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arxiv: 2607.00786 · v1 · pith:EYXPLZB2new · submitted 2026-07-01 · 🪐 quant-ph

Bias-Preserving Gates and Quantum Error Correction With Dual-Rail Cat Codes

Debjyoti Biswas , Nikhil Sharma , Alberto Salvador , Rui Wang , Mats Granath , Adithi Udupa , Giulia Ferrini This is my paper

Pith reviewed 2026-07-02 11:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords dual-rail cat codebosonic encodingbias-preserving gatesquantum error correctionerasure errorsphoton losscat codesbeam-splitter interactions
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The pith

The dual-rail cat code combines cat and dual-rail encodings to support bias-preserving logical gates using only beam-splitter interactions while enabling deterministic photon-loss correction.

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

The paper introduces the dual-rail cat code by nesting a cat code inside a dual-rail structure. This aims to merge the bias-preserving logical gates of cat codes with the efficient single-photon-loss detection of dual-rail codes. Universal gates are constructed with beam-splitter interactions that keep the noise biased toward erasures. A deterministic correction scheme for photon loss is proposed by adding an outer repetition code. The construction avoids relative geometric phases during gates and permits simultaneous syndrome extraction without halting stabilization.

Core claim

The dual-rail cat code is formed by concatenating an inner cat code with an outer dual-rail structure. This yields a bosonic encoding that inherits bias-tailored error correction from the cat code and converts photon-loss errors into detectable erasures via the dual-rail layer. A universal gate set is built from beam-splitter interactions alone, with every logical operation preserving the erasure-biased noise. The code supports deterministic erasure detection and correction, produces no relative geometric phases in gates, and allows syndrome extraction to run simultaneously with stabilization.

What carries the argument

The dual-rail cat code (DRCC), a concatenation of an inner cat code and an outer dual-rail encoding, which transmits bias preservation and erasure detection through the logical operations.

If this is right

  • Universal logical gates are realized using only beam-splitter interactions.
  • All logical operations preserve the erasure-biased noise structure.
  • Deterministic single-photon-loss correction is achieved by further concatenation with an outer repetition code.
  • No relative geometric phases appear during the gate operations.
  • Syndrome extraction proceeds simultaneously without interrupting stabilization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction could be paired with other outer codes to reach higher logical distances while retaining the bias.
  • Bias preservation may reduce the number of physical resources needed for fault tolerance on hardware that already favors erasure errors.
  • Direct tests of the beam-splitter gates in a photonic or circuit-QED platform could check whether new error channels remain suppressed.

Load-bearing premise

That concatenating the inner cat code with the outer dual-rail structure inherits the advantages of both without introducing new dominant error channels or breaking the bias-preserving property of the gates.

What would settle it

An observation of a relative geometric phase arising during beam-splitter gate operations on dual-rail cat code states would show the phase-absence claim is incorrect.

Figures

Figures reproduced from arXiv: 2607.00786 by Adithi Udupa, Alberto Salvador, Debjyoti Biswas, Giulia Ferrini, Mats Granath, Nikhil Sharma, Rui Wang.

Figure 1
Figure 1. Figure 1: FIG. 1: Wigner function of the even and odd cat states [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: For DRCC-1 (upper panel), any single-photon [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Schematics for syndrome measurement and [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Syndrome measurement scheme with the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Upper bound of the logical error for two- and three-qubit repetition code with inner DRCC-1, as a function [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Performance of the DRCC-1 under the near-optimal Petz recovery channel against photon-loss noise with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Performance of the DRCC-2 under photon-loss noise. Here, we assume identical loss rates for both modes. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Dependence of the performance of the DRCC-2 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: A schematic of the QEC protocol using the [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: QEC with a standard dual-rail code [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
read the original abstract

