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arxiv: 2607.06404 · v1 · pith:ZEO2RGOZ · submitted 2026-07-07 · quant-ph

Bosonic quantum error-correcting codes with finite stellar rank

Rui Wang , Adithi Udupa , Timo Hillmann , Ulysse Chabaud , Alessandro Ferraro , Giulia Ferrini This is my paper

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classification quant-ph PACS 03.67.Pp42.50.Dv
keywords rankbosonicstellarunderencodingslossdephasingfinite
0
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The pith

Two photon additions beat break-even for bosonic error correction

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

This paper asks: how much non-Gaussian resource does a bosonic quantum error-correcting code actually need? The authors use stellar rank — the minimum number of photon additions, combined with arbitrary Gaussian operations, required to prepare a single-mode state — as a resource budget. They show two things. First, when standard code families (cat and GKP) are approximated at finite stellar rank, a trade-off emerges: codes that are best in the ideal infinite-resource limit can become suboptimal once preparation is constrained, because approximation error outweighs intrinsic code quality. Second, and more concretely, when encodings are directly optimized at fixed stellar rank rather than approximating known targets, the minimal rank needed to surpass break-even is surprisingly small: stellar rank k=2 suffices for all dephasing strengths studied, while photon loss demands increasing rank with the loss rate. The optimized codes also spontaneously develop noise-adapted geometry — grid-like phase-space structures under photon loss and rotation-symmetric structures under dephasing.

Core claim

The central object is stellar rank as an operational resource measure for bosonic QEC. The paper establishes that this measure, which counts photon additions, directly controls both the achievable logical fidelity and the phase-space geometry of optimized codes. The key quantitative finding is that stellar rank k=2 is sufficient to surpass break-even under dephasing noise of all strengths considered, while the required rank increases with photon-loss rate. A secondary discovery is that codewords with superior ideal error-correction properties need not be optimal under finite-rank constraints — the best code at rank k depends on the noise channel and the resource budget, not on the ideal-code

What carries the argument

Stellar rank (minimum photon additions to prepare a state), stellar fidelity (best achievable overlap with a target at rank k), Petz-recovery fidelity (surrogate optimization objective), optimal-recovery channel fidelity (final benchmark via semidefinite programming), and variational optimization over Gaussian-unitary-transformed finite-Fock-core states.

If this is right

  • Stellar rank provides a concrete experimental resource budget: if a platform can perform k photon additions, the paper predicts whether break-even is achievable for a given noise model and rate.
  • The reversal of code ordering under finite-rank constraints (e.g., more damped GKP outperforming less damped GKP at the same rank) means experimental code selection should account for preparation cost, not just ideal code properties.
  • The emergence of grid-like and rotation-symmetric structures from unconstrained optimization at fixed rank suggests these geometries are resource-optimal, not merely traditional — they arise even without imposing the symmetry a priori.
  • The k=2 threshold for dephasing provides a concrete target for photonic platforms where photon addition is the primary non-Gaussian resource.

Load-bearing premise

The encoding optimization uses the Petz-recovery fidelity as a cheaper stand-in for the true optimal-recovery fidelity. The Petz map is provably near-optimal only up to a bound that can be loose, so if the Petz fidelity is a poor proxy in the regimes studied, some better codes at low rank may have been missed.

What would settle it

If a code at stellar rank k=1 (or k=2 for photon loss at moderate rates) could be shown to surpass break-even under dephasing (or photon loss), the reported resource thresholds would shift downward. Conversely, if optimal-recovery benchmarking of the k=2 dephasing codes reveals they do not actually surpass break-even, the central quantitative claim fails.

Figures

Figures reproduced from arXiv: 2607.06404 by Adithi Udupa, Alessandro Ferraro, Giulia Ferrini, Rui Wang, Timo Hillmann, Ulysse Chabaud.

