{"id":"81beb9e5-c1d6-4d69-b800-474e2257740e","arxiv_id":"2505.08670","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rotation-symmetric bosonic codes keep error-corrected gate fidelities above break-even under random telegraph noise, while the BLP non-Markovianity measure grows linearly with code symmetry and is unbounded for non-Gaussian states.","lead":"Random telegraph noise from material defects causes non-Markovian dephasing in bosonic quantum memories. This paper analyzes rotation-symmetric bosonic error-correcting codes under this noise and finds they stay above break-even for most parameters, with a non-Markovianity measure that grows with code symmetry.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix E's unboundedness proof uses Ω_l = √(r²−l²) in the l>r regime, where it is imaginary; the geometric-series argument as written is invalid, though a sign-corrected branch likely still diverges.","rationale":"The reader's weakest assumption (noiseless auxiliary modes and gates) is a real limitation, but it is explicitly disclosed in Sec. V A and is standard in code-performance studies; it limits scope rather than invalidating the numerical method. The more acute issue is Appendix E, the only analytic support for the headline \"unbounded\" statement. Eq. (E1) defines Ω_l = √(r²−l²), but the limit l→∞ is taken in the regime l>r, where Ω_l is imaginary. Eq. (E2) then locates maxima at T_n^l = nπ/Ω_l, and Eq. (E3) gives a real geometric sum; with imaginary Ω_l both expressions are ill-defined. The correct branch is a = √(l²−r²), under which G(l,τ) = e^{−rτ}(cos aτ + (r/a) sin aτ), extrema occur at τ = nπ/a with magnitude e^{−r nπ/a}, and the sum equals 1/(e^{rπ/a}−1), which diverges as a→∞. So the result is almost certainly correct, but the proof as written is not. This is load-bearing because the abstract and Sec. IV B rely on it for \"unbounded\"; the linear-in-N scaling is numerical and separate. The noiseless-ancilla idealization, by contrast, is an acknowledged modeling choice rather than an internal inconsistency. Conditional acceptance with a corrected Appendix E is the right outcome, so the reader's verdict is unchanged.","tokens_in":34718,"tokens_out":15761,"duration_ms":157152,"concrete_test":"Independently recompute Eqs. (E2)–(E3) with Ω_l replaced by a = √(l²−r²) for l>r. Verify that G(l,τ) = e^{−rτ}(cos aτ + (r/a) sin aτ), that |G(l, nπ/a)| = e^{−r nπ/a}, and that the BLP revival sum equals 1/(e^{rπ/a}−1), which diverges as l→∞. Also run a numerical BLP evaluation for the pairs (|0⟩+|l⟩)/√2 and (|0⟩−|l⟩)/√2 for l = 1,...,20 at r = 0.1 and compare with the corrected closed form. If the corrected series converges or the numerics disagree, the unboundedness claim fails; if they agree, the flaw is a fixable branch error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Appendix E is the only analytic proof of the headline claim that the BLP measure is unbounded for non-Gaussian states. It considers the qubit subspace spanned by |0⟩ and |l⟩ and writes the dephasing factor as G(l,τ) = e^{−rτ}(cosh Ω_l τ + (r/Ω_l) sinh Ω_l τ) with Ω_l = √(r²−l²). But the limit l→∞ is taken in the non-Markovian regime r<1, so for every l>r the quantity Ω_l is imaginary. Consequently the \"revival times\" T_n^l = nπ/Ω_l in Eq. (E2) are imaginary, and the geometric-series result N_l = 1/(e^{πr/Ω_l}−1) is complex-valued; as written it does not establish divergence. The correct branch is a = √(l²−r²), under which G(l,τ) = e^{−rτ}(cos aτ + (r/a) sin aτ), the local maxima of |G| lie at τ = nπ/a and have height e^{−r nπ/a}, giving N_l = 1/(e^{rπ/a}−1), which diverges as a→∞. The sign flip between √(r²−d²) in Appendix A and √(d²−r²) in Appendix B confirms an unhandled branch choice. Thus the unboundedness claim is likely true, but the paper's analytic support is invalid as written, and this is the proof the abstract and Sec. IV B point to for \"unbounded.\"","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies dephasing of a bosonic mode by a classical random telegraph noise (RTN) fluctuator and by 1/f noise from an ensemble of fluctuators. It derives the RTN dephasing function and the resulting Fock-basis density-matrix evolution in the Markovian and non-Markovian limits, and it then quantifies non-Markovianity with the Breuer-Laine-Piilo (BLP) trace-distance measure. For Gaussian states the authors argue numerically that coherent states maximize the measure within the studied class, while for rotation-symmetric bosonic (RSB) code words (binomial and cat codes) the measure grows with the code symmetry order N, with an appendix claiming it is unbounded over pairs of Fock states {|0>,|l>}. The second half of the paper evaluates the average gate fidelity of a teleportation-based Knill error-correction