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arxiv: 2605.10932 · v1 · submitted 2026-05-11 · 🪐 quant-ph · cond-mat.mes-hall

Recognition: 3 theorem links

· Lean Theorem

Crystallographic Symmetry Generates Phononic Holonomic Gates with Biased-Erasure Channels

El Mustapha Mansouri, Keigo Arai

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:35 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords crystallographic symmetryphononic holonomic gatesbiased-erasure channelsnitrogen-vacancy centersstrain tensorLambda manifoldsquantum error correctionsuperadiabatic gates
0
0 comments X

The pith

Crystallographic symmetry fixes the strain interaction in Lambda systems to a scalar dot product, enabling phononic holonomic gates whose error channel supports 64% fewer data qubits in XZZX simulations.

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

The paper establishes that when a projected strain tensor and Lambda-transition operators share a multiplicity-one two-dimensional irreducible representation, crystal symmetry constrains their linear interaction to a simple scalar dot product. This constraint lets two phase-locked mechanical modes generate a circular strain field for complex Lambda-leg control without microwave near fields. On that manifold the authors build a superadiabatic echo-lune holonomic gate whose rotating-frame dynamics yield 99.88% conditional fidelity and an extracted channel with only 0.47% erasure probability plus 0.168% residual Z error. In XZZX code-capacity simulations the resulting biased-erasure model produces a nominal 64% reduction in required data qubits relative to an unstructured Rabi drive. The symmetry also filters noise so that A2-sector perturbations become optically distinguishable while transverse E-sector faults are echo-suppressed, preserving decoder-friendly structure.

Core claim

When the projected strain tensor and Lambda-transition operators share a multiplicity-one two-dimensional irreducible representation, symmetry fixes the linear strain interaction to a scalar dot product. Two phase-locked mechanical modes then synthesize a circular strain field that implements complex phononic Lambda-leg control. This manifold supports a superadiabatic echo-lune holonomic gate whose channel, extracted from nitrogen-vacancy simulations, contains 0.47% erasure probability and 0.168% residual Z error, yielding a 64% fit-extrapolated data-qubit reduction in XZZX code-capacity simulations while the bright-state structure parity-filters A2 noise and echo-suppresses transverse E-sey

What carries the argument

The multiplicity-one two-dimensional irreducible representation shared by the projected strain tensor and Lambda-transition operators, which symmetry-reduces their interaction to a scalar dot product and thereby organizes the gate errors into a biased-erasure channel.

If this is right

  • Complex Lambda-leg control becomes possible using only two phase-locked mechanical modes without local microwave fields.
  • A2-sector perturbations are parity-filtered into an optically distinguishable auxiliary state while transverse E-sector faults are echo-suppressed.
  • The extracted channel supports a 64% nominal reduction in data-qubit overhead for XZZX surface-code simulations.
  • Repeated-round detector-model diagnostics preserve the nominal distance-9 proxy while flagging missed erasures and crosstalk limits.

Where Pith is reading between the lines

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

  • The same symmetry principle may be applied to other orbital Lambda systems to co-design phononic actuation with leakage suppression and decoder bias.
  • Bright-projector phonon-bus diagnostics could be used to monitor the symmetry-protected error structure in real time.
  • The bias toward erasures and Z errors may lower the logical-error threshold or simplify decoder circuits compared with fully depolarizing noise.

Load-bearing premise

The projected strain tensor and Lambda-transition operators share a multiplicity-one two-dimensional irreducible representation in the physical device, with no additional perturbations breaking the symmetry that fixes the interaction.

What would settle it

A measured residual Z-error rate above 0.2% or an erasure probability above 1% in a device whose strain tensor and Lambda operators are independently verified to share the required irreducible representation.

Figures

Figures reproduced from arXiv: 2605.10932 by El Mustapha Mansouri, Keigo Arai.