Scalable fault-tolerant quantum computation requires quantum error-correcting codes that simultaneously support universal logical operations, suppress hardware-specific noise, and enable efficient handling of photon-loss errors. Bosonic encodings such as the dual-rail and cat codes each offer attractive features but also exhibit important limitations when used in isolation. The dual-rail code enables efficient single-photon-loss detection by converting leakage out of the computational subspace induced by photon-loss errors into an erasure error. In contrast, the cat code provides a resource-efficient, bias-tailored error-correction scheme with bias-preserving logical gate operations. Here, we introduce the dual-rail cat code (DRCC), a concatenated bosonic encoding that combines an inner cat code with an outer dual-rail structure, thereby inheriting and enhancing the advantages of both constituent codes. We analyse the error-correction properties of the DRCC and propose a deterministic single-photon-loss correction protocol by concatenating it with an outer repetition code. Exploiting the code's intrinsic noise bias, we construct a universal set of logical gates using only beam-splitter interactions and demonstrate that all logical operations preserve the erasure-biased noise structure. The DRCC offers several distinctive advantages, including the absence of relative geometric phases during gate operations, deterministic erasure detection and correction, and simultaneous syndrome extraction without interrupting stabilisation. These features make the DRCC a promising bosonic code for hardware-efficient, bias-preserving, and erasure-resilient fault-tolerant quantum computation.

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

2 major / 1 minor

Summary. The manuscript introduces the dual-rail cat code (DRCC), a concatenated bosonic encoding that places an inner cat code inside an outer dual-rail structure. It claims to analyze the resulting error-correction properties, propose a deterministic single-photon-loss correction protocol obtained by further concatenation with an outer repetition code, construct a universal gate set realized solely with beam-splitter interactions that preserves the erasure-biased noise structure, and identify distinctive operational advantages including the absence of relative geometric phases, deterministic erasure detection, and simultaneous syndrome extraction without interrupting stabilization.

Significance. If the explicit constructions and error-channel analysis hold, the DRCC would constitute a concrete bosonic-code architecture that simultaneously achieves bias preservation, hardware-efficient photon-loss handling, and erasure resilience. Such a combination addresses two central obstacles to scalable bosonic fault tolerance and could therefore influence subsequent experimental and theoretical work on concatenated bosonic encodings.

major comments (2)
  1. [Abstract] Abstract and main text: the central claim that concatenation 'inherits and enhances the advantages of both constituent codes' without introducing new dominant error channels is stated without any derivation of the effective logical error channel, any explicit beam-splitter gate Hamiltonians, or any threshold calculation; the load-bearing assertion that bias preservation survives concatenation therefore cannot be verified from the supplied material.
  2. [Abstract] The deterministic single-photon-loss correction protocol obtained by concatenating the DRCC with an outer repetition code is described only at the level of the abstract; no syndrome-extraction circuit, no repetition-code stabilizer measurements, and no analysis of the resulting logical error rate appear, leaving the claim of 'deterministic erasure detection and correction' unsubstantiated.
minor comments (1)
  1. The acronym DRCC is introduced in the abstract but the title uses the plural 'Dual-Rail Cat Codes'; consistent nomenclature should be adopted throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting areas where additional detail would strengthen the presentation. We address each major comment below and will revise the manuscript accordingly to make the supporting derivations and circuits explicit.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main text: the central claim that concatenation 'inherits and enhances the advantages of both constituent codes' without introducing new dominant error channels is stated without any derivation of the effective logical error channel, any explicit beam-splitter gate Hamiltonians, or any threshold calculation; the load-bearing assertion that bias preservation survives concatenation therefore cannot be verified from the supplied material.

    Authors: We agree that the abstract and introductory claims would benefit from explicit supporting derivations. The manuscript contains an analysis of the DRCC error-correction properties and constructs the universal gate set via beam-splitter interactions while showing preservation of the erasure-biased structure. To make bias preservation after concatenation fully verifiable, we will add a new subsection deriving the effective logical error channel and providing the explicit beam-splitter Hamiltonians. No numerical threshold calculation is performed in the present work, which focuses on code construction and gate design rather than full fault-tolerance thresholds; we will state this scope limitation clearly in the revision. revision: yes

  2. Referee: [Abstract] The deterministic single-photon-loss correction protocol obtained by concatenating the DRCC with an outer repetition code is described only at the level of the abstract; no syndrome-extraction circuit, no repetition-code stabilizer measurements, and no analysis of the resulting logical error rate appear, leaving the claim of 'deterministic erasure detection and correction' unsubstantiated.

