pith. sign in

arxiv: 2606.09379 · v1 · pith:RAMB4WG3new · submitted 2026-06-08 · 🪐 quant-ph

Entanglement-assisted continuous-variable concatenated codes for encoding qubits or oscillators

Nihar Ranjan Dash , Sanjoy Dutta , R. Srikanth , Subhashish Banerjee This is my paper

Pith reviewed 2026-06-27 16:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglement-assisted codesGKP codesconcatenated codescontinuous-variable error correctionquadrature variance suppressionbosonic codesrepetition codes
0
0 comments X

The pith

Concatenating GKP codes with n-qubit EA repetition codes suppresses both position and momentum quadrature variances by a factor of 1/n using n-1 entangled modes.

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

The paper combines two ideas: entanglement-assisted stabilizer codes improve error-correction rates over standard codes, and GKP codes can be concatenated with qubit codes to lower logical failure rates. It first builds an EA outer code around a GKP inner code for encoding qubits into oscillators. It then builds the reverse concatenation, with a GKP outer code around an EA inner code, for encoding oscillators into oscillators. The central construction is the n-qubit EA repetition code that consumes n-1 maximally entangled modes and reduces the variances of both quadratures on a data mode by exactly 1/n. A sympathetic reader would care because the scheme offers a concrete way to use pre-shared entanglement to protect continuous-variable information more efficiently.

Core claim

We propose an EA version of the non-Gaussian oscillator-into-oscillators concatenated code that chains a GKP outer code with an EA-stabilizer inner code. As an example we present a GKP code concatenated with a three-qubit EA repetition code that uses two maximally entangled modes and suppresses the variances of both position and momentum quadrature errors of a data mode. Furthermore, we generalize the latter example to a family of GKP code concatenated with a n-qubit EA repetition code that uses n-1 emodes and suppresses the variances of both position and momentum quadrature errors of a data mode by a factor 1/n.

What carries the argument

The n-qubit entanglement-assisted repetition code using n-1 maximally entangled modes, concatenated with a GKP code as outer code.

If this is right

  • The same n-qubit EA repetition inner code works for both the qubit-into-oscillator and oscillator-into-oscillator concatenations.
  • Variance suppression applies equally to position and momentum quadratures.
  • The construction uses exactly n-1 entangled modes for any n.
  • The scheme is presented as an explicit generalization of the three-qubit case.

Where Pith is reading between the lines

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

  • The linear scaling with n suggests that adding more entangled modes yields proportionally better protection, which could be tested by increasing n in numerical simulations of the concatenated channel.
  • Because the construction works in either concatenation order, it may allow flexible placement of the GKP layer depending on whether the dominant noise is at the qubit or oscillator level.
  • The explicit use of repetition codes leaves open whether other EA stabilizer codes could produce stronger suppression or higher rates when concatenated with GKP.

Load-bearing premise

The constructions assume perfect maximally entangled modes and ideal GKP states are available.

What would settle it

An explicit calculation of the output quadrature variances when the input entangled modes have finite squeezing or the GKP states have finite envelope would show whether the suppression factor remains exactly 1/n.

Figures

Figures reproduced from arXiv: 2606.09379 by Nihar Ranjan Dash, R. Srikanth, Sanjoy Dutta, Subhashish Banerjee.

Figure 1
Figure 1. Figure 1: FIG. 1: Encoding circuits to realize the GKP Bell pairs via [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Encoding circuit for [[3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The Gaussian encoding circuit of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) Logical failure probability of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Entanglement-assisted (EA) stabilizer codes enhance the rate of error correction in relation to codes with no pre-shared entanglement. Meanwhile, bosonic error-correcting codes, such as the Gottesman-Kitaev-Preskill (GKP) code, can be concatenated with qubit stabilizer codes to significantly reduce the logical failure probability of those stabilizer codes. First, we combine the above two concepts to propose an EA version of the qubit-into-oscillators concatenated code that chains an EA-stabilizer (outer) code with a GKP (inner) code. As an example we present a three-qubit EA-repetition concatenated with a GKP code. Second, we propose an EA version of the non-Gaussian oscillator-into-oscillators concatenated code that chains a GKP (outer) code with an EA-stabilizer (inner) code. As an example we present a GKP code concatenated with a three-qubit EA repetition code that uses two maximally entangled modes (emodes) and suppresses the variances of both position and momentum quadrature errors of a data mode. Furthermore, we generalize the latter example to a family of GKP code concatenated with a $n$-qubit EA repetition code that uses ${n-1}$ emodes and suppresses the variances of both position and momentum quadrature errors of a data mode by a factor ${1/n}$.

