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Fidelity Analysis of Adiabatically Driven Donor Spins as Two-Qubit and Ququart Systems

T0 review · 3 major / 9 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Native ququart gates cut donor-spin errors 40–50% vs encoded qubits

desk verdict Solid comparative simulation; structural advantage is real, quantitative magnitude is noise-model-dependent read the letter →

arxiv 2607.07586 v1 pith:D26KSERJ submitted 2026-07-08 quant-ph

Brian Michon , James Keppens , Ashutosh Kinikar , George Simion , Kristof Moors , Bart Sorée This is my paper
classification quant-ph
keywords donorspinquditququartrandomizedbenchmarkingsiliconphosphorusCliffordgroupleakagechargenoise
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 argues that operating a phosphorus-donor spin system in silicon as a genuine four-level qudit (a ququart) is structurally more efficient than forcing the same four levels to behave as two encoded qubits. The central object is the Si:P donor spin system, whose electron and nuclear spin define four computational basis states. The authors compare two ways to run Clifford gates on this system: the native ququart Clifford group C4, which treats all four levels as one unit, and the encoded two-qubit Clifford group C2⊗2, which partitions the four levels into two logical qubits. Using numerical leakage-aware randomized benchmarking under a charge-noise model (a single two-level fluctuator producing random telegraph noise), the authors find that C4 consistently achieves 40–50% lower lower-bound error rates than C2⊗2. The mechanism is circuit complexity: the native ququart gate set requires fewer ESR pulses, fewer EDSR pulses, and fewer displacement ramps per Clifford element, so the system spends less time exposed to noise and leakage. The authors also show that adiabatic displacement ramps, which shuttle the electron between the ionization point (for EDSR) and the interface (for ESR), are essential for suppressing leakage outside the computational subspace, with a K-adiabatic ramp informed by avoided crossings performing best. The advantage is attributed to the structure of the gate sets themselves rather than to any particular choice of driving parameters.

What carries the argument

The comparison hinges on two gate sets defined on the same four-level Hilbert space: C4 (768 elements, generated by the qudit Fourier gate F4, phase gate S4, and Z4) versus C2⊗2 (11,520 elements, generated by Hadamard, S, and CNOT gates on two encoded qubits). Both are decomposed into Givens rotations (two-level subspace rotations) implemented by ESR and EDSR pulses, with virtual Z-gates. Leakage-aware randomized benchmarking extracts a population-transfer decay parameter r_PT that bounds the average gate fidelity from below, accounting for reversible leakage into excited orbital states. Adiabatic displacement ramps (linear, raised cosine, and K-adiabatic profiles) shuttle the electron to a

What would settle it

If the number of ESR pulses, EDSR pulses, and displacement ramps per Clifford element were found to be comparable between C4 and C2⊗2 under a different decomposition scheme, or if the noise model were changed such that the additional operations required by C2⊗2 did not meaningfully increase error exposure, the 40–50% advantage would not hold.

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Extended reading notes

Core claim

The native ququart Clifford group C4 requires fewer physical operations (ESR pulses, EDSR pulses, and displacement ramps) per Clifford element than the encoded two-qubit Clifford group C2⊗2, and this structural economy translates into a consistent 40–50% reduction in lower-bound error rates under charge noise. The comparison is made through leakage-aware randomized benchmarking on a Si:P donor spin system, where adiabatic ramps suppress leakage by positioning the electron at the ionization point only during EDSR control and near the interface during ESR control.

Load-bearing premise

The noise model uses a single symmetric two-level fluctuator with a fixed amplitude and switching frequency, acting as a quasistatic detuning offset per experimental shot. Real Si:P devices exhibit 1/f charge noise with multiple fluctuators and distributed parameters, so the quantitative magnitude of the 40–50% advantage could shift under different noise spectra, though the structural advantage from fewer operations would likely persist.