Figure 1
Figure 1. Figure 1: FIG. 1. Stellar fidelity as a function of stellar rank [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Approximation of GKP codewords with finite stellar rank. The optimized stellar fidelity is plotted versus stellar rank [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Error-correcting performance of finite-stellar-rank approximations of cat and GKP codewords. The optimal channel infidelity [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Optimized error-correction performance, average photon number, and phase-space structure of finite-stellar-rank bosonic encodings [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Stellar fidelity as a function of stellar rank [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Stellar fidelity as a function of stellar rank [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Channel infidelity [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Wigner functions of the maximally mixed encoded states for [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Wigner functions of the maximally mixed encoded states obtained from optimized encodings under a fixed stellar-rank constraint. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

Bosonic quantum error correction (QEC) relies on non-Gaussian bosonic encodings whose preparation cost is a central practical constraint. In this work, we use stellar rank as a resource measure to design and benchmark bosonic codes under finite non-Gaussian resources. For fixed cat and Gottesman--Kitaev--Preskill (GKP) code families, we show that finite stellar rank creates a trade-off among state approximability, energy, and logical protection under photon loss and photon-number dephasing, evaluated with optimal recovery. This trade-off implies that codewords with better ideal error-correction properties need not be optimal once finite-rank preparation constraints are imposed. Going beyond fixed-target codewords, we directly optimize bosonic encodings at fixed stellar rank, revealing noise-adapted code structures and concrete resource thresholds. Grid-like encodings emerge under photon loss, whereas approximately rotation-symmetric encodings arise under dephasing. In the optimized search, stellar rank k=2 suffices to surpass break-even for all dephasing strengths considered, while under photon loss the required rank increases with the loss rate. These results establish stellar rank as an operationally meaningful resource measure for bosonic QEC under practical state-preparation constraints.

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 / 7 minor

Summary. This manuscript investigates bosonic quantum error-correcting codes under a finite stellar-rank constraint, where stellar rank counts the minimal number of photon additions needed to prepare a non-Gaussian state. The authors first compute stellar fidelities for cat and GKP code families, deriving analytic expressions and showing that finite-rank approximations create a trade-off between state fidelity, energy, and logical performance under photon loss and dephasing. They then go beyond fixed code families by directly optimizing encodings at fixed stellar rank using the Petz-recovery fidelity as a surrogate objective, with final benchmarking via optimal-recovery SDP. The main findings are that stellar rank is an operationally meaningful resource measure: k=2 suffices to surpass break-even for all dephasing strengths considered, while photon loss requires increasing rank with the loss rate. The work bridges non-Gaussian resource theory and practical QEC performance, with analytic derivations, numerical optimization, and phase-space visualization of noise-adapted codes.

Significance. The paper makes a solid contribution by connecting the resource-theoretic notion of stellar rank to operational QEC performance, a question of genuine practical importance for bosonic code implementation in resource-constrained platforms. Strengths include: (1) analytic expressions for stellar fidelities of cat and GKP codes (Eqs. 25, 28, Appendices B, C), reducing the optimization to Gaussian parameters only; (2) a clear demonstration that high stellar fidelity does not guarantee good QEC performance, and the consequent reversal of code ordering under finite-rank constraints (Sec. III.B, Fig. 3); (3) direct variational optimization of encodings at fixed stellar rank, producing noise-adapted code structures (grid-like for loss, rotation-symmetric for dephasing) with concrete resource thresholds (Sec. IV, Fig. 4); (4) the important methodological choice to use the Petz fidelity only as a search surrogate and benchmark all reported results with the optimal-recovery SDP, which makes the sufficiency claims rigorous. The plateau analysis (Appendix C) provides useful physical intuition. The work is within the scope of a serious quantum information journal.