    Authors: We acknowledge that the protocol is introduced at a high level. The manuscript proposes the outer repetition-code concatenation for deterministic correction, but the explicit circuit, stabilizer measurements, and logical-error-rate analysis are not yet included. We will add a dedicated section containing the syndrome-extraction circuit, the repetition-code stabilizers, and a brief analysis of the resulting logical error rate to substantiate the deterministic erasure detection and correction claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on abstract inspection

full rationale

The abstract presents a proposal for the dual-rail cat code (DRCC) as a concatenated bosonic encoding, followed by direct analysis of its error-correction properties, a deterministic photon-loss correction protocol, and construction of bias-preserving logical gates via beam-splitter interactions. No equations, fitted parameters, or self-citations are referenced. Claims rest on explicit construction and analysis of the new code rather than reduction to prior fitted quantities or self-referential definitions. The central assertions (inheritance of advantages without new dominant channels, bias preservation) are presented as outcomes of the proposed structure itself, with no load-bearing steps that collapse by construction to inputs. This is the expected honest non-finding for a proposal paper whose full derivations have not yet been shown to contain the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review performed on abstract only; full manuscript details on assumptions and parameters unavailable. The DRCC itself is treated as a newly postulated encoding.

invented entities (1)
  • dual-rail cat code (DRCC) no independent evidence
    purpose: Concatenated bosonic encoding combining inner cat code with outer dual-rail structure for bias-preserving operations and erasure correction
    Newly introduced construction whose properties are asserted in the abstract.

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Reference graph

Works this paper leans on

81 extracted references · 31 canonical work pages · 3 internal anchors

  1. [1]

    Joint parity check For the syndrome detection of leakage errors with the standard dual-rail code, Ref.[20] uses a joint parity check scheme with an auxiliary (transmon) qubit and the fol- lowing unitary operation: ˆUPC = ˆI⊗ |g⟩⟨g|+e −jπ(ˆn1+ˆn2) ⊗ |e⟩⟨e|.(77) However, using this unitary would require stopping the stabilisation, even if we consider the di...

  2. [2]

    11 Following Ref

    Computational basis measurement To complete the discussion on the universal control of DRCC-1 qubits, we discuss measurements in the compu- tational (Z) basis, denoted byM Z. 11 Following Ref. [48], we first consider the resonator- assisted interaction Hamiltonian ˆH r MZ = J 2 (ˆn1 −ˆn2)(ˆar + ˆa† r),(81) where ˆar is the annihilation operator of the aux...

  3. [3]

    Hence, the code in Eq.(84) approximately satisfies the KL condition for all first-order photon-loss errors

    Similarly, one can verify that ˆ¯P2ˆa† i ˆaj ˆ¯P2 = 0 for all i̸=j. Hence, the code in Eq.(84) approximately satisfies the KL condition for all first-order photon-loss errors. Consequently, the DRCC-1 concatenated with the two- qubit repetition code corrects all first-order photon-loss errors in the largeαregime. We now focus on a simplified QEC protocol ...

  4. [4]

    The KL conditions can be studied using the same first-order photon-loss and first-order dephasing opera- tors considered for the DRCC-1. With the first-order photon loss and dephasing operators and for the BS an- gleδ=ϕ=π/4, we obtain the following relations ˆP4ˆa† 1ˆa1 ˆP4 = ˆP4( ˆU † BSˆa† 1ˆa1 ˆUBS) ˆP4 = 1 2 ˆP4(ˆa† 1 + ˆa† 2)(ˆa1 + ˆa2) ˆP4 = 1 2(⟨ˆn...