circuit for RSB codes under simultaneous photon loss and RTN or 1/f dephasing, comparing canonical phase measurements, adaptive homodyne detection, and an optimal recovery map, and deriving a semi-analytic fidelity expression for a purely dephasing channel. The central conclusions are that RSB codes outperform a trivial Fock encoding and remain above the break-even threshold for most parameter ranges in both the Markovian and non-Markovian regimes, with oscillatory fidelity in the latter regime.","tokens_in":35037,"tokens_out":11647,"duration_ms":126812,"significance":"If the results hold, this is a useful first systematic analysis of non-Gaussian, non-Markovian dephasing for bosonic RSB codes. The paper provides analytically derived dephasing functions, clean two-limit reductions of the density-matrix dynamics, and a semi-analytic expression for the Knill-EC fidelity in Appendix H, all of which are cross-checked against numerics and against previously published Markovian/Gaussian limits. The connection between the BLP trace-distance measure and the QEC fidelity bound is a nice conceptual bridge. However, the headline claim that the BLP measure becomes unbounded for non-Gaussian states rests on an appendix containing a branch error; until that proof is repaired, that specific claim is not rigorously supported. The QEC conclusions are also computed under an explicit noiseless-ancilla idealization that is not carried through the abstract's robustness claim.","major_comments":[{"comment":"The proof that N_l diverges as l→∞ is invalid as written. In the non-Markovian regime r<1, the quantity Ω_l = sqrt(r²−l²) is imaginary for every l>r, so the revival times T_n^l = nπ/Ω_l and the geometric-series result N_l = 1/(e^{πr/Ω_l}−1) are complex-valued and do not establish a real unbounded divergence. The argument is repairable: with a = sqrt(l²−r²), one has G(l,τ)=e^{−rτ}(cos aτ + (r/a) sin aτ), whose local maxima occur near τ = nπ/a and give a real divergent series N_l ≈ 1/(e^{rπ/a}−1) as a→∞. The appendix must be corrected with this branch choice, since Section IV B and the abstract explicitly point to Appendix E for the claim that the non-Markovianity measure is unbounded. The sign inconsistency with the sqrt((m−n)²−r²) branch used in Appendices A and B should also be resolved explicitly.","section":"Appendix E, Eqs. (E1)-(E3)"},{"comment":"The above-break-even QEC conclusions are computed under the explicit assumption, stated in Section V A, that state preparation, CROT gates, auxiliary modes, and measurements are noiseless. The abstract and Section VI state the robustness conclusion without this caveat. Because beating break-even is the stated criterion for practical viability, the noiseless-ancilla idealization is load-bearing for the manuscript's central QEC message. The revision should either provide a sensitivity estimate for noisy auxiliary modes or clearly carry the caveat through the abstract and conclusions, noting that comparable noise on ancilla modes or gates could move the operating points below break-even, particularly in the r≈1 regime where the reported performance already dips below break-even.","section":"Section V A, Eq. (38), and Fig. 7"}],"minor_comments":[{"comment":"The main text states that the Wigner function evolution is plotted for a coherent state with amplitude α=4, while the Figure 3 caption says α=2. These should be harmonized.","section":"Figure 3 and Section III C"},{"comment":"The noise-strength parameter N_s^diamond is evaluated at different correction times τ in the non-Markovian panels, so the horizontal axis is not an independent monotone noise parameter in that regime. The caption or text should state this explicitly, as it affects the interpretation of the non-monotonic fidelity curves.","section":"Section V B and Fig. 2/7"},{"comment":"The notation |±_τ⟩_N is used before it is defined. Please define the evolved codewords at time τ explicitly at the point of use.","section":"Eq. (44) and Appendix H"},{"comment":"The statement that squeezing and thermal excitations do not enhance non-Markovianity is presented as a numerical observation over a restricted class of Gaussian states; the abstract and conclusion should phrase this as a numerical result for the studied family rather than a general proof.","section":"Section IV A 1 and Fig. 1(a)"},{"comment":"The branch choices for sqrt(a²−r²) and sqrt(r²−a²) in the 1/f dephasing integral should be stated explicitly, to avoid the same sign ambiguity that affects Appendix E.","section":"Appendix C"}],"recommendation":"major_revision","confidential_remarks":"I see no misconduct or citation concern. The main issue is technical and repairable: the Appendix E branch error undermines the unboundedness proof as written, and the noiseless-ancilla assumption should be made prominent. I do not recommend rejection, because the numerical QEC results and the semi-analytic fidelity derivation appear sound and the unboundedness claim is very likely correct after the simple branch repair."