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 / 0 minor

Summary. The manuscript proposes two classes of entanglement-assisted concatenated continuous-variable codes. The first combines an EA-stabilizer outer code with a GKP inner code to encode qubits into oscillators, illustrated by a three-qubit EA-repetition example. The second combines a GKP outer code with an EA-stabilizer inner code to encode oscillators into oscillators, illustrated by a GKP code concatenated with a three-qubit EA repetition code that uses two emodes; this is generalized to an n-qubit EA repetition code using (n-1) emodes that suppresses both position and momentum quadrature error variances of a data mode by a factor of exactly 1/n.

Significance. If the constructions and the exact 1/n suppression are rigorously established under the stated ideal-resource model, the work would supply a concrete family of EA concatenated codes that leverage pre-shared entanglement to improve quadrature variance suppression in bosonic systems. The explicit n-parameter generalization is a potential strength if accompanied by explicit derivations.

major comments (2)
  1. [generalization to n-qubit EA repetition code] The generalization to the n-qubit EA repetition outer code (final paragraph of the abstract and corresponding section of the main text) asserts that the construction suppresses both quadrature variances by exactly 1/n. No explicit derivation, stabilizer tableau, or noise-propagation calculation is referenced that demonstrates how the (n-1) emodes produce this precise factor for both quadratures simultaneously; the central scaling claim therefore rests on unshown steps.
  2. [EA repetition concatenated with GKP (oscillator-into-oscillators construction)] The constructions assume perfect GKP states and ideal maximally entangled modes. The manuscript supplies no calculation showing how finite squeezing, loss, or entanglement infidelity propagates through the concatenation and degrades the claimed 1/n factor. Because the headline result is presented as a direct consequence of the ideal model, this omission is load-bearing for any claim of practical utility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below and will update the manuscript to strengthen the presentation under the ideal-resource model.

read point-by-point responses

  1. Referee: The generalization to the n-qubit EA repetition outer code (final paragraph of the abstract and corresponding section of the main text) asserts that the construction suppresses both quadrature variances by exactly 1/n. No explicit derivation, stabilizer tableau, or noise-propagation calculation is referenced that demonstrates how the (n-1) emodes produce this precise factor for both quadratures simultaneously; the central scaling claim therefore rests on unshown steps.

    Authors: We agree that an explicit derivation is required to rigorously support the generalization. In the revised manuscript we will insert a new subsection containing the full stabilizer tableau for the n-qubit EA repetition code together with a complete noise-propagation calculation. This will show, step by step, how the (n-1) emodes simultaneously reduce both quadrature variances by the exact factor 1/n under the stated ideal model. revision: yes

  2. Referee: The constructions assume perfect GKP states and ideal maximally entangled modes. The manuscript supplies no calculation showing how finite squeezing, loss, or entanglement infidelity propagates through the concatenation and degrades the claimed 1/n factor. Because the headline result is presented as a direct consequence of the ideal model, this omission is load-bearing for any claim of practical utility.

    Authors: The manuscript is framed entirely within the ideal-resource model (perfect GKP states and ideal entanglement), as stated in the abstract and introduction; the 1/n claim is derived strictly under those assumptions. No degradation analysis is needed to support the stated theoretical result. We will nevertheless add a short paragraph in the conclusions noting that extensions to finite squeezing and loss constitute valuable future work. revision: partial

Circularity Check

0 steps flagged

No circularity: constructions are forward proposals under ideal resources

full rationale

The paper proposes concatenated EA-GKP code families as explicit constructions. The 1/n variance suppression is stated as a direct structural consequence of the n-qubit EA repetition outer code acting on ideal GKP states and perfect emodes; no equations, fitted parameters, or self-referential definitions appear that would reduce the claimed scaling to an input by construction. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are invoked in the abstract. The work is a standard constructive proposal rather than a derivation that loops back on itself, so the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum information assumptions rather than new postulates; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard stabilizer formalism and GKP code properties hold for the inner and outer codes.
    The constructions chain EA-stabilizer and GKP codes, which presupposes the established definitions and error-correction properties of both.
  • domain assumption Maximally entangled modes (emodes) can be prepared and maintained perfectly.
    The n-qubit EA repetition example explicitly uses (n-1) emodes to achieve the 1/n suppression.