  5. [5]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A64, 012310 (2001)

    2001

  6. [6]

    A. L. Grimsmo and S. Puri, Quantum Error Correction with the Gottesman-Kitaev-Preskill Code, PRX Quan- 21 tum2, 020101 (2021), arXiv:2106.12989 [quant-ph]

    work page arXiv 2021

  7. [7]

    S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frattini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, S. T. Flammia, and S. M. Girvin, Bias- preserving gates with stabilized cat qubits, Science Ad- vances6, 10.1126/sciadv.aay5901 (2020)

    work page doi:10.1126/sciadv.aay5901 2020

  8. [8]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ul- trahigh Error Threshold for Surface Codes with Biased Noise, Physical Review Letters120, 050505 (2018)

    2018

  9. [9]

    J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The XZZX surface code, Na- ture Communications12, 2172 (2021)

    2021

  10. [10]

    Srivastava, A

    B. Srivastava, A. Frisk Kockum, and M. Granath, The XYZ2 hexagonal stabilizer code, Quantum6, 698 (2022)

    2022

  11. [11]

    A. Dua, A. Kubica, L. Jiang, S. T. Flammia, and M. J. Gullans, Clifford-deformed surface codes, PRX Quantum 5, 010347 (2024)

    2024

  12. [12]

    Denys and A

    A. Denys and A. Leverrier, Quantum error-correcting codes with a covariant encoding, Phys. Rev. Lett.133, 240603 (2024)

    2024

  13. [13]

    Hillmann and F

    T. Hillmann and F. Quijandr´ ıa, Quantum error cor- rection with dissipatively stabilized squeezed-cat qubits, Physical Review A107, 032423 (2023)

    2023

  14. [14]

    Q. Xu, G. Zheng, Y.-X. Wang, P. Zoller, A. A. Clerk, and L. Jiang, Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits, npj Quantum Information9, 78 (2023)

    2023

  15. [15]

    Mirrahimi, Z

    M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, Dynamically protected cat-qubits: a new paradigm for universal quan- tum computation, New Journal of Physics16, 045014 (2014)

    2014

  16. [16]

    Guillaud and M

    J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Phys. Rev. X9, 041053 (2019)

    2019

  17. [17]

    Puttermanet al., Hardware-efficient quantum error correction via concatenated bosonic qubits, Nature638, 927–934 (2025)

    H. Puttermanet al., Hardware-efficient quantum error correction via concatenated bosonic qubits, Nature638, 927–934 (2025)

    2025

  18. [18]

    Kruckenhauser, M

    A. Kruckenhauser, M. Yuan, H. Zheng, M. Mamaev, P. Zeng, X. Mao, Q. Xu, T. V. Zache, L. Jiang, R. van Bi- jnen, and P. Zoller, Dark spin-cat states as biased qubits, Physical Review Letters135, 10.1103/w9zh-jwsx (2025)

    work page doi:10.1103/w9zh-jwsx 2025

  19. [19]

    I. L. Chuang and Y. Yamamoto, Simple quantum com- puter, Physical Review A52, 3489–3496 (1995)

    1995

  20. [20]

    I. L. Chuang and Y. Yamamoto, Quantum Bit Regen- eration, Phys. Rev. Lett.76, 4281 (1996), arXiv:quant- ph/9604031 [quant-ph]

    work page arXiv 1996

  21. [21]

    Knill, R

    E. Knill, R. Laflamme, and G. J. Milburn, A scheme for efficient quantum computation with linear optics, Nature 409, 46 (2001)

    2001

  22. [22]

    Knill, Bounds on the probability of success of postse- lected nonlinear sign shifts implemented with linear op- tics, Physical Review A68, 064303 (2003)

    E. Knill, Bounds on the probability of success of postse- lected nonlinear sign shifts implemented with linear op- tics, Physical Review A68, 064303 (2003)

    2003

  23. [23]

    Kubica, A

    A. Kubica, A. Haim, Y. Vaknin, H. Levine, F. Brand˜ ao, and A. Retzker, Erasure qubits: Overcoming the t 1 limit in superconducting circuits, Physical Review X13, 041022 (2023)

    2023

  24. [24]