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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

3 major / 9 minor

Summary. This manuscript presents a numerical fidelity analysis comparing native ququart Clifford group C4 operation versus encoded two-qubit Clifford group C2^⊗2 operation on a Si:P donor spin system. The Hamiltonian (Eqs. 1-3) follows Tosi et al. [26], with control via ESR pulses near the interface and EDSR pulses at the ionization point, connected by adiabatic displacement ramps. Three ramp types (linear, raised cosine, K-adiabatic) are compared. The authors employ leakage-aware randomized benchmarking following Chen & Baldwin [37] to extract population-transfer decay parameters r_PT under a single symmetric two-level fluctuator (TLF) noise model (Eq. 5, A_TLF = 10 V/m, f_RTN = 10 Hz). The central finding is that C4 consistently achieves approximately 40-50% lower lower-bound error rates ε^LB_PT compared to C2^⊗2, attributed to reduced circuit complexity (fewer ESR pulses, EDSR pulses, and ramps per Clifford element, as shown in Fig. 3). The authors argue this advantage is structural rather than an artifact of driving conditions.

Significance. The manuscript addresses a well-motivated question: whether native qudit operation provides fidelity advantages over encoded qubit operation within the same Hilbert space, specifically for donor spin systems. The use of leakage-aware RB (following Ref. [37]) is appropriate for the system under study, where leakage outside the computational subspace is non-negligible. The comparison protocol is fair in that the same random seeds and noise realizations are used across gate sets. The resource-count analysis (Fig. 3) provides a transparent, group-theoretic explanation for the observed fidelity difference. The K-adiabatic ramp construction (Eqs. 9-11) is a reasonable adaptation of prior work [26] to the multi-ramp context. The structural argument—that C4 requires fewer operations per Clifford element (768 vs 11,520 elements)—is largely noise-independent and is the strongest aspect of the paper. However, the quantitative 40-50% figure is specific to the chosen noise model parameters.

major comments (3)
  1. Sec. II.A, Eq. (5) and Sec. V.A: The noise model uses a single symmetric TLF with A_TLF = 10 V/m and f_RTN = 10 Hz, placing it deep in the quasi-static regime (mean dwell time ~100 ms >> gate durations ~microseconds). The authors acknowledge this (Sec. V.A: 'the TLF therefore acts primarily as a detuning offset during a given RB shot'). In this regime, per-gate error scales approximately with gate duration times noise sensitivity, and total sequence error scales with total operation count. This means the 40-50% quantitative advantage is largely a consequence of the resource counts shown in Fig. 3, and the specific A_TLF value mainly sets the overall error scale. The manuscript should state more explicitly that the quantitative figure is specific to this noise model and that the structural advantage (fewer operations) is the noise-independent claim. A brief sensitivity analysis varying A_
  2. Sec. V.C, Fig. 4d: The lower-bound error ε^LB_PT = 1-(1+3r_PT)/4 is plotted for both groups as a function of ramp duration τ. The text states 'there is a consistent 40-50% reduction of the lower bound error for C4 compared to C2^⊗2.' However, the error bars (99% confidence intervals, shown as shaded regions in panels a-c but not explicitly in panel d) are not reported for panel d. Given that the absolute error rates are on the order of 3-8% (from Table I), the statistical significance of the 40-50% reduction should be verified. Please add confidence intervals or error bars to Fig. 4d and confirm that the reduction is statistically significant across the full range of τ.
  3. Sec. IV, Fig. 3: The resource-count comparison (panels a-d) shows distributions over all Clifford elements, but the connection between resource counts and the RB error rates is only qualitatively stated. The text says 'C2^⊗2 requires not only more ESR pulses, but also more EDSR pulses and more ramps' (Sec. V.C). Since ESR and EDSR pulses have different noise sensitivities (ESR at interface vs EDSR at ionization point), and ramps contribute nonadiabatic leakage, a more quantitative decomposition of the total error into contributions from each operation type would strengthen the causal claim. At minimum, the mean operation counts (ESR, EDSR, ramps) per Clifford for each group should be reported numerically, not only visually in the violin plots.
minor comments (9)
  1. Sec. II.A, Eq. (5): The notation switches between A_TLF (in Eq. 5 and Sec. V.A) and A_RTN (in Sec. V.A text: 'A_RTn = 10 V/m'). Please use consistent notation.
  2. Sec. V.B, Fig. 4 caption: Panel (a) is labeled 'S_PT,4(m)' but the y-axis label reads 'S_PT, 4 (m) (%)'. The subscript formatting is inconsistent across the figure.
  3. Sec. III.B, Eq. (6): The function s(α) is introduced, but in Eq. (10) the notation switches to s as the integration variable for ∆E_z(s). This is potentially confusing since s(t) in Eq. (5) denotes the TLF state. Consider using a different symbol for one of these.
  4. Table I: The column header 'S(m) / S_PT(m)' is ambiguous—presumably S(m) is the standard survival and S_PT(m) is the leakage-aware version, but this should be stated explicitly. Also, the F_RB column appears to report standard RB fidelity while F_PT reports the leakage-aware bounds; the relationship to Eqs. (27) and (32)-(33) should be clarified.
  5. Sec. V.B: The text mentions 'n_seeds = 10 and n_trials = 150' but the caption of Fig. 4 says 'n_seeds = 10, n_trials = 150.' In Sec. V.C, the text says '10 RB seeds of 150 trials each.' These are consistent but the total sample size of 1500 per τ should be stated once clearly.
  6. Sec. IV, Eq. (14): The gate decomposition U_Gate(ϕ1,...,ϕ4; θ1,...,θ6) uses subscripts on Y_i that refer to the {|i-1⟩,|i⟩} subspace, but the relationship between the index i (1-3) and the six θ parameters is not immediately clear. A brief clarification would help readers unfamiliar with Ref. [21].
  7. Sec. II: The detuning field is defined as ∆E_z ≡ E_z - E^0_z, but the orbital Hamiltonian (Eq. 2) uses (E_z - E^0_z)/h. The factor of h (Planck's constant) in Eq. (2) versus its absence in the definition of ∆E_z should be clarified—presumably ∆E_z is in units where h is absorbed, but this is not stated.
  8. Fig. 2b: The y-axis label 'P_initial' is unclear. It should be 'Survival probability' or 'P_survival' for consistency with the text.
  9. Sec. V.A: The phrase 'and, which ideally leads to the final state being |ψ_init⟩ = |0⟩' contains a grammatical error (stray comma after 'and').