pith-pipeline@v0.9.1-grok · 5783 in / 1377 out tokens · 25925 ms · 2026-06-27T16:22:00.807308+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 2 canonical work pages

  1. [1]

    substitution code

    than the computational GKP state Eq. (2) [28, 29]. Due to non-Gaussianity of the (canonical) GKP state, it is used in the encoding of the oscillator-into- oscillators codes to correct Gaussian errors, in accordance with the no-go result that asserts that correction of Gaussian errors can not be done by only Gaussian resources [26]. The canonical GKP state...

  2. [2]

    ∞ −∞ dζ (1) q dζ (2) q Pσ(ζ(1) q )P(ζ(2) q )P(3ζ(∗) −ζ (1) q −ζ (2) q ), =3N 3

    The encoded state for theC GKP ▷C[[n,1,n;n−1]] code is defined as: |ψL⟩=(U G enc ⊗I B)(|ψGKP⟩ ⊗ |Φ+ GKP⟩⊗(n−1) AB ),(37) whereU G enc is the relevant Gaussian encoder for theC GKP ▷ C[[n,1,n;n−1]] code. Here, the first mode is data mode, and the rest (n−1) emodes are initialized with GKP Bell pairs. Analogous to Eqs. (16) and (25), we can write encoders f...

  3. [3]

    Gottesman,Stabilizer codes and quantum error correction, Ph.D

    D. Gottesman,Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology (1997)

    1997

  4. [4]

    D. A. Lidar and T. A. Brun,Quantum Error Correction(Cam- bridge University Press, 2013)

    2013

  5. [5]

    Laflamme, C

    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Perfect quantum error correcting code, Phys. Rev. Lett.77, 198 (1996)

    1996

  6. [6]

    P. W. Shor, Scheme for reducing decoherence in quantum com- puter memory, Phys. Rev. A52, R2493 (1995)

    1995

  7. [7]

    A. M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett.77, 793 (1996)

    1996

  8. [8]

    S. L. Braunstein, Error correction for continuous quantum vari- ables, Phys. Rev. Lett.80, 4084 (1998)

    1998

  9. [9]

    Lloyd and J.-J

    S. Lloyd and J.-J. E. Slotine, Analog quantum error correction, Phys. Rev. Lett.80, 4088 (1998)

    1998

  10. [10]

    Lee and H

    S.-W. Lee and H. Jeong, Near-deterministic quantum telepor- tation and resource-efficient quantum computation using linear optics and hybrid qubits, Phys. Rev. A87, 022326 (2013)

    2013

  11. [11]

    Kapit, Hardware-efficient and fully autonomous quantum er- ror correction in superconducting circuits, Phys

    E. Kapit, Hardware-efficient and fully autonomous quantum er- ror correction in superconducting circuits, Phys. Rev. Lett.116, 150501 (2016)

    2016

  12. [12]

    Y . Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eick- busch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, T. Tsunoda, I. L. Chuang, N. Wiebe, and S. M. Girvin, Hybrid oscillator-qubit quantum processors: Instruction set architec- tures, abstract machine models, and applications, PRX Quan- tum7, 010201 (2026)

    2026

  13. [13]

    Gottesman, A

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

    2001

  14. [14]

    A. L. Grimsmo and S. Puri, Quantum error correction with the gottesman-kitaev-preskill code, PRX Quantum2, 020101 (2021)

    2021

  15. [15]

    Fukui, A

    K. Fukui, A. Tomita, and A. Okamoto, Analog quantum error correction with encoding a qubit into an oscillator, Phys. Rev. Lett.119, 180507 (2017)

    2017

  16. [16]

    Wang, Quantum error correction with the gkp code and concatenation with stabilizer codes (2019), arXiv:1908.00147 [quant-ph]