Figure 1
Figure 1. Figure 1: FIG. 1. Synthetic rotation isomorphism and hardware blueprint. (a) Three-dimensional schematic: a simply-supported dia [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Symmetry-to-decoder logic of the proposal. A multiplicity-one shared two-dimensional irrep fixes the linear strain [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Composite NGQC trajectory. (a) Mixing angle [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Geometric robustness of the holonomic gate against [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Counter-diabatic validation. (a) Sweep 5: noiseless (solid) and noisy (dashed) fidelity versus counter-diabatic parame [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Gate-time compression for Regime A (Sweep 7, [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Transverse-floor validation envelope for the extracted [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Decoder-level threshold separation under the extracted biased-erasure noise model. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
read the original abstract

Solid-state processors require control layers whose errors are legible to quantum-error-correction decoders. We show that crystallographic symmetry can provide such a layer in strain-active Lambda manifolds. When the projected strain tensor and Lambda-transition operators share a multiplicity-one two-dimensional irreducible representation, symmetry fixes the linear strain interaction to a scalar dot product. Two phase-locked mechanical modes synthesize a circular strain field, enabling complex phononic Lambda-leg control without local microwave near fields. On this manifold we construct a superadiabatic echo-lune holonomic gate using Lambda-leg control and a resonant double-quantum counterdiabatic tone. Rotating-frame simulations of a nitrogen-vacancy center give 99.88% conditional average fidelity in 1.833 microseconds, or 99.40% when leakage is counted as error. A resonant gigahertz high-overtone bulk acoustic resonator analysis translates the Hamiltonian into Rabi-rate, linewidth, and envelope-tracking requirements. The bright-state structure organizes noise: A2-sector perturbations are parity-filtered into an optically distinguishable auxiliary state, whereas transverse E-sector faults are echo suppressed and retained as a decoder stress axis. The extracted channel has 0.47% erasure probability and 0.168% residual Z error. In XZZX code-capacity simulations, this biased-erasure model yields a nominal 64% fit-extrapolated data-qubit reduction relative to an unstructured Rabi baseline. Repeated-round detector-model diagnostics preserve the nominal distance-9 proxy and identify missed erasures, transverse floors, leakage/flag timing, and strong crosstalk as validation limits. Extensions to orbital Lambda systems and bright-projector phonon-bus diagnostics identify crystallographic symmetry as a principle for co-designing phononic actuation, leakage, noise bias, and quantum decoding.

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

3 major / 3 minor

Summary. The manuscript claims that when the projected strain tensor and Lambda-transition operators in a strain-active Lambda manifold share a multiplicity-one two-dimensional irreducible representation of the crystallographic point group, symmetry fixes the linear strain interaction to a scalar dot product. This enables construction of superadiabatic echo-lune holonomic gates driven by two phase-locked mechanical modes in an NV-center setting, with rotating-frame simulations reporting 99.88% conditional average fidelity (99.40% including leakage) in 1.833 μs. The resulting channel is extracted as having 0.47% erasure probability and 0.168% residual Z error; XZZX code-capacity simulations then show a nominal 64% fit-extrapolated reduction in data-qubit overhead relative to an unstructured Rabi baseline, with detector-model diagnostics preserving distance-9 proxy.

Significance. If the multiplicity-one irrep condition holds and the simulations are validated, the work supplies a concrete symmetry principle for co-designing phononic actuation, leakage pathways, and biased noise channels that are legible to quantum decoders. The separation of A2-sector perturbations (parity-filtered to an auxiliary state) from E-sector faults (echo-suppressed) and the explicit translation of the Hamiltonian into resonator Rabi-rate, linewidth, and envelope requirements constitute useful engineering guidance. The XZZX overhead reduction is a falsifiable prediction that could be tested in existing NV platforms.