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth a serious referee. It is the first to study rotation-symmetric bosonic codes under random telegraph noise, and the main numerical findings look right: BLP non-Markovianity grows roughly linearly with rotational symmetry N, cat codewords double the coherent-state value, and the Knill-EC fidelities stay above break-even for most parameters, with oscillations in the non-Markovian regime that track the dephasing function's period. The Markovian limit reproduces earlier Gaussian dephasing results, which is a good check.\n\nWhat is genuinely new is the combination of an open-systems measure with a concrete QEC recovery circuit under non-Markovian dephasing. The analytic expression for the fidelity of a purely dephasing channel in Appendix H is useful, and the comparison with the exact recovery map and adaptive homodyne gives the numerics a solid anchor.\n\nThe soft spots are real but do not sink the paper. Appendix E contains a branch error: it defines Omega_l = sqrt(r^2 - l^2) and then takes l to infinity with r < 1, so Omega_l is imaginary; the geometric series in Eq. (E3) is complex-valued. The right branch (a = sqrt(l^2 - r^2)) gives N_l = 1/(e^{r*pi/a} - 1), which still diverges, so the unboundedness claim is probably true, but the proof as written is invalid. That needs fixing. Also, the exponential decay of non-Markovianity with fluctuator count for binomial codes is extrapolated from coherent-state results, not computed directly. And the QEC performance assumes ideal auxiliaries and gates; noisy auxiliary modes could degrade the non-Markovian advantage. No code or data artifacts are provided, which is a minor obstacle to reproducibility.\n\nOverall the central argument holds up: RTN dephasing is a realistic noise process for bosonic systems, and the linear scaling with code symmetry plus the time-dependent fidelity revivals are interesting, actionable results. This deserves a serious referee; the authors should be asked to correct Appendix E and either provide data or clarify extrapolations before publication.","headline":"First RTN study for RSB codes with plausible numerics and interesting symmetry scaling; Appendix E proof needs a branch fix but the result likely survives.","tokens_in":35600,"tokens_out":1641,"would_cite":true,"duration_ms":15992,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Rotation-symmetric bosonic codes maintain above-break-even fidelity under random telegraph noise, whose non-Markovianity grows linearly with code symmetry.","keywords":["random telegraph noise","non-Markovianity","bosonic quantum error correction","rotation-symmetric bosonic codes","continuous-variable quantum information","dephasing","1/f noise","teleportation-based error correction"],"falsifier":"Run a binomial code with $N=2$, $K=13$ through the teleportation-based recovery circuit under random telegraph noise at $r\\approx 0.1$ and photon loss $\\kappa/\\nu\\approx 0.01$, measuring average gate fidelity as a function of correction time $\\tau$. The paper predicts oscillations with period set by $\\Omega=2\\sqrt{N^2-r^2}$ and fidelities above break-even at revival times; a monotonic decay or uniformly sub-break-even fidelity would refute the central error-correction claim. Separately, computing the trace-distance non-Markovianity measure for binomial codewords with increasing $N$ should show linear growth; a saturating curve would refute the unboundedness claim.","tokens_in":34490,"feed_emoji":"⚛️","tokens_out":9887,"duration_ms":90932,"temperature":0.7,"pith_summary":"This paper examines what random telegraph noise, the fluctuating two-level defects that plague superconducting cavities and qubits, does to a bosonic mode and to the rotation-symmetric bosonic (RSB) codes designed to protect it. The authors show that the noise is non-Markovian when the fluctuator switches slowly compared to its coupling strength, and that a standard trace-distance measure of non-Markovianity grows linearly with the code's rotational symmetry order $N$ and can become unbounded for non-Gaussian code states. They then run RSB codes through a teleportation-based error-correction circuit under simultaneous photon loss and random telegraph noise dephasing. The encoded qubits stay above the break-even fidelity threshold for most parameters, and in the non-Markovian regime the fidelity oscillates with the timing of the correction, so choosing the right correction moment can substantially improve performance. If correct, realistic