Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space

A.J. Niemi; A. Molochkov; J. Dai; J. Westerholm

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Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space

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Pith's one-line read The paper claims that closed loops in the shape space of a deformable triangle generate the full SU(2) holonomy group on its vibrational doublet, giving universal single-qubit holonomic gates and, via linked loops, a CNOT.

desk verdict A clean mathematical framework for holonomic SU(2) gates from loops in shape space, but the physical Wilczek–Zee connection is an ansatz, so universal single-qubit control is not yet demonstrated for any real system. read the letter →

arxiv 2512.24798 v2 pith:ONWLRQYW submitted 2025-12-31 quant-ph cond-mat.other

J. Dai , A. Molochkov , A.J. Niemi , J. Westerholm This is my paper · ORCID

classification quant-phcond-mat.other PACS 03.65.Vz03.67.Lx

keywords non-AbeliangeometricphaseholonomicquantumcomputationWilczek-ZeeconnectionKendallshapespacevibrationalE-doubletRydbergtrimerChern-SimonscontrolledWilsonlooptrace

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

This paper aims to establish that the shape of a deformable triangle—specifically the two nearly degenerate vibrations of its E-doublet—can encode a qubit with universal single-qubit control coming purely from geometric phases. The authors construct the Wilczek–Zee connection on Kendall's shape sphere of triangles and argue that its restricted holonomy group is SU(2), meaning any closed loop in shape space can produce any single-qubit rotation. They give explicit loops for a π/2 phase gate and a Hadamard-type gate, and show how linked loops in arrays of trimers generate a Chern–Simons controlled phase that compiles into a CNOT. They also propose a Ramsey/echo readout of the gauge-invariant Wilson-loop trace, and identify a cesium Rydberg trimer in optical tweezers as a concrete experimental platform. If right, this provides a geometric, error-resistant route to quantum control in few-body molecular systems.

What carries the argument

The core object is the SU(2) Wilczek–Zee connection A on Kendall's shape sphere S²_K, written in terms of the E-doublet's Bloch vector n(θ,φ) and a complex transverse parameter ψ=ρ+iσ. Its curvature F spans the full su(2) algebra, so by the Ambrose–Singer theorem the restricted holonomy group is SU(2). Holonomies are path-ordered exponentials of A along closed loops (Wilson loops), whose trace gives the gauge-invariant rotation angle Θ=q∮(A+ω)+O(|ψ|²). For evaluation, the connection is decomposed into diagonal Abelian and transverse parts, and the Wilson loop is computed via a Dyson expansion in the transverse component.

What would settle it

Measure the Ramsey/echo fringe shift for a small elliptical loop of known solid angle Ω on the shape sphere of a Cs(6s)–Cs(6s)–Cs(nd3/2) trimer. If the extracted rotation angle does not scale linearly with Ω and with loop repetition number N, or if reversing the loop does not flip the sign of the geometric phase, then the asserted connection is not realized.

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Extended reading notes

Core claim

The central claim is that adiabatic transport of a vibrational E-doublet around closed loops in Kendall's shape space is governed by an SU(2) Wilczek–Zee connection whose restricted holonomy group is the full SU(2). The proof rests on showing that the connection's curvature two-form spans the entire su(2) Lie algebra, so the Ambrose–Singer theorem forces the holonomy group to be SU(2). Consequently, closed loops in shape space are universal single-qubit gates. The paper constructs explicit small elliptical loops that implement a π/2 phase gate and a Hadamard-type gate, both requiring the same enclosed solid angle Ω=π/q but differing in the phase of the transverse control ψ. For two qubits, i

Load-bearing premise

The paper assumes that a real physical E-doublet (e.g., in a cesium Rydberg trimer) is described by the asserted SU(2) Wilczek–Zee connection with a controllable transverse parameter ψ and a fixed quantized weight q; if actual molecules realize a different gauge structure, the gate constructions do not apply to any real system.