major comments (3)
  1. [symmetry analysis and Hamiltonian construction] The central claim that the projected strain tensor and Lambda-transition operators share a multiplicity-one 2D irrep (thereby fixing the interaction to a scalar dot product) is asserted in the abstract and the Hamiltonian-construction section but is not accompanied by an explicit character-table decomposition, projection-operator calculation, or effective-Hamiltonian expansion confirming absence of additional invariants at leading order. This assumption is load-bearing for the entire gate construction, the biased-erasure channel extraction, and the subsequent code-capacity claims.
  2. [simulation results and XZZX code-capacity section] §5 (simulation results): the reported 99.88% conditional fidelity and 0.47%/0.168% channel parameters are given without error bars, convergence checks against time-step or basis truncation, or direct comparison to an experimental NV strain-coupling baseline; the 64% overhead reduction is therefore an unvalidated extrapolation.
  3. [noise-sector analysis] The robustness statement that 'no additional perturbations break the symmetry' is not quantified; a concrete test (e.g., inclusion of higher-order strain or electric-field terms and re-extraction of the channel bias) is needed to support the claim that the E-sector faults remain echo-suppressed under realistic device conditions.
minor comments (3)
  1. [resonator analysis] The Rabi-rate, linewidth, and envelope-tracking requirements extracted from the high-overtone bulk acoustic resonator analysis should be tabulated with explicit numerical values and units for reproducibility.
  2. [gate construction] Notation for the 'bright-state structure' and 'Lambda-leg control' is introduced without a compact diagram or operator definitions; a single figure summarizing the relevant states and drives would improve clarity.
  3. [fidelity definitions] The abstract states '99.40% when leakage is counted as error' but the main text does not specify the precise leakage model or the optical-readout fidelity assumed for the auxiliary state; this should be stated explicitly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough and constructive review. The comments identify key areas where the manuscript can be strengthened, particularly by making the symmetry argument explicit, validating the numerical results, and quantifying robustness to perturbations. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [symmetry analysis and Hamiltonian construction] The central claim that the projected strain tensor and Lambda-transition operators share a multiplicity-one 2D irrep (thereby fixing the interaction to a scalar dot product) is asserted in the abstract and the Hamiltonian-construction section but is not accompanied by an explicit character-table decomposition, projection-operator calculation, or effective-Hamiltonian expansion confirming absence of additional invariants at leading order. This assumption is load-bearing for the entire gate construction, the biased-erasure channel extraction, and the subsequent code-capacity claims.

    Authors: We agree that an explicit demonstration strengthens the central claim. In the revised manuscript, we have added Appendix A containing the character table for the C_{3v} crystallographic point group of the NV center, the projection operators for the two-dimensional E irreducible representation, and the full effective-Hamiltonian derivation. This shows that under the multiplicity-one condition, the linear strain interaction reduces exactly to a scalar dot product, with no additional independent invariants at leading order. The derivation confirms the absence of other coupling terms that would otherwise mix the Lambda legs. revision: yes

  2. Referee: [simulation results and XZZX code-capacity section] §5 (simulation results): the reported 99.88% conditional fidelity and 0.47%/0.168% channel parameters are given without error bars, convergence checks against time-step or basis truncation, or direct comparison to an experimental NV strain-coupling baseline; the 64% overhead reduction is therefore an unvalidated extrapolation.

    Authors: The reported fidelities and channel parameters are obtained from deterministic rotating-frame simulations. We have added convergence analysis in the Supplementary Information, demonstrating that the results are stable to within 0.01% under variations in time-step size (from 0.1 ns to 1 ns) and basis truncation (up to 20 levels in the Lambda manifold). Since the simulations are coherent and noise-free, traditional statistical error bars are not applicable; instead, we report the numerical precision. A direct comparison to experimental baselines is provided by referencing measured strain-coupling rates in NV centers from the literature (e.g., 1-10 MHz/GHz strain). The 64% overhead reduction is a model-based prediction from the extracted biased-erasure channel and is labeled as such in the revised text; we view it as a testable hypothesis for future experiments. revision: partial

  3. Referee: [noise-sector analysis] The robustness statement that 'no additional perturbations break the symmetry' is not quantified; a concrete test (e.g., inclusion of higher-order strain or electric-field terms and re-extraction of the channel bias) is needed to support the claim that the E-sector faults remain echo-suppressed under realistic device conditions.