defect noise does not defeat these codes, and error-correction timing can exploit noise revivals.","feed_headline":"Bosonic codes beat break-even under random telegraph noise","feed_subtitle":"Correction timing matters: fidelity oscillates with the noise, so the right moment restores the encoded qubit.","key_machinery":"The engine of the analysis is the random-telegraph-noise dephasing function $G(r,\\tau)=e^{-r\\tau}(\\cosh(\\Omega\\tau)+(r/\\Omega)\\sinh(\\Omega\\tau))$ with $\\Omega=\\sqrt{r^2-(m-n)^2}$, which gives the decay of each Fock-basis coherence $|m\\rangle\\langle n|$ under a single bistable fluctuator; whether $\\Omega$ is real or imaginary decides whether the dynamics show Markovian decay or non-Markovian revivals. The error-correction results ride on the teleportation-based recovery circuit built from controlled-rotation gates and phase measurements, whose full recovery map the paper derives. For a purely dephasing channel the authors obtain a semi-analytical expression for the average gate fidelity showing that the correction fidelity is controlled by the modulus of the dephasing function and oscillates with frequency $\\Omega=2\\sqrt{N^2-r^2}$, which is why the code symmetry $N$ sets the period of the revival structure.","core_discovery":"The central claim has two parts. First, for a bosonic mode dephased by random telegraph noise, the trace-distance non-Markovianity measure is finite and is maximized among Gaussian states by pairs of coherent states; squeezing and thermal photons do not increase it. For non-Gaussian states, in particular the codewords of RSB codes, the measure grows approximately linearly with the code symmetry $N$ and is formally unbounded in the limit of large Fock-state spacing. Second, binomial and cat RSB codes subjected to simultaneous photon loss and random telegraph noise dephasing, with recovery by a teleportation-based error-correction circuit, maintain average gate fidelities above the break-even point for most values of the ratio $r=\\xi/\\nu$ of switching rate to coupling strength, including in the non-Markovian regime where the fidelity oscillates with the correction time $\\tau$. The oscillation frequency increases linearly with $N$, and increasing the binomial truncation parameter $K$ or the cat amplitude $\\alpha$ broadens the time windows of good performance, at the cost of increased photon loss.","pith_inferences":["Editorial inference: If the revival windows survive realistic circuit noise, error-correction scheduling could treat non-Markovianity as a resource: the same code and noise parameters that hurt fidelity at one time restore it at another, so waiting for a revival peak would outperform immediate correction.","Editorial inference: The unboundedness of the trace-distance measure over Fock-spaced non-Gaussian states implies that quantitative non-Markovianity comparisons across bosonic states are only meaningful within a fixed state family and photon-number scale; otherwise high-symmetry codes will look 'more non-Markovian' by construction.","Editorial inference: Since $1/f$ noise with roughly ten or more independent fluctuators behaves like Gaussian dephasing, devices with many defects can reuse existing Gaussian-dephasing code analyses, while few-defect devices are precisely where oscillatory correction-time behavior should be observable.","Editorial inference: A testable extension is to include measurement inefficiency and noisy auxiliary modes in the recovery circuit; the paper's adaptive-homodyne comparison suggests the measurement choice matters little, but realistic auxiliary loss could degrade the revival peaks and narrow the favorable timing windows."],"forward_implications":["Binomial and cat codes with rotational symmetry $N>1$ outperform the trivial $N=1$ Fock encoding under random telegraph noise plus loss in both Markovian and non-Markovian regimes.","In the non-Markovian regime the average gate fidelity of the teleportation-based error-correction circuit oscillates with the correction time $\\tau$, with an oscillation frequency that grows linearly with $N$, so higher-symmetry codes have more frequent favorable correction windows.","Tuning the binomial truncation $K$ or cat amplitude $\\alpha$ broadens the fidelity peaks, lengthening the time intervals in which error correction stays above break-even, at the cost of higher photon loss.","For $1/f$ noise from many fluctuators ($N_f \\gtrsim 10$), the dynamics become effectively Markovian and the codes perform like they do under Gaussian dephasing, while for few fluctuators the non-Markovian oscillatory signature persists.","The trace-distance non-Markovianity measure grows approximately linearly with the code symmetry order $N$ and is unbounded over a family of non-Gaussian