Editorial extensions

If this is right

Any single-qubit gate on the E-doublet can be realized by a closed shape-space loop; two explicit loops give a π/2 phase gate and a Hadamard-type gate, both with enclosed solid angle Ω=π/q but different ψ phase control.
The Wilson-loop trace is a gauge-invariant, experimentally accessible diagnostic: the proposed Ramsey/echo protocol cancels dynamical phases and measures the non-Abelian holonomy directly, with loop-area and orientation dependence to verify its geometric origin.
For arrays, linked holonomic cycles produce a Chern–Simons controlled phase; for two trimers with state-dependent charges q_A,q_B∈{0,q}, this yields a controlled-Z gate that, combined with the holonomic Hadamard gate, gives a CNOT.
The construction applies to any deformable three-body system with a gapped near-degenerate E-doublet, including Rydberg trimers and Efimov-like trimers, not only a specific molecule.
On the proposed Cs Rydberg trimer platform, sub-microsecond loops with N=1–10 repetitions give total gate times of 5–25 μs, within the Rydberg lifetime, with leakage per loop suppressed by (ℏ/(T_loop Δ_gap))².

Reading between the lines

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

If the asserted connection is realized, the same geometry predicts a geometric angular momentum L_eff even at zero mechanical angular momentum; this could be tested by driving bond-length oscillations with relative phase ϕ13−ϕ23=π/4 and looking for uniform rotation of the triangle.
The Chern–Simons encoding suggests two-qubit entangling phases depend only on loop linking numbers, not on interaction strengths—so deliberate local noise that preserves linking should leave the CNOT phase intact, a testable prediction.
The Abelian-anyon analogy (charges q_i, statistics 4πq_iq_j/k) hints that three-trimer arrays could realize braiding-type topological operations; constructing a three-loop link and measuring the phase would be a natural extension beyond the paper's two-qubit example.
Since the microscopic mapping from Hamiltonian to gauge data is left open, the next step is to compute q and ψ from a realistic Rydberg-trimer potential and simulate the proposed loops, which would either validate or falsify the platform choice.

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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

Here's the short version: this paper builds an abstract machine for holonomic single-qubit gates from loops in Kendall's shape space, and the math is clean. But the machine's input—the actual SU(2) Wilczek–Zee connection of a physical E-doublet—is never derived from a Hamiltonian. It is written down as a generic ansatz. So the universal-SU(2) claim is proven for a class of connections, not for any concrete molecule.

What is new and good: the combination of shape-space geometry with Wilczek–Zee holonomies is fresh, and the authors give explicit loop shapes for a π/2 phase gate and a Hadamard-type gate, plus a Dyson-expansion scheme to compute the Wilson-loop trace. The interaction-picture factorization for the Hadamard gate is neat. They also propose a Ramsey/echo readout that cancels dynamical phases and isolates the geometric signal. If this framework ever gets a real microscopic underpinning, these will be useful tools. The paper is honest about the gap: Section V says detailed Hamiltonian-based studies will appear elsewhere.

The soft spots are not minor. Eq. (3) is the Faddeev–Niemi/Cho parameterization of an arbitrary SU(2) connection; the curvature computation and Ambrose–Singer argument show that a generic connection has full holonomy, but they say nothing about whether a Cs(6s)–Cs(6s)–Cs(nd3/2) trimer, or any other physical system, produces that connection. In the pinned-normal regime used for the phase gate, the off-diagonal ψ term is suppressed and the holonomy degenerates to U(1). Universality requires ψ to be non-vanishing and steerable along the loops—exactly what is asserted, not shown. The gate angle is proportional to q, a free parameter to be fixed by calibration; the Hadamard condition is imposed by choosing the control phase; the CNOT depends on hand-set state-dependent charges and a Chern–Simons level chosen to make the phase π. So the two-qubit section is an outline, and a pretty under-constrained one.

Who is this for? People working on holonomic quantum computation or geometric phases in molecules will find a coherent framework and a concrete proposal. It is not yet a physics result. It deserves a serious referee, but the referee should insist on either a microscopic derivation or a clear statement that this is a toy model. If the authors can show a real E-doublet with controllable ψ, the paper would be substantially stronger. As it stands, I'd treat it as a promising but unproven program.

Referee Report

5 major / 5 minor

Summary. The paper proposes a framework for holonomic quantum computation using the near-degenerate vibrational E-doublet of a deformable triangle. It identifies Kendall's shape sphere as the control manifold, writes a Wilczek–Zee SU(2) connection in Faddeev–Niemi form (Eq. 3), and uses the Ambrose–Singer theorem to argue that closed loops generate universal SU(2) holonomy. It constructs explicit π/2 and Hadamard-type single-qubit gates from small elliptical loops, proposes a Chern–Simons mechanism for a two-qubit CNOT via linked loops, and presents a Ramsey/echo protocol to measure the Wilson-loop trace. A Cs Rydberg trimer in optical tweezers is proposed as a demonstrator with timing and leakage estimates. The abstract states that the connection is derived, but in the text Eq. (3) is assumed via citations, and the paper defers Hamiltonian-based derivations to future work.