    Authors: We have addressed this by performing additional numerical tests. In the revised §5, we include simulations with quadratic strain terms and electric-field perturbations at magnitudes typical for NV devices in diamond (up to 1 kV/cm). The channel parameters are re-extracted, yielding an erasure probability of 0.47% with a variation of less than 0.05% and residual Z error of 0.168% ± 0.02%. The E-sector faults remain echo-suppressed, with the symmetry protection intact. These results are now quantified and support the robustness claim under realistic conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; symmetry fixing and simulations are independent

full rationale

The paper's derivation starts from the explicit assumption that the projected strain tensor and Lambda-transition operators share a multiplicity-one 2D irrep of the point group; representation theory then fixes the linear interaction to a scalar dot product. This assumption is stated as a precondition rather than derived from the gate or channel results. The superadiabatic echo-lune holonomic gate is constructed on that manifold, and rotating-frame simulations of the NV-center Hamiltonian directly produce the reported fidelities (99.88% conditional, 99.40% with leakage) and extracted channel (0.47% erasure, 0.168% residual Z). XZZX code-capacity simulations then use that channel model to obtain the 64% fit-extrapolated qubit reduction. No step renames a fitted parameter as a prediction, no self-citation chain is load-bearing for the central claim, and the quantitative outputs are simulation results rather than tautological re-expressions of the input symmetry condition. The work is therefore self-contained against its own simulation benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that crystallographic symmetry fixes the strain-Lambda interaction under the stated irrep condition, plus simulation parameters that translate the Hamiltonian into device requirements.

free parameters (1)
  • Rabi rates, linewidths, and envelope tracking parameters
    Translated from the Hamiltonian into explicit requirements for the high-overtone bulk acoustic resonator; values chosen to achieve the reported fidelity.
axioms (1)
  • domain assumption When the projected strain tensor and Lambda-transition operators share a multiplicity-one two-dimensional irreducible representation, symmetry fixes the linear strain interaction to a scalar dot product.
    Invoked to enable complex phononic Lambda-leg control without local microwave fields.

pith-pipeline@v0.9.0 · 5626 in / 1377 out tokens · 36920 ms · 2026-05-12T03:35:54.338171+00:00 · methodology

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Reference graph

Works this paper leans on

119 extracted references · 119 canonical work pages

  1. [1]

    This is not a full actuator-transfer- function simulation

    Effective-spurion robustness diagnostic We also ran a compact robustness diagnostic that perturbs the idealA 2 → |0⟩map at the effectiveΛ- Hamiltonian level. This is not a full actuator-transfer- function simulation. It asks whether the parity-filter fractions degrade continuously under detuning, arm- amplitude, and arm-phase spurions. For a dressed-state...

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  2. [2]

    They depend on T1,T 1ρ, surface-spin spectral density, control imperfec- tions, mechanical mode splitting, quadrature calibration, and erasure-detection efficiency

    What symmetry does not determine The numerical probabilitiesp era,p Z,p dep, andp XY are not fixed by representation theory. They depend on T1,T 1ρ, surface-spin spectral density, control imperfec- tions, mechanical mode splitting, quadrature calibration, and erasure-detection efficiency. In particular, the ME- CCE/KMC bath in the Regime-A simulation is n...

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  3. [3]

    The voltage values in Ta- ble S3 useQ L = 10 4 andηsh = 10−3, equivalently ηshQL ≃10

    is an upper bound. The voltage values in Ta- ble S3 useQ L = 10 4 andηsh = 10−3, equivalently ηshQL ≃10. This givesVRF ≃0.018 V,0.18 V,0.91 V forε0 = 10−6,10−5,5×10−5, respectively. Diamond HBAR resonators routinely achieveQ≳10 4 at GHz frequencies: MacQuarrieet al.demonstrated mechani- cally driven NV spin transitions with a diamond res- onator atQ >10 4...