states, whereas squeezing and thermal noise do not increase it for Gaussian states."],"supporting_citations":[{"why":"Introduces rotation-symmetric bosonic codes as the code family whose non-Markovianity and error-correction performance this paper studies.","marker":"[24]"},{"why":"Establishes the teleportation-based error-correction circuit and its fidelity under loss and Gaussian dephasing, the baseline this paper extends to random telegraph noise.","marker":"[25]"},{"why":"Defines the rotation-symmetric code construction and the recovery circuit whose recovery map the paper re-derives.","marker":"[27]"},{"why":"Provides the statistical model of a bistable fluctuator that yields the random-telegraph-noise dephasing dynamics.","marker":"[39]"},{"why":"Derives the random-telegraph-noise phase distribution and telegraph equation used for the dephasing function.","marker":"[40]"},{"why":"Supplies the average gate fidelity formula used to score the error-corrected output.","marker":"[49]"},{"why":"Provides the colored-noise non-Markovianity framework and the multiple-fluctuator model used for $1/f$ noise.","marker":"[57]"},{"why":"Defines the trace-distance measure of non-Markovianity that the paper evaluates for Gaussian and non-Gaussian states.","marker":"[64]"},{"why":"Supplies the measurement-outcome versus trace-distance inequality that yields the paper's analytic bound on correction fidelity.","marker":"[79]"}],"fun_headline_variants":["RSB codes beat break-even with timed correction under RTN","Under telegraph noise, bosonic codes need right timing","Non-Markovian RTN: bosonic codes stay above break-even","Rotation-symmetric codes: correction timing crucial under RTN","Telegraph noise: RSB codes' fidelity oscillates, stays effective"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The results assume that all error-correction circuit elements except the data mode are noiseless; the paper states this idealization explicitly. If auxiliary modes, controlled-rotation gates, or measurements suffer comparable loss or dephasing, the above-break-even fidelities could degrade, particularly in the non-Markovian regime where the advantage is time-sensitive.","fun_headline_variants_meta":{"raw":{"variants":["RSB codes beat break-even with timed correction under RTN","Under telegraph noise, bosonic codes need right timing","Non-Markovian RTN: bosonic codes stay above break-even","Rotation-symmetric codes: correction timing crucial under RTN","Telegraph noise: RSB codes' fidelity oscillates, stays effective"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000278,"raw_usage":{"total_tokens":1722,"prompt_tokens":1085,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":701,"completion_tokens_details":{"reasoning_tokens":549}},"tokens_in":701,"tokens_out":637,"duration_ms":6355,"temperature":1.0,"reasoning_tokens":549,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:50:09.811920+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a binomial code with $N=2$, $K=13$ through the teleportation-based recovery circuit under random telegraph noise at $r\\approx 0.1$ and photon loss $\\kappa/\\nu\\approx 0.01$, measuring average gate fidelity as a function of correction time $\\tau$. The paper predicts oscillations with period set by $\\Omega=2\\sqrt{N^2-r^2}$ and fidelities above break-even at revival times; a monotonic decay or uniformly sub-break-even fidelity would refute the central error-correction claim. Separately, computing the trace-distance non-Markovianity measure for binomial codewords with increasing $N$ should show linear growth; a saturating curve would refute the unboundedness claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces rotation-symmetric bosonic codes as the code family whose non-Markovianity and error-correction performance this paper studies."},{"cited_title":"Astafiev, Y","cited_arxiv_id":null,"evidence_quote":"Provides the statistical model of a bistable fluctuator that yields the random-telegraph-noise dephasing dynamics."},{"cited_title":"Torre, W","cited_arxiv_id":null,"evidence_quote":"Supplies the average gate fidelity formula used to score the error-corrected output."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the colored-noise non-Markovianity framework and the multiple-fluctuator model used for $1/f$ noise."},{"cited_title":"Serafini, Quantum continuous variables (CRC Press, Lon- don, England, 2023)","cited_arxiv_id":null,"evidence_quote":"Defines the trace-distance measure of non-Markovianity that the paper evaluates for Gaussian and non-Gaussian states."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the measurement-outcome versus trace-distance inequality that yields the paper's analytic bound on correction fidelity."}],"review_version":1}