major comments (3)
  1. Sec. IV.A, Eqs. (33)–(36): The encoding optimization uses the Petz-recovery fidelity as a surrogate for the optimal-recovery channel fidelity. The authors acknowledge that joint SDP-variational optimization was computationally intractable. The Petz bound (Eq. 34) guarantees F̃_opt ≤ F_opt, so the surrogate is conservative for benchmarking. However, the surrogate guides the variational search itself: if the Petz fidelity is a poor proxy for the optimal-recovery fidelity in the regimes studied, the search may miss encodings where Petz fidelity is low but optimal-recovery fidelity is high. This particularly affects the necessity side of the threshold claims (e.g., 'k=2 suffices for dephasing break-even' is a valid sufficiency claim, but the possibility that k=1 could also suffice if a better search objective were used is not excluded). The authors should more explicitly discuss this asymmmh
  2. Sec. IV.B, Fig. 4: The claim that 'k=2 suffices to surpass break-even for all dephasing strengths considered' is a central result, but the figure only shows three dephasing strengths (κϕt = 10⁻³, 10⁻², 10⁻¹). The phrase 'all dephasing strengths considered' is technically accurate but could be misread as a broader statement. The authors should clarify the range of noise strengths for which this claim has been verified, and ideally comment on whether the trend is expected to hold for stronger dephasing.
  3. Sec. IV.B: The Fock-space truncation at d_phys = 4k+2 (for k≥2) is stated without a systematic convergence check. The tail penalty and the cutoff check (population in highest six Fock levels below 10⁻⁵) are mentioned, but no data is provided showing that increasing d_phys does not change the optimized fidelities. Since the Gaussian-transformed core states can have support beyond 4k+2, a brief convergence analysis (even for one representative case) would strengthen the reliability of the reported thresholds.
minor comments (7)
  1. Sec. III intro: The statement that 'optimized stellar fidelities may not appear strictly monotonic with stellar rank' due to numerical optimization is important but buried. A brief forward-reference to the non-monotonicities visible in Fig. 2 (GKP, Δ=0.2) would help the reader.
  2. Eq. (34): The bound is written as ½(1−F̃_opt) ≤ 1−F_opt ≤ 1−F̃_opt. A citation to the specific theorem or derivation would be helpful; Ref. [64] is cited but the bound's tightness conditions are not discussed.
  3. Sec. IV.B: The comparison with the binomial code (Eq. 39) at k=4 is useful, but the binomial code has stellar rank k=2 by construction (core states |0⟩+|4⟩ and |2⟩), not k=4. The text says 'with same stellar rank' which is ambiguous. Please clarify whether the comparison is at equal stellar rank or equal k.
  4. Fig. 4: The marker color encoding mean photon number is effective but the color scale is not quantified in the caption. A colorbar or explicit range would improve readability.
  5. Sec. II.C, Eq. (17): The notation N₀ for the normalization constant could be confused with the rotational symmetry order N used elsewhere. Consider using a different symbol.
  6. Appendix C: The plateau analysis is clear and well-motivated. The large-amplitude example (α=7, Fig. 5) is beyond the QEC parameter regime; a brief note that this is illustrative rather than operationally relevant would prevent confusion.
  7. The Acknowledgements section discloses use of GPT-5.5 for manuscript polishing. This is appropriate and transparent.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. All three major comments identify legitimate points regarding the Petz-surrogate asymmetry, the scope of the dephasing claim, and Fock-space convergence. We address each below and describe the corresponding revisions.

read point-by-point responses

  1. Referee: Sec. IV.A, Eqs. (33)–(36): The Petz-recovery fidelity is used as a surrogate during the variational search, but if the Petz fidelity is a poor proxy for the optimal-recovery fidelity, the search may miss encodings where Petz fidelity is low but optimal-recovery fidelity is high. This particularly affects the necessity side of threshold claims (e.g., k=1 could also suffice for dephasing break-even if a better search objective were used). The authors should more explicitly discuss this asymmetry.