    J. D. Teoh, P. Winkel, H. K. Babla, B. J. Chap- man, J. Claes, S. J. de Graaf, J. W. O. Garmon, W. D. Kalfus, Y. Lu, A. Maiti, K. Sahay, N. Thakur, T. Tsunoda, S. H. Xue, L. Frunzio, S. M. Girvin, S. Puri, and R. J. Schoelkopf, Dual-rail encoding with superconducting cavities, Proceedings of the Na- tional Academy of Sciences120, e2221736120 (2023), https...

    work page doi:10.1073/pnas.2221736120 2023

  25. [25]

    S. Gu, A. Retzker, and A. Kubica, Fault-tolerant quan- tum architectures based on erasure qubits, Phys. Rev. Res.7, 013249 (2025)

    2025

  26. [26]

    Chang, S

    K. Chang, S. Singh, J. Claes, K. Sahay, J. Teoh, and S. Puri, Surface code with imperfect erasure checks, PRX Quantum6, 040355 (2025)

    2025

  27. [27]

    Bias-preserving and error- detectable entangling operations in a superconduct- ing dual-rail system

    N. Mehta, J. D. Teoh, T. Noh, A. Agrawal, A. Ander- son, B. Birdsall, A. Brahmbhatt, W. Byrd, M. Cacioppo, A. Cabrera,et al., Bias-preserving and error-detectable entangling operations in a superconducting dual-rail sys- tem, arXiv preprint arXiv:2503.10935 (2025)

    work page arXiv 2025

  28. [28]

    R. G. Ahmed, A. Udupa, and G. Ferrini, Multimode ro- tationally symmetric bosonic codes from group-theoretic construction, Physical Review Research8, 10.1103/xt48- fxt5 (2026)

    work page doi:10.1103/xt48- 2026

  29. [29]

    J. D. Teoh, N. Cottet, P. Winkel, L. D. Burkhart, L. Frunzio, and R. J. Schoelkopf, Robust quantum communication through lossy microwave links (2025), arXiv:2509.18547 [quant-ph]

    work page arXiv 2025

  30. [30]

    V. G. Matsos, C. H. Valahu, T. Navickas, A. D. Rao, M. J. Millican, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, Robust and deterministic preparation of bosonic logical states in a trapped ion, Phys. Rev. Lett. 133, 050602 (2024)

    2024

  31. [31]

    Wang, D.-Y

    J. Wang, D.-Y. Huang, X.-L. Zhou, Z.-M. Shen, S.-J. He, Q.-Y. Huang, Y.-J. Liu, C.-F. Li, and G.-C. Guo, Ultra- fast high-fidelity state readout of single neutral atom, Phys. Rev. Lett.134, 240802 (2025)

    2025

  32. [32]

    L. H. Bohnmann, D. F. Locher, J. Zeiher, and M. M¨ uller, Bosonic quantum error correction with neu- tral atoms in optical dipole traps, Physical Review A 111, 10.1103/physreva.111.022432 (2025)

    work page doi:10.1103/physreva.111.022432 2025

  33. [33]

    V. V. Albert, S. O. Mundhada, A. Grimm, S. Touzard, M. H. Devoret, and L. Jiang, Pair-cat codes: autonomous error-correction with low-order nonlinearity, Quantum Science and Technology4, 035007 (2019)

    2019

  34. [34]

    A. J. Boon, O. Landon-Cardinal, and N. Quesada, Gener- alised all-optical cat correction (2026), arXiv:2603.03263 [quant-ph]

    work page arXiv 2026

  35. [35]

    A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Quan- tum computing with rotation-symmetric bosonic codes, Physical Review X10, 011058 (2020)

    2020

  36. [36]

    Y. Zeng, F. Quijandr´ ıa, C. Gneiting, and F. Nori, Quan- tum error correction with superpositions of squeezed fock states (2025), arXiv:2510.04209 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  37. [37]

    D. J. Brod and M. Oszmaniec, Classical simulation of linear optics subject to nonuniform losses, Quantum4, 267 (2020)

    2020

  38. [38]