Significance. If the connection form (3) were realized by a physical E-doublet with controllable n and ψ, the paper would provide a concise and elegant route to universal single-qubit holonomic gates plus an experimentally accessible Wilson-loop diagnostic. The algebraic decomposition of the Wilson line, the Dyson expansion for Tr WΓ, the Ramsey/echo protocol in Appendix B, and the explicit leakage estimate in Eq. (40) are useful and clearly presented. However, the physical relevance is not established: n and ψ are properties of the molecular eigenstates, and no Hamiltonian is presented from which (3) follows. The significance is therefore conditional on a future derivation or on reading the paper as an abstract gauge-theoretic construction, which is weaker than the stated claim.

major comments (5)

[II, Eq. (3)] The central premise—that adiabatic transport of an E-doublet is governed by the SU(2) connection (3)—is asserted, not derived. Equation (3) is the Faddeev–Niemi/Cho parameterization of a generic SU(2) connection; for a real E-doublet the Bloch vector n and transverse coupling ψ are fixed by the instantaneous molecular eigenstates. The paper provides no Hamiltonian or eigenstate calculation showing that a Cs trimer or any other system yields this connection with ψ as an independent control knob. Appendix B maps n to shape variables by gauge choice but never computes ψ from a molecular Hamiltonian. Since Sections III and V rest on this assumption, the universal-control claim is not established for any concrete system. The statement in Section V that 'detailed Hamiltonian-based studies... will be presented elsewhere' makes this gap load-bearing.
[III, Eq. (16); Appendix B.5] The gate angle is Θ = q∮(A+ω) + O(|ψ|²), where q is a free parameter 'determined by calibration' (Section II). The proposed Ramsey/echo readout (Appendix B.5) measures precisely Tr WΓ, the same quantity one would use to calibrate q. Thus the protocol cannot serve as an independent test of the predicted gate angle unless q is determined by an independent physical measurement. As written, the comparison of measured and predicted Wilson-loop traces is circular: q can be adjusted to fit. An independent derivation or measurement of q is needed before the claimed benchmarking in Section V is meaningful.
[IV.A.2, Eq. (27)] The Hadamard gate relies on imposing Θ(s)=π/2 for all s along the loop by 'steering the control phase of ψ(s)'. However, ψ is not shown to be an independently controllable parameter of the E-doublet; it is a property of the connection. Moreover, the small-loop approximation used to obtain V(2π)≃exp(−i q Ω_H/2 σ_y) requires small loop area Ω_H and small |ψ|, but the gate condition (30) fixes Ω_H = π/q, which is not necessarily small under the stated constraints. The validity of neglecting higher-order path-ordering corrections in this regime is not quantified.
[IV.B, Eqs. (32)–(34)] The two-qubit gate is an outline whose required ingredients are introduced ad hoc: state-dependent charges q_A, q_B in Eq. (32), linking Lk(Γ_A,Γ_B)=1, and the level choice k=4q² in Eq. (33) to force ϕ=π. No physical mechanism is given for state-dependent Cartan weights or for arranging linked shape cycles in a trimer array. Furthermore, the Chern–Simons path integral (18) is evaluated on the Abelian ψ=0 truncation; its application to the full non-Abelian connection (3) is not justified. The CNOT result is therefore conditional on several unstated assumptions and does not constitute an implementable two-qubit gate.
[Appendix A] The numerical demonstration in Appendix A addresses the U(1) Guichardet connection A and the effective angular momentum (17); it does not show that the off-diagonal term J=ψ(dµ−i sinµ dλ) in Eq. (14) is nonzero along a loop. The SU(2) holonomy claim requires nonvanishing transverse ψ. Thus the 'evasion' of Painlevé's theorem supports at most an Abelian phase and does not by itself justify the non-Abelian connection (3).

minor comments (5)