    Authors: The referee correctly identifies an asymmetry in our methodology: the Petz fidelity guides the variational search, while the optimal-recovery SDP is used only for final benchmarking. We agree that this creates a potential gap on the necessity side of our threshold claims. Specifically, our statement that k=2 suffices for dephasing break-even is a valid sufficiency claim, but we cannot rule out the possibility that k=1 might also suffice if a different search objective (or direct SDP-variational optimization) were used and happened to find an encoding that the Petz surrogate undervalues. We note two mitigating factors. First, the Petz bound (Eq. 34) guarantees 1/2(1 - F_tilde_opt) <= 1 - F_opt <= 1 - F_tilde_opt, so the surrogate is not uncorrelated with the true objective; it provides a two-sided bound. Second, for k=1 we did perform the optimization and benchmarked the result with the SDP, finding that k=1 does not surpass break-even for the dephasing strengths considered. However, since the search itself was Petz-guided, we cannot certify that no k=1 encoding exists that would surpass break-even under a different search strategy. We will revise the manuscript to state this limitation explicitly: (i) our threshold claims are sufficiency claims, not necessity claims; (ii) the Petz-guided search could in principle miss encodings where the Petz fidelity is low but the optimal-recovery fidelity is high; (iii) ruling out k=1 for dephasing break-even would require either joint SDP-variational optimization or an exhaustive search over the k=1 parameter space, which is beyond our current computational capabilities. We believe this is an honest and important qualification that strengthens the paper. revision: yes

  2. Referee: Sec. IV.B, Fig. 4: The claim that 'k=2 suffices to surpass break-even for all dephasing strengths considered' could be misread as a broader statement. The authors should clarify the range of noise strengths and comment on whether the trend is expected to hold for stronger dephasing.

    Authors: We agree that the phrasing could be misread. The claim has been verified for three dephasing strengths: kappa_phi * t = 10^{-3}, 10^{-2}, and 10^{-1}. We will revise the text in Section IV.B and the corresponding figure caption to state the parameter range explicitly, replacing 'all dephasing strengths considered' with 'all three dephasing strengths considered (kappa_phi * t = 10^{-3}, 10^{-2}, 10^{-1})' at the first occurrence, and clarifying the range in the abstract as well. Regarding whether the trend is expected to hold for stronger dephasing: at very large dephasing rates, the break-even threshold itself becomes increasingly stringent (the unencoded qubit degrades rapidly), so the required code structure must provide stronger phase-space separation. Our results at kappa_phi * t = 10^{-1} already show that k=2 suffices at this relatively strong dephasing rate, with the optimized encoding developing approximate pi-rotation symmetry. For even stronger dephasing, the trend may or may not continue, as the code would need to balance larger phase-space separation against the finite stellar-rank budget. We will add a brief comment to this effect, noting that extrapolation to stronger dephasing is not straightforward and would require further investigation. revision: yes

  3. Referee: Sec. IV.B: The Fock-space truncation at d_phys = 4k+2 (for k>=2) is stated without a systematic convergence check. The tail penalty and cutoff check are mentioned, but no data is provided showing that increasing d_phys does not change the optimized fidelities. A brief convergence analysis would strengthen the reliability of the reported thresholds.

    Authors: This is a fair point. The cutoff d_phys = 4k+2 was chosen to provide sufficient Hilbert-space support for Gaussian-transformed core states while keeping the optimization tractable, and the tail penalty and cutoff check (population in the highest six Fock levels below 10^{-5}) serve as internal consistency indicators. However, we agree that a direct convergence check would strengthen the reliability of the reported thresholds. We will perform a convergence analysis for at least one representative case (e.g., k=4 under photon loss at kappa*t = 10^{-2} and under dephasing at kappa_phi*t = 10^{-2}) by re-running the optimization with enlarged cutoffs (e.g., d_phys = 4k+4 and d_phys = 4k+6) and verifying that the optimal-recovery fidelities do not change appreciably. The results of this check will be reported in a brief addition to Section IV.B or in an appendix. We expect convergence to hold based on the tail population criterion already being satisfied, but we agree that explicit verification is the right scientific practice. revision: yes

Circularity Check

0 steps flagged

No significant circularity; stellar rank and stellar fidelity are independently defined quantities from prior work, and all QEC performance claims are benchmarked with external optimal-recovery SDP.

