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Performance of rotation-symmetric bosonic codes in the presence of random telegraph noise

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Pith's one-line read Rotation-symmetric bosonic codes maintain above-break-even fidelity under random telegraph noise, whose non-Markovianity grows linearly with code symmetry.

desk verdict 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. read the letter →

arxiv 2505.08670 v2 pith:LXMTPPXC submitted 2025-05-13 quant-ph

Adithi Udupa , Timo Hillmann , Rabsan Galib Ahmed , Andrea Smirne , Giulia Ferrini This is my paper · ORCID
classification quant-ph
keywords randomtelegraphnoisenon-Markovianitybosonicquantumerrorcorrectionrotation-symmetriccodescontinuous-variableinformationdephasing1/fteleportation-based
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Appendix E, Eqs. (E1)-(E3)] 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.
  2. [Section V A, Eq. (38), and Fig. 7] 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.
minor comments (5)
  1. [Figure 3 and Section III C] 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.
  2. [Section V B and Fig. 2/7] 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.
  3. [Eq. (44) and Appendix H] The notation |±_τ⟩_N is used before it is defined. Please define the evolved codewords at time τ explicitly at the point of use.
  4. [Section IV A 1 and Fig. 1(a)] 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.
  5. [Appendix C] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RTN dephasing function drives all derived results, and self-citations are only baseline checks.

full rationale

No circularity found. The dephasing function in Eq. (7) is derived in Appendix A from the RTN autocorrelation or the telegraph equation and is then used as the single input for both the BLP analysis (Sec. IV) and the fidelity formula (Appendix H); the observed linear-in-N growth of N*_BLP for RSB codewords is an analytic consequence of the codewords' Fock support at multiples of N entering G(kN,lN,tau), not a quantity fitted to that growth. The QEC results are numerical circuit simulations cross-checked against an independently derived analytic expression (Fig. 11), and the Markovian-limit comparison with Refs. [25,27] is a validation baseline: Ref. [25] is a published PRX Quantum result, and neither it nor any other self-citation supplies a premise needed for the new RTN claims. No parameter is fitted to the target non-Markovianity or fidelity. Appendix E's use of Omega_l = sqrt(r^2 - l^2) for l > r is a branch-handling or correctness concern, not a circularity, because the claimed unboundedness would not reduce to an input; it is therefore outside the circularity score.

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

The central claims rest on standard open-quantum-system models (classical RTN, 1/f noise from independent fluctuators) and on idealized EC circuit assumptions. No free parameters are fitted to data; all parameters are physical inputs or scanned design parameters. No new physical entities are introduced. The main fragilities are the idealized noiseless EC circuit and the numerical rather than analytic Gaussian-state optimization.

assumptions (5)
  • domain assumption The fluctuator is a classical bistable switch with statistics <c(tau1)c(tau2)> = e^{-2r|tau1-tau2|} and the cluster factorization <c(tau1)c(tau2)...c(tau_n)> = <c(tau1)c(tau2)><c(tau3)...c(tau_n)> for ordered times.
    Used in Appendix A to derive the RTN dephasing function Eq. (7). This is the standard classical RTN model for two-level fluctuators; the final dephasing function is cross-checked with the telegraph-equation method in the same appendix.
  • domain assumption The bosonic mode couples to the fluctuator only through dephasing, H = epsilon a-dagger a + nu c(t) a-dagger a, with no dissipative or longitudinal coupling.
    Sec. III A, Eq. (1). This restricts the noise to a non-dissipative dephasing channel; the paper does not treat amplitude damping from the fluctuator itself, only the separate photon-loss channel.
  • ad hoc to paper The Knill-EC circuit elements (state preparation, CROT gates, auxiliary modes, measurements) are noiseless; only the data mode undergoes loss and RTN dephasing.
    Sec. V A: 'We assume idealized conditions for state preparation, gate operations, and auxiliary qubits [25, 27] to isolate the effects of these dephasing channels.' This is load-bearing for the QEC performance claim; if auxiliaries are also noisy, the reported fidelities could change.
  • domain assumption 1/f noise is modeled by Nf independent fluctuators with switching rates drawn from P(xi) proportional to 1/xi over [xi_min, xi_max] and uniform coupling nu/sqrt(Nf).
    Sec. III B, Eqs. (10)-(11). This is a standard model for 1/f noise from TLS ensembles; the chosen range [10^{-4}, 10^{4}] for r is said to be experimentally relevant [29,32].
  • domain assumption For the Gaussian-state optimization, the global maximum of N_BLP over all Gaussian states is attained by coherent states with opposite phases and equal amplitudes, with no squeezing or thermal excitation.
    Sec. IV A and Appendix D. This is established numerically over the ten-parameter family Eq. (21), not proven analytically; the paper states the result for 'the class of Gaussian states represented by Eq.(21)'.

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Pith. "Pith review of Performance of rotation-symmetric bosonic codes in the presence of random telegraph noise." pith.science (2026). https://pith.science/paper/LXMTPPXC

@misc{pith2026250508670,
  author       = {Pith},
  title        = {Pith review of: Performance of rotation-symmetric bosonic codes in the presence of random telegraph noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXMTPPXC}},
  note         = {Machine review of arXiv:2505.08670}
}
abstract

Decoherence in quantum devices, such as qubits and resonators, is often caused by bistable fluctuators modeled as random telegraph noise (RTN), leading to significant dephasing. We analyze the impact of individual and multiple fluctuators on a bosonic mode in continuous variable systems, identifying non-Markovian behavior governed by two timescales: the fluctuator switching rate ($\xi$) and coupling strength ($\nu$). Using the Breuer-Piilo-Laine (BLP) measure, we show that for Gaussian states, squeezing and thermal fluctuations do not enhance non-Markovianity. In contrast, for non-Gaussian states, the measure becomes unbounded. For rotation-symmetric bosonic (RSB) codes, known for their error correction advantages, non-Markovianity grows linearly with code symmetry. We evaluate the performance of RSB codes under simultaneous loss and RTN dephasing. For a teleportation-based Knill error-correction circuit, the codes perform robustly in the Markovian limit. In the non-Markovian regime, the performance depends on the time the error correction is performed for a given codeword. The average gate fidelity of the error-corrected state in this case exhibits oscillations as a function of time due to the oscillatory nature of the dephasing function of the RTN noise; however, for most of the parameter ranges, the values stay above the break-even point. Extending to multiple fluctuators that produce $1/f$ noise, we observe that non-Markovianity decays with increasing fluctuator count, while the performance of RSB codes remains effective with increasing number of fluctuators.

Figures

Figures reproduced from arXiv: 2505.08670 by the authors.

Figure 1
Figure 1. FIG. 1. (a) For Gaussian states, we show that for a given pair of coherent states with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Upper panel: Average gate fidelity [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Evolution of the Wigner function of a coherent state with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of the trace distance between pairs of quantum [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Non-Markovianity measure [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Teleportation-based Knill-EC circuit for bosonic codes. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Infidelity of the error-corrected state recovered using the Knill-EC circuit, shown for various values of the parameter [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Infidelity from the Knill EC circuit for a binomial code with [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The optimisation procedure consists of maximising over the following ten parameters: [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (a) The Wigner negativity exhibits oscillations with [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The analytical expression for the average gate fidelity of the [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Evolution of the trace distance between pairs of quantum states: (a) coherent states with [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]

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