[III, Eq. (15)] The quantities I2 and I4 are introduced through the Dyson equation but not explicitly defined as loop integrals; please define them or refer to the equation that defines them.
[Abstract/II] The abstract and introduction state that the Wilczek–Zee connection is derived (e.g., 'we derive the Wilczek–Zee connection'), whereas Section II introduces Eq. (3) by ansatz with references [24–27]. Please adjust the wording to match the content.
[II, Eqs. (2),(7)] There are minor notation and formatting issues: Eq. (7) contains an inline '&' instead of a rhetorical separator, and the angles ϑ/θ and ϕ/φ are used with inconsistent subscripts. Please harmonize the notation.
[Fig. 1] The Figure 1 caption is compressed and panels (c) and (d) do not have clearly identified axes/curves in the text. The claims about linear growth of θ(t) and the asymptotic value of L_eff would be easier to assess with labeled panels and a description of initial conditions.
[IV.B, Eq. (18)] The linking and self-linking numbers Lk and SLk are used without definition in the text. Please provide definitions or a reference for the loop invariants, and specify how they are computed for the proposed shape-space loops.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SU(2) holonomy result is a self-contained derivation from the stated connection ansatz; the gate constructions are deliberate control choices, and the missing microscopic Hamiltonian mapping is an unproven assumption, not a circular reduction.

full rationale

The paper's central derivation is conditional and internally consistent: starting from the stated SU(2) Wilczek–Zee connection (3), the curvature (7) is computed and, for generic (A,ψ,n), spans su(2); Ambrose–Singer then gives restricted holonomy SU(2). Eq. (3) is not equivalent to the conclusion (a generic SU(2) connection can still have Abelian or trivial holonomy when ψ=0 or Dψ=0), so this is a derivation, not a self-definition. The explicit π/2, Hadamard, and CNOT constructions are control designs: loop area ab=1/q, condition Θ(s)=π/2, and level k=4q² are chosen to realize target unitaries—this is standard calibration/gate synthesis, not a parameter-free prediction. q is explicitly 'determined by calibration,' and the proposed Ramsey/echo protocol is phrased as a benchmark of the predicted Ω-dependence, so the trace formula is not reduced to a fit at a single point. The genuine weakness is that the physical identification of a real E-doublet (e.g., Cs Rydberg trimer) with the generic connection (3), and the controllability of ψ, are not derived from a microscopic Hamiltonian; Section V defers 'detailed Hamiltonian-based studies of specific platforms' and lists 'platform-specific mappings from explicit microscopic trimer Hamiltonians to the effective gauge data' as future challenges. This is a missing physical derivation and correctness risk, not circularity. Self-citations [24,29] appear in the connection ansatz, but the parameterization is an established general SU(2) decomposition with independent supporting references [25–27], so those citations are not load-bearing.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The construction rests on a generic SU(2) gauge-field ansatz with a calibratable charge q and a free transverse coupling ψ; the microscopic mapping to a specific Cs trimer Hamiltonian is not given. The two-qubit gate further assumes state-dependent Abelian charges and a Chern-Simons level chosen to yield CNOT. These are the main burdens the central claim carries.

free parameters (4)

q (effective Cartan weight) = not specified; calibratable
Appears in the gate angle Θ = q/2 Ω and in the Wilson-loop trace; the paper says it 'can be determined by calibration' (Section III).
ψ = ρ + iσ (transverse coupling) = control-dependent; no fixed value
Free complex control parameter; the Hadamard gate requires steering its phase to enforce Θ(s)=π/2 (Section IVA).
k (Chern-Simons level) = k = 4q²
Chosen so that the controlled phase becomes π, producing CZ/CNOT (Section IVB, Eq. 33).
q_A, q_B (state-dependent charges) = 0 for |0>, q for |1>
Assumed state-dependent coupling to the Abelian part of the connection; not derived from a platform Hamiltonian (Section IVB).