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the structural resource-count argument as the strongest, noise-independent aspect of the paper and raises three points: (1) the quantitative 40-50% figure is specific to the chosen noise parameters and should be stated more explicitly, with a sensitivity analysis; (2) confidence intervals are missing from Fig. 4d and statistical significance should be verified; (3) mean operation counts should be reported numerically and a quantitative error decomposition by operation type would strengthen the causal claim. We agree with all three points and will revise accordingly.

read point-by-point responses

  1. Referee: Sec. II.A, Eq. (5) and Sec. V.A: The noise model uses a single symmetric TLF with A_TLF = 10 V/m and f_RTN = 10 Hz, placing it deep in the quasi-static regime. The 40-50% quantitative advantage is largely a consequence of resource counts, and the specific A_TLF value mainly sets the overall error scale. The manuscript should state more explicitly that the quantitative figure is specific to this noise model and that the structural advantage is the noise-independent claim. A brief sensitivity analysis varying A_TLF is requested.

    Authors: The referee's analysis is correct. In the quasi-static regime (mean dwell time ~100 ms >> gate durations ~microseconds), per-gate error scales approximately with gate duration times noise sensitivity, and total sequence error scales with total operation count. The 40-50% figure is therefore largely a consequence of the resource counts shown in Fig. 3, with A_TLF setting the overall error scale rather than the ratio between the two gate sets. We will revise Sec. V.A and the conclusion to state explicitly that (i) the quantitative 40-50% reduction is specific to the chosen noise model parameters (A_TLF = 10 V/m, f_RTN = 10 Hz), and (ii) the structural advantage—fewer operations per Clifford element for C4 versus C2^⊗2—is the noise-independent claim. We will also add a brief sensitivity analysis varying A_TLF over a range of values (e.g., 1-50 V/m) to demonstrate that the ratio of error rates between the two groups remains approximately constant while the absolute error scale changes, confirming that A_TLF sets the overall scale but not the relative advantage. revision: yes

  2. Referee: Sec. V.C, Fig. 4d: The error bars (99% confidence intervals) are not reported for panel d. Given that the absolute error rates are on the order of 3-8%, the statistical significance of the 40-50% reduction should be verified. Please add confidence intervals or error bars to Fig. 4d and confirm that the reduction is statistically significant across the full range of τ.

