifier":"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.","tokens_in":17736,"feed_emoji":"⚛️","tokens_out":1038,"duration_ms":25896,"temperature":0.7,"pith_summary":"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.","feed_headline":"One AOD rotates a surface-code block in log strokes","feed_subtitle":"Three binary shears replace quadratic atom-by-atom moves for transversal Hadamard","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Single AOD rotates surface-code block via log shears","Paeth shears yield 90° atom rotation in 3 log strokes","Binary AOD sweeps cut rotation to logarithmic strokes","One AOD pair shears d×d block for transversal Hadamard","Log-stroke AOD shears replace quadratic atom moves"],"cache_read_input_tokens":128,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Single AOD rotates surface-code block via log shears","Paeth shears yield 90° atom rotation in 3 log strokes","Binary AOD sweeps cut rotation to logarithmic strokes","One AOD pair shears d×d block for transversal Hadamard","Log-stroke AOD shears replace quadratic atom moves"]},"model":"grok-4.5","effort":"low","cost_usd":0.004282,"raw_usage":{"total_tokens":1280,"prompt_tokens":795,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":42824000,"prompt_tokens_details":{"text_tokens":795,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":415,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":795,"tokens_out":70,"duration_ms":7083,"temperature":1.0,"reasoning_tokens":415,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T16:19:09.224888+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}