full rationale

The paper's central claims about stellar rank as a resource measure for bosonic QEC are not circular. Stellar rank [Ref. 37, Chabaud et al.] and stellar fidelity [Refs. 38, 39] are independently defined mathematical quantities — the number of Q-function zeros and the maximal fidelity over bounded-rank states, respectively — with their own derivations that do not depend on the QEC results of the present paper. While Chabaud appears as a co-author on both the present paper and the cited foundational works [37, 38, 39], these cited results are independently verifiable mathematical definitions and theorems, not fitted parameters or ansätze tuned to the present paper's outputs. The QEC performance benchmarks use an external tool: optimal recovery via semidefinite programming from Ref. [42, Kosut & Lidar], which is an independent method with no author overlap. The Petz-recovery fidelity used as a surrogate optimization objective (Eq. 36) is a standard construction from Ref. [63, Petz] with a near-optimality bound from Ref. [64, Zheng et al.], neither of which has author overlap. The final reported performance numbers (Fig. 4) use the SDP-based optimal recovery, not the Petz surrogate, so the headline claims are verified by an independent benchmark. The one minor self-citation concern is that the stellar rank framework originates from overlapping authors, but this is a genuine mathematical framework with independent derivations, not a fitted input renamed as a prediction. No step in the derivation chain reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

No new physical entities are postulated. Stellar rank, stellar fidelity, cat codes, GKP codes, Petz recovery, and the noise channels are all pre-existing constructs from the cited literature. The paper's contribution is the framework combining them, not new ontology.

free parameters (5)
  • Cat code amplitude α = 1.537, 2.345, 2.94 (sweet-spot values from Ref. [60])
    Chosen as physically motivated target values; not fitted to QEC performance in this paper.
  • GKP damping parameter Δ = 0.3, 0.25, 0.2
    Controls finite-energy approximation of ideal GKP states; chosen to span a range of approximation difficulty.
  • Gaussian unitary parameters (ζ, β) for stellar fidelity = optimized per target state and rank k
    Optimized via BFGS over squeezing (r ∈ [0,3], θ ∈ [0,2π]) and displacement (|β| ∈ [0,3]) to maximize stellar fidelity.
  • Core-state coefficients {c_n} and Gaussian parameters for encoding optimization = optimized per noise channel and rate
    Variational parameters of the encoding isometry, optimized via Adam (lr=3×10^{-4}, 4000 steps, 60 restarts) to maximize Petz-recovery fidelity.
  • Photon-number penalty and tail penalty strengths = not specified
    Introduced ad hoc to suppress solutions near Fock cutoff; strengths not reported.
axioms (5)
  • domain assumption Stellar rank r*(ψ) equals the minimal number of photon additions required to generate |ψ⟩ given arbitrary Gaussian operations.
    Invoked throughout via Eq. (17); established in Ref. [37]. Used as the resource measure underlying the entire framework.
  • domain assumption Stellar fidelity f*_k(ψ) = max_{G∈G} ⟨ψ|G†Π_k G|ψ⟩ correctly quantifies the best achievable fidelity with a target state at stellar rank ≤ k.
    Invoked via Eq. (20); established in Ref. [38]. Used to benchmark approximation quality of cat and GKP codewords.
  • ad hoc to paper The Petz-recovery fidelity is a sufficiently accurate surrogate for the optimal-recovery fidelity to guide encoding optimization.
    Invoked in Sec. IV.A (Eq. 36) to make the variational optimization tractable. Justified by the near-optimality bound Eq. (34), but the tightness of this bound in the studied regime is not verified.
  • ad hoc to paper Fock-space truncation at d_phys = 4k+2 (for k≥2) captures the physically relevant support of Gaussian-transformed finite-rank core states.
    Invoked in Sec. IV.B. Chosen for computational tractability; verified only via a tail-population check (< 10^{-5} in highest six levels) for selected solutions.
  • standard math Photon loss and dephasing channels (Appendix A2-A3) adequately model the dominant noise in bosonic oscillator systems.
    Standard Lindblad models used throughout bosonic QEC literature; not specific to this paper.

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Reference graph

Works this paper leans on

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