    Papiˇ c, A

    M. Papiˇ c, A. Auer, and I. de Vega, Fast estimation of physical error contributions of quantum gates, arXiv preprint arXiv:2305.08916 (2023)

    work page arXiv 2023

  39. [39]

    Knill, R

    E. Knill, R. Laflamme, and L. Viola, Theory of quantum error correction for general noise, Phys. Rev. Lett.84, 2525 (2000)

    2000

  40. [40]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, 2000)

    2000

  41. [41]

    D. P. DiVincenzo, The physical implementation of quan- tum computation, Fortschritte der Physik48, 771–783 (2000)

    2000

  42. [42]

    Zanardi and L

    P. Zanardi and L. Campos Venuti, Coherent quantum dy- namics in steady-state manifolds of strongly dissipative systems, Phys. Rev. Lett.113, 240406 (2014). 22

    2014

  43. [43]

    Facchi and S

    P. Facchi and S. Pascazio, Quantum zeno subspaces, Phys. Rev. Lett.89, 080401 (2002)

    2002

  44. [44]

    V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Geom- etry and response of lindbladians, Phys. Rev. X6, 041031 (2016)

    2016

  45. [45]

    Arenz, D

    C. Arenz, D. Burgarth, P. Facchi, V. Giovannetti, H. Nakazato, S. Pascazio, and K. Yuasa, Universal con- trol induced by noise, Phys. Rev. A93, 062308 (2016)

    2016

  46. [46]

    M. Yuan, Q. Xu, and L. Jiang, Construction of bias- preserving operations for pair-cat codes, Physical Review A106, 10.1103/physreva.106.062422 (2022)

    work page doi:10.1103/physreva.106.062422 2022

  47. [47]

    These states are distinct from the logical states of DRCC-1

    Another class of solutions is obtained by choosingC 00 = C11 = 0, which leads to |ξ⟩=    X n hkn|2n,2n+ 1⟩, C 01 ̸= 0, C 10 = 0, X n hkn|2n+ 1,2n⟩, C 01 = 0, C 10 ̸= 0, (116) where ˆHkn = α4k √ (2n)!(2n+1)! . These states are distinct from the logical states of DRCC-1. Apart from one cav- ity containing even numbers, the other has odd num- bers of ph...

  48. [48]

    Khanahmadi and K

    M. Khanahmadi and K. Mølmer, Environment-assisted generation of non-gaussian wavepacket quantum states (2025), arXiv:2504.04513 [quant-ph]

    work page arXiv 2025

  49. [49]

    Gerry and P

    C. Gerry and P. Knight,Introductory Quantum Optics (Cambridge University Press, 2004)

    2004

  50. [50]

    M. V. Berry, Quantal phase factors accompanying adia- batic changes, Proceedings of the Royal Society of Lon- don. A. Mathematical and Physical Sciences392, 45 (1984), https://royalsocietypublishing.org/rspa/article- pdf/392/1802/45/65511/rspa.1984.0023.pdf

    work page arXiv 1984

  51. [51]

    V. V. Albert, C. Shu, S. Krastanov, C. Shen, R.-B. Liu, Z.-B. Yang, R. J. Schoelkopf, M. Mirrahimi, M. H. De- voret, and L. Jiang, Holonomic quantum control with continuous variable systems, Physical Review Letters 116, 10.1103/physrevlett.116.140502 (2016)

    work page doi:10.1103/physrevlett.116.140502 2016

  52. [52]

    A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K. Tuckett, and S. Puri, Practical Quantum Error Correc- tion with the XZZX Code and Kerr-Cat Qubits, PRX Quantum2, 030345 (2021), arXiv:2104.09539 [quant-ph]

    work page arXiv 2021

  53. [53]

    A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Fault- tolerant linear optical quantum computing with small- amplitude coherent states, Phys. Rev. Lett.100, 030503 (2008)

    2008

  54. [54]