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  4. [4]

    Consequently, any second-order en- ergy shift generated by the DQ channel shifts|±1⟩only relative to each other, never relative to|0⟩

    DQ operator structure In the{|+1⟩,|0⟩,|−1⟩}eigenbasis ofSz the DQ opera- tors take the matrix form S2 x−S2 y =   0 0 1 0 0 0 1 0 0  ,{Sx,Sy}=   0 0−i 0 0 0 i0 0  ,(J1) so that the raising/lowering combinations are (S2 x−S2 y)±i{Sx,Sy}= 2|∓1⟩⟨±1|.(J2) Crucially, both operators haveidentically zeromatrix el- ements involving|0⟩. Consequently, any se...

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  5. [5]

    Inserting Eq

    DQ coupling from a circularly polarized strain drive The DQ part of the Udvarhelyi spin-strain Hamilto- nian [36] is Hε2 h = h16 2 [ εxz (S2 y−S2 x) +εyz{Sx,Sy} ] .(J3) For aσ+ circularly polarized shear drive at frequencyω, εxz =ε0 cosωtandεyz =ε0 sinωt. Inserting Eq. (J1) and definingΩ DQ≡h16ε0 gives H(σ+) DQ =−hΩ DQ 2 [ eiωt|+1⟩⟨−1|+e−iωt|−1⟩⟨+1| ] .(J...

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  6. [6]

    The simulation operates in the doubly-rotating frame defined by |+1⟩→e−iω+t|+1⟩,|−1⟩→e−iω−t|−1⟩,|0⟩→|0⟩

    Two-drive architecture and rotating frame The gate protocol uses two strain drives: Drive 1:ω+ = 2π(D+γeBz), g + = Ω DQ 2 cos(θ/2), (J6) 33 Drive 2:ω−= 2π(D−γeBz), g −= Ω DQ 2 sin(θ/2), (J7) whereθ(t)is the polar angle of the Bloch-sphere tra- jectory. The simulation operates in the doubly-rotating frame defined by |+1⟩→e−iω+t|+1⟩,|−1⟩→e−iω−t|−1⟩,|0⟩→|0⟩....

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    Second-order energy shifts For a monochromatic off-resonant perturbationV(t) = Aeiωt+A†e−iωtacting on states that are degenerate in the rotating frame, the standard AC Stark (light-shift) formula [38, 40] gives the second-order energy shift of state|n⟩: δE(2) n = ∑ m̸=n [|⟨m|A|n⟩|2 ℏω +|⟨m|A†|n⟩|2 −ℏω ] .(J11) This is the dispersive limit of the driven tw...

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    Total effective Hamiltonian Summing the contributions: δE+1 =−g2 + ℏΩ− + g2 − ℏΩ + , δE−1= + g2 + ℏΩ− −g2 − ℏΩ + , δE0 = 0.(J18) The common-mode shift vanishes identically:(δE+1 + δE−1)/2 = 0. The effective Hamiltonian restricted toQ is therefore HDQ eff =δE+1|+1⟩⟨+1|+δE−1|−1⟩⟨−1| = (g2 − Ω + −g2 + Ω− ) Sz.(J19) Substituting g+ = Ω DQ 2 cosθ 2, g −= Ω DQ ...

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  9. [9]

    This is the Hamiltonian-level input to the matched- environment 3C-SiC benchmark in Appendix N, Sec

    C3v Stark-scaling substitution for 3C-SiC For anyC 3v spin-1 platform using the same single- quantumΛ-leg mechanism, the strain required to reach the target mechanical Rabi rate is ε0 = Ω m |h26|,(J21) and the off-resonant DQ coupling generated by the same strain is Ω DQ =|h16|ε0 = ⏐⏐⏐⏐ h16 h26 ⏐⏐⏐⏐ Ω m.(J22) Thus δAC = 1 4D ⏐⏐⏐⏐ h16 h26 ⏐⏐⏐⏐ 2 Ω 2 m.(J23...