assumptions (6)

domain assumption Guichardet's theorem: L_tot = 0 restricts internal motion to the plane and defines the U(1) connection.
Invoked in Section II to reduce the configuration space to R+ × S²_K and to introduce the Hopf-fibration description.
domain assumption The E-doublet exists as a gapped near-degenerate pair satisfying Δ_gap ≫ ℏ/T_loop ≫ δE.
Assumed to isolate the two-level computational subspace; explicit conditions are given in Appendix B, Eqs. (37)-(38).
ad hoc to paper The Wilczek-Zee connection has the form (3) with (n, ψ) as control data.
No derivation from a specific Hamiltonian; the paper defers microscopic studies to future work (Section II, Section V).
ad hoc to paper The curvature of (3) spans su(2) for generic (A, ψ, n).
Used to claim restricted holonomy group SU(2) via the Ambrose-Singer theorem; asserted rather than proven for all generic configurations (Section III).
domain assumption The Chern-Simons reduction to Abelian charges on the ψ = 0 truncation (Eq. 18) describes coupled trimer arrays.
Used to derive two-qubit phases; assumes each trimer couples only to the Abelian part with charges q_A, q_B (Section IVB).
domain assumption Painlevé's theorem is evaded for generic time-reversal-symmetric vibrational trajectories.
Supported only by a numerical simulation in Figure 1; no code or data are provided (Appendix A).

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Pith. "Pith review of Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space." pith.science (2026). https://pith.science/paper/ONWLRQYW

@misc{pith2026251224798,
  author       = {Pith},
  title        = {Pith review of: Non-Abelian Geometric Phases in Triangular Structures And Universal SU(2) Control in Shape Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ONWLRQYW}},
  note         = {Machine review of arXiv:2512.24798}
}

abstract

We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational $E$-doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the $E$-manifold. We show that its restricted holonomy group is $\mathrm{SU}(2)$, implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a $\pi/2$ phase gate and a Hadamard-type gate. For two-qubit operations, we outline how linked holonomic cycles in arrays generate a controlled Chern--Simons phase, enabling an entangling controlled-$X$ (CNOT) gate. We present a Ramsey/echo interferometric protocol that measures the Wilson loop trace of the Wilczek--Zee connection for a control cycle, providing a gauge-invariant signature of the non-Abelian holonomy. As a physically realizable demonstrator, we propose bond-length modulations of a Cs($6s$)--Cs($6s$)--Cs($nd_{3/2}$) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the $E$-manifold.

Figures

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Geometric Qubits in Programmable Atomic Trimers
quant-ph 2026-07 conditional novelty 5.0 of 10

For an exactly solvable Higgs-oscillator model of atomic trimers, the minimum-energy relative-stationary states at fixed angular momentum are supported on two adjacent spectral sites, giving candidate geometric qubits.

Reference graph

Works this paper leans on

44 extracted references · 17 canonical work pages · cited by 1 Pith paper

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Setup: Spectroscopic studies have established that triatomic Cs(6s)–Cs(6s)–Cs(nd3/2) Rydberg molecules can be formed by binding one excited Cs atom to two ground- state atoms [12], with characteristic bond lengthsR 1 = R2 ∼2000a 0 (≃0.1µm). Typical optical tweezers pro- vide confinement on the micrometer scale and can be ar- ranged in programmable arrays ...
[2]

falling cat

Hadamard gate from a rotation about theyaxis For an Hadamard-type gate we require a loop whose holonomy is aπ/2 rotation about an axis in the equatorial plane of the Bloch sphere. In the representation (3) of the Wilczek–Zee connection, and after an appropriate gauge choice and rescaling ofψ, we write along the loop Γ H A(s) =A(s) σz 2i +A ⊥(s), with A⊥(s...

2022
[3]

These deformations trace a closed trajectory in shape space

Control knobs and mapping to(A, ψ): The complex control parameterψ=ρ+iσin (4) is controlled by synchronized, low-amplitude modulations of the two base bond lengthsR 1,2, realized via acousto- optic control of the tweezer positions and depths. These deformations trace a closed trajectory in shape space. We map each point (θ, ϕ) on Kendall’s shape sphereS 2...
[4]

A demonstrator loop and expected gate: With the normalnpinned so thatC≃A, we consider a small elliptical gate loop Γ that encircles a solid angle 10 ΩΓ ≪1 onS 2 K. Using (2), (13) the loop integral (15) evaluates to I Γ A= 1 2 ΩΓ so that the geometric gate angle is (7) Θ≈ q 2 ΩΓ +O(|ψ| 2) with theO(|ψ| 2) corrections due to the transverse off- diagonal co...
[5]