    A. P. Lund, T. C. Ralph, and H. L. Haselgrove, Fault- tolerant linear optical quantum computing with small- amplitude coherent states, Physical Review Letters100, 10.1103/physrevlett.100.030503 (2008)

    work page doi:10.1103/physrevlett.100.030503 2008

  55. [55]

    Messinger, V

    A. Messinger, V. Torggler, B. Klaver, M. Fellner, and W. Lechner, Fault-tolerant quantum computing with the parity code and biased-noise qubits, Physical Review Ap- plied23, 10.1103/physrevapplied.23.044032 (2025)

    work page doi:10.1103/physrevapplied.23.044032 2025

  56. [56]

    D. Ruiz, J. Guillaud, A. Leverrier, M. Mirrahimi, and C. Vuillot, Ldpc-cat codes for low-overhead quantum computing in 2d (2024), arXiv:2401.09541 [quant-ph]

    work page arXiv 2024

  57. [57]

    Gouzien, D

    E. Gouzien, D. Ruiz, F.-M. Le R´ egent, J. Guillaud, and N. Sangouard, Performance analysis of a repetition cat code architecture: Computing 256-bit elliptic curve log- arithm in 9 hours with 126 133 cat qubits, Phys. Rev. Lett.131, 040602 (2023)

    2023

  58. [58]

    Zheng, W

    G. Zheng, W. He, G. Lee, and L. Jiang, Near-optimal performance of quantum error correction codes, Physical Review Letters132, 250602 (2024)

    2024

  59. [59]

    Tsunoda, J

    T. Tsunoda, J. D. Teoh, W. D. Kalfus, S. J. de Graaf, B. J. Chapman, J. C. Curtis, N. Thakur, S. M. Girvin, and R. J. Schoelkopf, Error-detectable bosonic entan- gling gates with a noisy ancilla, PRX Quantum4, 020354 (2023)

    2023

  60. [60]

    A. S. Fletcher, P. W. Shor, and M. Z. Win, Channel- adapted quantum error correction for the amplitude damping channel, IEEE Transactions on Information Theory54, 5705 (2008)

    2008

  61. [61]

    H. K. Ng and P. Mandayam, Simple approach to approx- imate quantum error correction based on the transpose channel, Phys. Rev. A81, 062342 (2010)

    2010

  62. [62]

    Barnum and E

    H. Barnum and E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity, Journal of Mathematical Physics43, 2097 (2002)

    2097

  63. [63]

    X. Mao, Q. Xu, and L. Jiang, Optimized four-qubit quan- tum error correcting code for amplitude damping channel (2025), arXiv:2411.12952 [quant-ph]

    work page arXiv 2025

  64. [64]

    M. M. Wilde, From classical to quantum shannon theory, arXiv preprint arXiv:1106.1445 (2011)

    work page arXiv 2011

  65. [65]

    Biswas, G

    D. Biswas, G. M. Vaidya, and P. Mandayam, Noise- adapted recovery circuits for quantum error correction, Phys. Rev. Res.6, 043034 (2024)

    2024

  66. [66]

    Biswas and P

    D. Biswas and P. Mandayam, Universal syndrome-based recovery for noise-adapted quantum error correction (2025), arXiv:2510.08719 [quant-ph]

    work page arXiv 2025

  67. [67]

    Anonymous, Petz recovery maps of single-qubit decoher- ence channels in an ion trap quantum processor, Physical Review A 10.1103/7f8x-n2np (2025)

    work page doi:10.1103/7f8x-n2np 2025

  68. [68]

    Singh, R

    G. Singh, R. S. Sahani, V. Jagadish, L. Lautenbacher, N. K. Bernardes, and K. Dorai, Realizing the petz recov- ery map on an nmr quantum processor, Physical Review A113, 10.1103/xd6k-swv7 (2026)

    work page doi:10.1103/xd6k-swv7 2026

  69. [69]

    M. A. Nielsen, The entanglement fidelity and quantum error correction (1996), arXiv:quant-ph/9606012 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  70. [70]