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  10. [10]

    apply identically to both implementation paths de- scribed below once their attainableΩm and transfer func- tion are specified. a. Path A: Membrane mode-topology calibration path. This path implements the mode-pair geometry that tests the holonomic strain topology. Appendix B gives the thin-plate strain-topology guide, but the local Rabi rate is not calib...

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    Boundary conditions and displacement limit The simply-supported (SS) membrane has an effec- tive spring constantk SS mech = 4310 N m−1(Sec. A). In the original slope-scale estimate this would permit ac- cess to the fullΩ m = 2.22 MHzatx 0 = 5 nm. Ap- pendix B uses the membrane modes as a strain-topology guide rather than as a calibrated transverse-shear p...

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  12. [12]

    Electrostatic degeneracy tuning Standardelectron-beamlithographyintroducesdimen- sional asymmetries (δ∼1%) that break the(1,2)/(2,1) modal degeneracy by≈0.47 MHz(Sec. C). Degener- acy is restored via electrostatic spring-softening: a DC biasV DC = 21.6 Vacross thed= 200 nmvacuum gap selectively softens the stiffer mode [63], with a ro- bust13.4 Vsafety ma...

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  13. [13]

    TABLE S6

    Voltage budget The voltage budget (Table S6) demonstrates that the system isdisplacement-limited, not voltage-limited. TABLE S6. Voltage budget for the simply-supported mem- brane with full-area electrode andδ= 1%asymmetry. Component Value Breakdown limitV bd 35.0V DC tuningV DC 21.6V Safety margin3.0V AC headroomV max AC 10.4V RequiredV AC atΩ m = 2.22 MHz∼5mV

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    Coherence parameters:T 1 = 1 ms[42],T 1ρ= 500µs[43] (isotopi- cally purified12C diamond with enrichment>99.9%[44, 46]), andT∗ 2∼10µs

    NV center requirements The simulations assume a single NV center at depth dNV = 20 nmbelow the membrane surface, achievable with standard nitrogen ion implantation [41]. Coherence parameters:T 1 = 1 ms[42],T 1ρ= 500µs[43] (isotopi- cally purified12C diamond with enrichment>99.9%[44, 46]), andT∗ 2∼10µs. Appendix N: Extended parametric studies This section ...

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    mechanical Rabi frequency (Sweep 1) Sweep 1 evaluates fidelity versus mechanical Rabi fre- quencyΩ m = 0.1–2.5 MHzunder worst-case noise (τc = 10 ns)

    Fidelity vs. mechanical Rabi frequency (Sweep 1) Sweep 1 evaluates fidelity versus mechanical Rabi fre- quencyΩ m = 0.1–2.5 MHzunder worst-case noise (τc = 10 ns). With DRAG active, the fidelity is nearly flat: F= 99.40%–99.49%, a variation of0.09%[Fig. S5(a)], confirmingthatcounter-diabaticcorrectiondecouplesthe fidelity from the adiabatic parameter

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    S5(b)]: fidelity is flat to0.03%, confirming that the DC tuning mechanism does not require extreme precision

    Frequency detuning tolerance (Sweep 3) Sweep 3 scans residual frequency detuning∆f= 0– 500 kHz[Fig. S5(b)]: fidelity is flat to0.03%, confirming that the DC tuning mechanism does not require extreme precision

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    S5(d)): with DRAG active, the clamped sweep remains within99.43%–99.47%, about0.02–0.06per- centage points below the simply-supported optimum

    Boundary-condition sensitivity (Sweep 4) Sweep 4 repeats Sweep 1 under clamped boundary conditions (kclamp mech = 9530 N m−1,Ω max m = 1.004 MHz; Fig. S5(d)): with DRAG active, the clamped sweep remains within99.43%–99.47%, about0.02–0.06per- centage points below the simply-supported optimum. Sweep 4.1 confirms that without DRAG, the SS mem- brane shows o...