    Y . Wang, Quantum error correction with the gkp code and concatenation with stabilizer codes (2019), arXiv:1908.00147 [quant-ph]

    work page arXiv 2019

  17. [17]

    Lin and K

    M. Lin and K. Noh, Exploring the quantum capacity of a gaus- sian random-displacement channel using gottesman-kitaev- preskill codes and maximum-likelihood decoding, Phys. Rev. A111, 052445 (2025)

    2025

  18. [18]

    Fukui, A

    K. Fukui, A. Tomita, and A. Okamoto, Tracking quantum error correction, Phys. Rev. A98, 022326 (2018)

    2018

  19. [19]

    Fukui, A

    K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, High-threshold fault-tolerant quantum computation with analog quantum error correction, Phys. Rev. X8, 021054 (2018)

    2018

  20. [20]

    Vuillot, H

    C. Vuillot, H. Asasi, Y . Wang, L. P. Pryadko, and B. M. Ter- hal, Quantum error correction with the toric gottesman-kitaev- preskill code, Phys. Rev. A99, 032344 (2019)

    2019

  21. [21]

    Noh and C

    K. Noh and C. Chamberland, Fault-tolerant bosonic quan- tum error correction with the surface–gottesman-kitaev-preskill code, Phys. Rev. A101, 012316 (2020)

    2020

  22. [22]

    K. Noh, C. Chamberland, and F. G. Brand ˜ao, Low-overhead fault-tolerant quantum error correction with the surface-gkp code, PRX Quantum3, 010315 (2022)

    2022

  23. [23]

    Raveendran, N

    N. Raveendran, N. Rengaswamy, F. Rozpedek, A. Raina, L. Jiang, and B. Vasic, Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound, Quantum6, 767 (2022)

    2022

  24. [24]

    Xiao and X

    H. Xiao and X. Chen, Quantum convolutional codes concate- nated with the GKP code for correcting continuous errors, Quantum Inf. Process.21, Paper No. 198, 15 (2022)

    2022

  25. [25]

    Li and D

    Z. Li and D. Su, Correcting biased noise using gottesman- kitaev-preskill repetition code with noisy ancilla, Phys. Rev. A 109, 052420 (2024)

    2024

  26. [26]

    Rozpedek, K

    F. Rozpedek, K. Noh, Q. Xu, S. Guha, and L. Jiang, Quantum repeaters based on concatenated bosonic and discrete-variable quantum codes, npj Quantum Information7, 102 (2021)

    2021

  27. [27]

    Chamberland, K

    C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Camp- bell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. Brand˜ao, Building a fault- tolerant quantum computer using concatenated cat codes, PRX Quantum3, 010329 (2022)

    2022

  28. [28]

    K. Noh, S. M. Girvin, and L. Jiang, Encoding an oscillator into many oscillators, Phys. Rev. Lett.125, 080503 (2020)

    2020

  29. [29]

    F. Guo, F. Mueller, and Y . Liu, Concatenated dual displacement code for continuous-variable quantum error correction, Phys. Rev. Res.8, 023158 (2026)

    2026

  30. [30]

    Y . Xu, Y . Wang, E.-J. Kuo, and V . V . Albert, Qubit-oscillator concatenated codes: Decoding formalism and code compari- son, PRX Quantum4, 020342 (2023)

    2023

  31. [31]

    J. Wu, A. J. Brady, and Q. Zhuang, Optimal encoding of oscil- lators into more oscillators, Quantum7, 1082 (2023)

    2023

  32. [32]

    Zhuang, J

    Q. Zhuang, J. Preskill, and L. Jiang, Distributed quantum sens- ing enhanced by continuous-variable error correction, New Journal of Physics22, 022001 (2020)

    2020

  33. [33]

    B. Zhou, A. J. Brady, and Q. Zhuang, Enhancing distributed sensing with imperfect error correction, Phys. Rev. A106, 012404 (2022)

    2022

  34. [34]

    B.-H. Wu, Z. Zhang, and Q. Zhuang, Continuous-variable quan- tum repeaters based on bosonic error-correction and teleporta- tion: architecture and applications, Quantum Science and Tech- nology7, 025018 (2022)

    2022

  35. [35]

    T. Brun, I. Devetak, and M.-H. Hsieh, Correcting quantum er- rors with entanglement, Science314, 436–439 (2006)

    2006

  36. [36]