    Authors: We agree that confidence intervals should be shown in Fig. 4d. The 99% confidence intervals are available from the fitting procedure used for panels a-c but were omitted from panel d for visual clarity. We will add shaded confidence regions to Fig. 4d. Based on our data (10 seeds × 150 trials = 1500 samples per τ value), the confidence intervals for C4 and C2^⊗2 do not overlap across the full range of τ shown, confirming that the 40-50% reduction is statistically significant. We will state this explicitly in the revised text. revision: yes

  3. Referee: Sec. IV, Fig. 3: The connection between resource counts and RB error rates is only qualitatively stated. A more quantitative decomposition of the total error into contributions from each operation type would strengthen the causal claim. At minimum, the mean operation counts (ESR, EDSR, ramps) per Clifford for each group should be reported numerically, not only visually in the violin plots.

    Authors: We agree that the mean operation counts should be reported numerically. We will add a table reporting the mean ESR pulses, EDSR pulses, and ramps per Clifford element for both C4 and C2^⊗2 alongside the existing Fig. 3. Regarding the quantitative error decomposition by operation type: a full decomposition would require separate RB experiments isolating each operation type (ESR-only, EDSR-only, ramp-only sequences), which is not straightforward within the Clifford RB framework since each Clifford element contains a mixture of operation types. However, we can provide a partial decomposition by noting that ESR pulses operate near the interface (low charge-noise sensitivity), EDSR pulses operate at the ionization point (high sensitivity), and ramps contribute primarily nonadiabatic leakage. We will add a paragraph in Sec. V.C that qualitatively discusses these different noise sensitivities and their relative contributions to the total error, and we will note that a full quantitative decomposition is a direction for future work. We believe the numerical mean counts plus this discussion sufficiently strengthen the causal claim for the present manuscript. revision: partial

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; central claim follows from group structure and independent simulation

full rationale

The paper's derivation chain is substantially self-contained against external benchmarks. The Hamiltonian (Eqs. 1–3) is taken from Tosi et al. (Ref [26], external). The gate decomposition (Eq. 14) and phase values come from Seifert et al. (Ref [21], external). The leakage-aware RB framework (Eqs. 28–33) comes from Chen & Baldwin (Ref [37], external). The K-adiabatic ramp construction (Eqs. 9–11) cites Ref [26] (external). The central claim—a 40–50% lower-bound error reduction for C4 vs C2^⊗2—follows from two independent inputs: (1) the mathematical fact that C4 has 768 elements vs 11,520 for C2^⊗2, which via the decomposition of Eq. 14 directly yields fewer ESR pulses, EDSR pulses, and ramps per Clifford element (Fig. 3, a straightforward counting exercise), and (2) numerical RB simulation under a specified noise model (single symmetric TLF, A_TLF = 10 V/m, f_RTN = 10 Hz) that is a modeling choice, not a fit to data that is then re-predicted. The noise parameters are chosen, not fitted to the target quantity. The self-citations present (Refs [18], [19], [31] share co-authors with this paper) concern peripheral topics—a qudit stabilizer simulator, qudit Clifford group definitions, and flopping-mode qubit modeling—none of which are load-bearing for the central comparison. No step in the derivation chain reduces to its own inputs by construction.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

No new physical entities are postulated. The paper works entirely within the established Si:P donor spin Hamiltonian from Ref [26].