    Universal Quantum Computing with Arbitrary Continuous-Variable Encoding

    H.-K. Lau and M. B. Plenio, Universal Quantum Com- puting with Arbitrary Continuous-Variable Encoding, Phys. Rev. Lett.117, 100501 (2016), arXiv:1605.09278 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  71. [71]

    Ouyang, Avoiding coherent errors with rotated con- catenated stabilizer codes, npj Quantum Information7, 10.1038/s41534-021-00429-8 (2021)

    Y. Ouyang, Avoiding coherent errors with rotated con- catenated stabilizer codes, npj Quantum Information7, 10.1038/s41534-021-00429-8 (2021)

    work page doi:10.1038/s41534-021-00429-8 2021

  72. [72]

    V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. M. Girvin, B. M. Terhal, and L. Jiang, Performance and structure of single-mode bosonic codes, Phys. Rev. A97, 032346 (2018)

    2018

  73. [73]

    Reinken, S

    A. Soro and A. F. Kockum, Chiral quantum optics with giant atoms, Physical Review A105, 10.1103/phys- reva.105.023712 (2022)

    work page doi:10.1103/phys- 2022

  74. [74]

    Shanahan and D

    P. Shanahan and D. Ruiz, Elevator codes: Concatena- tion for resource-efficient quantum memory under biased noise (2026), arXiv:2601.10786 [quant-ph]

    work page arXiv 2026

  75. [75]

    Blasiak, K

    P. Blasiak, K. Penson, and A. Solomon, The boson nor- mal ordering problem and generalized bell numbers, An- nals of Combinatorics7, 127 (2003)

    2003

  76. [76]

    cos2 δ N− N+ 2 sin2 δ N+ N− 2# |α|2, (A4) ⟨1L|ˆn1|1L⟩=

    J. D. Chadwick, M. H. Teo, J. Viszlai, W. Yang, and F. T. Chong, Erasure minesweeper: exploring hybrid- 23 erasure surface code architectures for efficient quantum error correction (2025), arXiv:2505.00066 [quant-ph]. Appendix A: DRCC-1 with beam-splitted modes against pure dephasing errors The BS operation ˆUBS commutes with photon loss errors, resulting...

    work page arXiv 2025

  77. [77]

    (32)–(35)

    Photon loss The Knill–Laflamme (KL) conditions for single-photon loss are given in Eqs. (32)–(35). More generally, for an ℓ-th order photon loss acting on thei-th mode, one finds ˆ¯Pˆaℓ i ˆ¯P∝ ( 0, ℓodd, ˆ¯P, ℓeven. (B1) We now consider joint photon-loss processes acting on both cavities. Forℓ-th order loss in mode 1 andk-th order loss in mode 2, we obtai...

  78. [78]

    Dephasing We have seen, by analysing the KL condition for first- order dephasing in either cavity, that dephasing errors induce a logical ˆZerror in the DRCC-1 qubit. To analyse higher-order dephasing processes, we use the following identity ˆnℓ = ℓX p=1 S(ℓ, p)ˆa†pˆap,(B4) where S(ℓ, p) = 1 p! pX k=0 (−1)p+k p k kℓ (B5) are the Stirling numbers of the se...

  79. [79]

    We begin by proving the following lemma

    LogicalY We show that the operator ˆG− 12 generates the logical gate ˆXL for evenNand the logical gatei ˆYL for oddN in the multimode RSB code. We begin by proving the following lemma. 25 Lemma 1.For the operator ˆG− 12, the following relation holds: ejπ ˆG− 12/2ˆa† i e−jπ ˆG− 12/2 = (−1)iˆa† i+1.(D1) Proof.Using the Baker–Campbell–Hausdorff (BCH) for- mu...

  80. [80]

    In particular, we revisit the gate con- struction introduced in Ref

    Logical ˆZ We now discuss an alternative implementation of the logical ˆZL gate and its extension to arbitrary rotational symmetry orderN. In particular, we revisit the gate con- struction introduced in Ref. [24], but with a code vector that excludes the beam splitter. Lemma 2.For a two-mode RSB code with arbitrary ro- tational symmetry orderN, the logica...

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