    T. A. Brun, I. Devetak, and M.-H. Hsieh, Catalytic quantum error correction, IEEE Transactions on Information Theory60, 3073–3089 (2014)

    2014

  37. [37]

    M. M. Wilde, H. Krovi, and T. A. Brun, Entanglement-assisted quantum error correction with linear optics, Phys. Rev. A76, 052308 (2007)

    2007

  38. [38]

    N. C. Menicucci, Fault-tolerant measurement-based quantum computing with continuous-variable cluster states, Phys. Rev. Lett.112, 120504 (2014)

    2014

  39. [39]

    Glancy and E

    S. Glancy and E. Knill, Error analysis for encoding a qubit in an oscillator, Phys. Rev. A73, 012325 (2006)

    2006

  40. [40]

    Schmidt and P

    F. Schmidt and P. van Loock, Quantum error correction with 10 higher gottesman-kitaev-preskill codes: Minimal measure- ments and linear optics, Phys. Rev. A105, 042427 (2022)

    2022

  41. [41]

    Knill, Scalable quantum computing in the presence of large detected-error rates, Phys

    E. Knill, Scalable quantum computing in the presence of large detected-error rates, Phys. Rev. A71, 042322 (2005)

    2005

  42. [42]

    B. W. Walshe, B. Q. Baragiola, R. N. Alexander, and N. C. Menicucci, Continuous-variable gate teleportation and bosonic- code error correction, Phys. Rev. A102, 062411 (2020)

    2020

  43. [43]

    F. K. Marqversen, J. H. Wesenberg, N. T. Zinner, and U. L. Andersen, Performance analysis of gkp error correction (2025), arXiv:2505.14775 [quant-ph]

    work page arXiv 2025

  44. [44]

    Lai and T

    C.-Y . Lai and T. A. Brun, Entanglement increases the error- correcting ability of quantum error-correcting codes, Phys. Rev. A88, 012320 (2013)

    2013

  45. [45]

    C.-Y . Lai, T. A. Brun, and M. M. Wilde, Dualities and identities for entanglement-assisted quantum codes, Quantum informa- tion processing13, 957 (2014)

    2014

  46. [46]

    Gaitan,Quantum Error Correction and Fault Tolerant Quan- tum Computing(Taylor & Francis, 2008)

    F. Gaitan,Quantum Error Correction and Fault Tolerant Quan- tum Computing(Taylor & Francis, 2008)

    2008

  47. [47]

    N. R. Dash, S. Dutta, R. Srikanth, and S. Banerjee, Concate- nating quantum error-correcting codes with decoherence-free subspaces and vice versa, Phys. Rev. A109, 062411 (2024)

    2024

  48. [48]

    Lai and T

    C.-Y . Lai and T. A. Brun, Entanglement-assisted quantum error- correcting codes with imperfect ebits, Phys. Rev. A86, 032319 (2012)

    2012

  49. [49]

    N. J. Cerf, G. Leuchs, and E. S. Polzik,Quantum information with continuous variables of atoms and light(World Scientific, 2007)

    2007

  50. [50]

    S. Puri, A. Grimm, P. Campagne-Ibarcq, A. Eickbusch, K. Noh, G. Roberts, L. Jiang, M. Mirrahimi, M. H. Devoret, and S. M. Girvin, Stabilized cat in a driven nonlinear cavity: A fault-tolerant error syndrome detector, Phys. Rev. X9, 041009 (2019)

    2019

  51. [51]

    P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macroscop- ically distinct quantum-superposition states as a bosonic code for amplitude damping, Phys. Rev. A59, 2631 (1999)

    1999

  52. [52]

    Hastrup and U

    J. Hastrup and U. L. Andersen, All-optical cat-code quantum error correction, Phys. Rev. Res.4, 043065 (2022)

    2022

  53. [53]

    M. H. Michael, M. Silveri, R. T. Brierley, V . V . Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quan- tum error-correcting codes for a bosonic mode, Phys. Rev. X6, 031006 (2016)

    2016

  54. [54]

    Laha and P

    P. Laha and P. van Loock, Arbitrary high-fidelity binomial codes from multiphoton spin-boson interactions, Phys. Rev. Res.8, 013237 (2026)

    2026