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    SATD eliminates the adiabatic- speed constraint regardless ofΩ m: the process fidelity isF avg = 99.50–99.85%across the three scenarios at Tgate = 2µs(Fig

    GHz HBAR extension (Sweep 9) Sweep 9 extends the gate-time analysis to GHz bulk acoustic resonator Rabi frequencies (Appendix H): con- servative (Ω m = 2.83 kHz), moderate (28.3 kHz), and optimistic (141.5 kHz). SATD eliminates the adiabatic- speed constraint regardless ofΩ m: the process fidelity isF avg = 99.50–99.85%across the three scenarios at Tgate ...

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    Quadrature drive robustness (Sweep 10) Sweep 10 scans amplitude ratior∈[0.90,1.10]and phase errorδφ∈[−10◦,+10◦]atT gate = 1.833µs (Fig. S7). The gate maintainsFavg≥99.5%over the cen- tral plateau (|r−1|≲4%,|δφ|≲4◦). Over the broader |r−1|≤6%,|δφ|≤6◦box the minimum sampled value is99.32%, and the full scan degrades gracefully to98.0% at the worst corner

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    It is re- tained in the validation ledger as Sweep 12 rather than counted among Sweeps 1–10

    Floquet multitone validation (Sweep 12) To test the rotating-wave and multitone assumptions directly, we added a Floquet-corrected lab-frame valida- tion beyond the ten core parametric sweeps. It is re- tained in the validation ledger as Sweep 12 rather than counted among Sweeps 1–10. The calculation propa- gates the Regime-A unitary using the same twoΛ-l...

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  21. [21]

    We keep 38 TABLE S7

    Matched-environment 3C-SiC C3v benchmark The 3C-SiC calculation is a controlled platform- design check, not a measured-device forecast. We keep 38 TABLE S7. Floquet/lab-frame multitone validation atTgate = 1.833µsandΩ m = 2.22 MHz.F avg is computed relative to the nominal rotating-frame unitary on the computational subspace. CaseF avg vs. RWA Added leakag...

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    Error reduction summary Figure S9 compares the baseline DRAG protocol with the composite NGQC + SATD protocol across the full gate-time range. The high-strain benchmark composite protocol achievesF avg = 99.88%atT gate = 1.833µs, a substantial error reduction over the baseline ceiling, confirming that the three mechanisms (singularity neu- 39 NV 3C-SiC 10...

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  23. [23]

    Noise model The Regime-A/SATD noise model is: Channel Probability Origin Erasurep era = 0.47%Detected|0⟩leakage: noise-mediated CD mismatch +T 1 Zdephasingp Z = 0.168%T 1ρ+A 2-sector residual Depolarizingp dep = 0.012%Undetected leakage X/Yflip numerical floor in the nominal extraction Echo-suppressedE-sector All rates are uniformly scaled by a common fac...

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  24. [24]

    CSS toric code.Standardd×dperiodic lattice withX-stabilisers on faces andZ-stabilisers on vertices, encoding one logical qubit in2d2 data qubits

    Code constructions a. CSS toric code.Standardd×dperiodic lattice withX-stabilisers on faces andZ-stabilisers on vertices, encoding one logical qubit in2d2 data qubits. Distances tested:d∈{3,5,7,9,11}. b. XZZX toric code.Same lattice with Hadamard rotations applied to vertical edges, converting each XXXXface stabiliser toXZZXand eachZZZZver- tex stabiliser...

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  25. [25]

    Table S11 gives XZZX toric code results.pL decreases monotonically withdat every tested scale includings= 100, confirming deeply sub-threshold operation

    Full data: toric codes Table S10 gives CSS toric code results.pL increases withdfors≥5, confirming above-threshold operation. Table S11 gives XZZX toric code results.pL decreases monotonically withdat every tested scale includings= 100, confirming deeply sub-threshold operation. The CSS-to-XZZX advantage ratio atd= 11grows from a factor of∼200ats= 50(p CS...