free parameters (8)
  • B0 = 0.4 T
    Static magnetic field, chosen for simulation, not fitted to data
  • A (hyperfine) = 117.53 MHz
    Hyperfine coupling constant for Si:P, taken from literature
  • A_TLF = 10 V/m
    RTN noise amplitude, chosen for simulation
  • f_RTN = 10 Hz
    TLF switching frequency, chosen for simulation
  • E_ac = 10 V/m
    EDSR driving amplitude, described as conservative
  • B1 = 100 μT
    ESR driving amplitude, described as conservative
  • ΔEz (interface) = 6.5 kV/m
    Detuning at interface operating point, chosen for simulation
  • τ (ramp duration) = 50 ns (primary)
    Ramp duration, swept in Fig. 4d
assumptions (5)
  • domain assumption The donor spin system is adequately described by the 8-dimensional Hamiltonian H = H_orb + H_B + H_A (Eq. 1-3) with a two-level charge qubit approximation for the orbital degree of freedom.
    Sec. II, Eq. 1-3. The orbital Hilbert space is truncated to two states {|i⟩, |d⟩}, neglecting higher orbital excitations except as leakage channels.
  • domain assumption Charge noise is adequately modeled by a single symmetric two-level fluctuator with random telegraph noise (Eq. 5).
    Sec. II.A. Real devices have 1/f noise and multiple fluctuators; the single-TLF model is a simplification.
  • domain assumption The rotating wave approximation (RWA) is valid for all ESR and EDSR pulses, making them effective two-level Givens rotations (Eq. 12).
    Sec. IV, Eq. 12. RWA requires driving frequencies much larger than Rabi frequencies, which should hold for the chosen parameters but is not explicitly verified.
  • domain assumption Virtual Z-gates have near-perfect fidelity (phase absorbed into driving fields).
    Sec. IV, after Eq. 14. This assumes instantaneous phase updates with no noise impact, which is standard but idealized.
  • standard math The gate decomposition of Eq. 14 (from Ref [21]) is universal for SU(4).
    Sec. IV, Eq. 14. This is a known mathematical result from Brylinski & Brylinski (Ref [33]).

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Pith. "Pith review of Fidelity Analysis of Adiabatically Driven Donor Spins as Two-Qubit and Ququart Systems." pith.science (2026). https://pith.science/paper/D26KSERJ

@misc{pith2026260707586,
  author       = {Pith},
  title        = {Pith review of: Fidelity Analysis of Adiabatically Driven Donor Spins as Two-Qubit and Ququart Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D26KSERJ}},
  note         = {Machine review of arXiv:2607.07586}
}
abstract

Donor spin systems host a native Hilbert space whose dimension exceeds that of a qubit, meaning they can be used as qudits. Here we study a \ce{Si{:}P} donor spin system through leakage-aware randomized benchmarking (RB) of native ququart $\mathcal{C}_4$ and encoded two-qubit $\mathcal{C}_2^{\otimes 2}$ Clifford groups. We implement adiabatic ramps to operate electron dipole spin resonance (EDSR) pulses at the ionization point, where the electron is shared halfway between the donor and the interface, and to operate electron spin resonance (ESR) pulses near the interface, motivated by the sensitivity of the effective magnetic field to charge noise at the ionization point. By placing the electron near the ionization point only during EDSR control and using sufficiently long displacement ramp durations, leakage outside the computational basis is strongly suppressed, which is crucial for optimized qudit control. We find in our analysis based on leakage RB that $\mathcal{C}_4$ consistently achieves $\sim 40$--$50\%$ lower (lower-bound) error rates $\varepsilon^{\mathrm{LB}}_\mathrm{PT}$ with respect to $\mathcal{C}_2^{\otimes 2}$, due to its reduced circuit complexity. These results indicate that donor spin qudits benefit from genuine qudit operation as opposed to imposed encoded qubit operation.

Figures

Figures reproduced from arXiv: 2607.07586 by the authors.

Figure 1
Figure 1. FIG. 1. Donor spin device schematic and operation. (a) Schematic visualization of the Si:P donor spin system. The excess [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Detuning field profile as a function of time for all ramp types in which the total ramp duration is kept fixed at [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Violin distributions indicating the resource counts for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Overview of RB results. Numerical metrics for leakage-aware RB are reported in Table [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.