Timing and adiabaticity: Adiabatic transport within theE-manifold requires the separation of scales as specified in (37) and (38), δE ≪ ℏ Tloop ≪∆ gap (39) HereT loop is the duration of a single small loop Γ on Kendall’s shape sphereS 2 K. The lower inequality ensures that the twoE-levels remain effectively degenerate over the course of one loop, while th...
[6]

This sequence cancels dynamical phases associated with the residual splittingδ E while doubling the geo- metric contribution of the Wilczek–Zee holonomy

Readout: gauge-invariant Wilson-loop trace A minimal readout protocol is a Ramsey/echo se- quence within theE-manifold, that proceeds as follows: (i) prepare anE-superposition; (ii) execute the loop Γ during the first free-precession window; (iii) apply aπ-pulse that exchanges|E (1)⟩ ↔ |E(2)⟩; (iv) execute the time-reversed loop Γ −1; (v) close with aπ/2 ...
[7]

Error channels and operating window: The dominant sources of error are: (i) spontaneous decay and dephasing of the Rydberg elec- tron over the total gate timeT gate, producing an error ∼1−e −Tgate/τR withτ R the lifetime of the Cs(nd 3/2) Rydberg state; (ii) nonadiabatic leakage out of theE-subspace during each loop, with a per-loop leakage probability th...
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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Setup: Spectroscopic studies have established that triatomic Cs(6s)–Cs(6s)–Cs(nd3/2) Rydberg molecules can be formed by binding one excited Cs atom to two ground- state atoms [12], with characteristic bond lengthsR 1 = R2 ∼2000a 0 (≃0.1µm). Typical optical tweezers pro- vide confinement on the micrometer scale and can be ar- ranged in programmable arrays ...

[2] [2]

falling cat

Hadamard gate from a rotation about theyaxis For an Hadamard-type gate we require a loop whose holonomy is aπ/2 rotation about an axis in the equatorial plane of the Bloch sphere. In the representation (3) of the Wilczek–Zee connection, and after an appropriate gauge choice and rescaling ofψ, we write along the loop Γ H A(s) =A(s) σz 2i +A ⊥(s), with A⊥(s...

2022

[3] [3]

These deformations trace a closed trajectory in shape space

Control knobs and mapping to(A, ψ): The complex control parameterψ=ρ+iσin (4) is controlled by synchronized, low-amplitude modulations of the two base bond lengthsR 1,2, realized via acousto- optic control of the tweezer positions and depths. These deformations trace a closed trajectory in shape space. We map each point (θ, ϕ) on Kendall’s shape sphereS 2...

[4] [4]

A demonstrator loop and expected gate: With the normalnpinned so thatC≃A, we consider a small elliptical gate loop Γ that encircles a solid angle 10 ΩΓ ≪1 onS 2 K. Using (2), (13) the loop integral (15) evaluates to I Γ A= 1 2 ΩΓ so that the geometric gate angle is (7) Θ≈ q 2 ΩΓ +O(|ψ| 2) with theO(|ψ| 2) corrections due to the transverse off- diagonal co...

[5] [5]

Timing and adiabaticity: Adiabatic transport within theE-manifold requires the separation of scales as specified in (37) and (38), δE ≪ ℏ Tloop ≪∆ gap (39) HereT loop is the duration of a single small loop Γ on Kendall’s shape sphereS 2 K. The lower inequality ensures that the twoE-levels remain effectively degenerate over the course of one loop, while th...

[6] [6]

This sequence cancels dynamical phases associated with the residual splittingδ E while doubling the geo- metric contribution of the Wilczek–Zee holonomy

Readout: gauge-invariant Wilson-loop trace A minimal readout protocol is a Ramsey/echo se- quence within theE-manifold, that proceeds as follows: (i) prepare anE-superposition; (ii) execute the loop Γ during the first free-precession window; (iii) apply aπ-pulse that exchanges|E (1)⟩ ↔ |E(2)⟩; (iv) execute the time-reversed loop Γ −1; (v) close with aπ/2 ...

[7] [7]

Error channels and operating window: The dominant sources of error are: (i) spontaneous decay and dephasing of the Rydberg elec- tron over the total gate timeT gate, producing an error ∼1−e −Tgate/τR withτ R the lifetime of the Cs(nd 3/2) Rydberg state; (ii) nonadiabatic leakage out of theE-subspace during each loop, with a per-loop leakage probability th...

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