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    Every asymmetric code (dr < dc) achieveszerologi- cal failures in 5000 trials at all tested scales including s= 100

    Full data: rectangular XZZX planar codes Table S12 gives rectangular XZZX planar code results. Every asymmetric code (dr < dc) achieveszerologi- cal failures in 5000 trials at all tested scales including s= 100. Only the square codes (7×7and9×9) exhibit nonzero failures, and only ats≥50. The Wilson score 95%confidence-level upper bound for a zero-failure ...

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    All non- erasure faults are grouped into an isotropic depolarizing residual, so the only structured resource retained by the decoderisknown-locationerasure

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    Transverse-floor validation envelope The square-code overhead quoted in the main text is a nominal code-capacity estimate for the extracted Regime-A channel. Because the nominalp XY compo- nent is a model-extracted numerical floor rather than a group-theoretic constant, we propagate explicitX/Y floors and reduced erasure-detection efficiency through the s...

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    Holonomy derivation Use the cycle-frequency HamiltonianK=H/hon HΛ = span{|0L⟩,|1L⟩,|a⟩}: K(t) = ∆(t)|a⟩⟨a| + 1 2 [Ω 0(t)|a⟩⟨0L|+ Ω1(t)|a⟩⟨1L|+ h.c.].(P1) The proportional-control command is Ω 0(t) = Ω(t) cosαcosϑ 2,(P2) Ω 1(t) = Ω(t) cosαe−iϕsinϑ 2,(P3) ∆(t) = Ω(t) sinα.(P4) With |bn⟩= cosϑ 2|0L⟩+eiϕsinϑ 2|1L⟩,(P5) |dn⟩= sinϑ 2|0L⟩−eiϕcosϑ 2|1L⟩,(P6) the ...

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    When ∫ T 0 Ω(t)dt= 1,(P10) the bright/auxiliary block evolves asexp[−i2πMα] = e−iπ(1+sinα)I. The induced logical gate is therefore UL(n,γ) =|dn⟩⟨dn|+e−iγ|bn⟩⟨bn|, γ=π(1+sinα), (P11) or, up to a global phase,UL .= exp[−iγn·σ/2]

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    Non-Abelian diagnostic The commutator follows directly from the Pauli repre- sentation: [U(n 1,γ1),U(n 2,γ2)] =−2isinγ1 2 sinγ2 2 (n1×n2)·σ. (P12) 44 Thus nonparallel nontrivial pulses do not commute. The numerical diagnostic appliesXπ/2Zπ/2andZ π/2Xπ/2to the same test states. Each ordered sequence has unit fi- delity to its own target in the ideal compil...

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    Relative delay, arm imbal- ance, phase skew, reflections, or detuning-channel mis- match rotateu 0 and therefore produce gate errors

    Transfer matrix and detuning channel The exact theorem requires the physical controls to preserve a fixed direction in command space, u(t) = (Ω 0,Ω 1,∆) T = Ω(t)u 0.(P13) We model the acoustic and detuning actuator by a base- band transfer matrix, uact(ω) =G(ω)ucmd(ω).(P14) Common-envelope filtering is harmless when G(ω)u0≃g(ω)u0 (P15) over the pulse band...

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    SiV single-shot bright-state benchmark (Regime E) Regime E applies the same bright-state compiler to the SiVorbitalΛmanifold. Thebenchmarkusesonlythetwo orbit-strainΛlegsandasynchronizedscalardetuning; no lower-doublet SATD actuator is assumed. The orbitalT1 model sends auxiliary-state population equally into the two logical states, producing in-subspace ...

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    Force-budget reminder A simple off-resonantΛelimination gives the desired force but also exposes the main feasibility constraint. With H ℏ = ∆|a⟩⟨a|+ [Ωc 2 +g 0ae−iδt ] |a⟩⟨b|+ h.c.,(Q6) adiabatic elimination of|a⟩gives Heff ℏ =−|Ωc|2 4∆ Pb−|g0|2 ∆ a†aPb− [Ω∗ cg0 2∆ ae−iδt+ h.c. ] Pb. (Q7) Thusf= Ω cg0/(2∆). In the Rabi-rate convention used in the main te...

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