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REVIEW 3 major objections 5 minor 59 references

A gate-based microwave quantum repeater using GKP grid states achieves entanglement generation and swapping success probabilities of about 0.75 and 0.58, beating the 1/2 ceiling set by ideal beamsplitter Bell-state measurements.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 14:32 UTC pith:45VOFCRY

load-bearing objection The repeater architecture is real and new, but Table I's "success probabilities" are fidelities, so the claimed crossing of the 1/2 beamsplitter ceiling is not established. the 3 major comments →

arxiv 2512.19896 v2 pith:45VOFCRY submitted 2025-12-22 quant-ph

Gate-Based Microwave Quantum Repeater Via Grid-State Encoding

Hany Khalifa , Matti Silveri This is my paper
classification quant-ph
keywords quantum repeatermicrowave cQEDGKP grid statesautonomous quantum error correctionentanglement generationentanglement swappingBell-state measurementsecret key rate
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes a microwave quantum repeater built from a transmon and two bosonic resonators per node, with information stored in GKP grid states protected by autonomous error correction. The key move is to replace probabilistic beamsplitter interference with deterministic gate operations: entanglement is generated by absorbing a single-photon wavepacket, and swapping is done with a bosonic controlled-Z gate plus homodyne measurements. The authors show that at a stationary damping time of 40 ms, the protocol reaches entanglement generation success probability of about 0.75 and swapping success of about 0.58, both above the 1/2 bound for ideal linear-beamsplitter Bell measurements. If the numbers hold, this would be the first microwave repeater to surpass that bound for both steps, and it uses only currently available cQED technology. A sympathetic reader would care because it offers a concrete path to secure chip-to-chip microwave communication and distributed quantum computing.

Core claim

The central claim is that a fully gate-based microwave quantum repeater can outperform beamsplitter-based repeaters by avoiding mode-mismatch and routing losses entirely. Each station prepares a transmon-GKP state, then uses a remote controlled-Z gate mediated by a round-trip single photon to generate entanglement deterministically; swapping is performed by a bosonic controlled-Z interaction between two stationary GKP codewords followed by two X-basis homodyne projections. The paper derives success probabilities for generation and swapping as functions of the codeword overlaps α and β, channel transmissivity η, and homodyne fidelity, and evaluates a secret key rate against a beamsplitter-bas

What carries the argument

The central objects are GKP grid states—bosonic codewords with wavefunctions peaked on a square lattice in phase space—and two gate-based mechanisms that replace beamsplitter interference. First, sequential entanglement generation: a transmon emits a single-photon wavepacket that travels to the receiving node, gets phase-shifted and reflected, and is absorbed back, realizing a deterministic remote controlled-Z gate; success is conditioned on successful absorption. Second, all-bosonic entanglement swapping: a cross-Kerr controlled-Z gate e^{-iQ_B Q_C} between two memory resonators, followed by homodyne X-basis projective measurements on each, yields a Bell-state measurement without joint inte

Load-bearing premise

The quoted success probabilities assume error-free transmon operations, including deterministic absorption and reflection of the photon wavepacket and ideal transmon measurements; if those operations have realistic errors, the remote controlled-Z is no longer deterministic and the generation and swapping probabilities would drop.

What would settle it

A concrete experiment: implement the sequential entanglement generation protocol in a cQED setup with measured transmon gate and readout fidelities, and compare the observed success probability to the paper's predicted curve as a function of stationary damping time. If the observed generation success probability at 40 ms damping falls below 0.5 once transmon errors are included, the claimed advantage over ideal beamsplitter Bell-state measurements would not hold.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • A microwave quantum repeater can surpass the 1/2 Bell-measurement limit for both generation and swapping, enabling higher secret key rates than beamsplitter-based microwave repeaters.
  • Losses are confined to stationary storage, so improving stationary damping time directly boosts performance: at 40 ms vs 25 ms, generation success rises from 0.60 to 0.75 and swapping from 0.36 to 0.58.
  • The protocol requires no microwave single-photon counters, only homodyne detectors and transmon operations, both mature in cQED.
  • The device can serve as a bosonic remote entangling gate on the logical subspace of autonomously error-corrected grid states, relevant for measurement-based quantum computing with cluster states.
  • Higher squeezing (e.g., 20 dB) would reduce cross-overlap errors to about 1e-5, further improving gate fidelities.

Where Pith is reading between the lines

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

  • If transmon gate and readout errors are included, the quoted success probabilities would likely drop; the authors acknowledge this, and a realistic error budget using measured gate fidelities would show whether the 0.75/0.58 figures survive.
  • The sequential absorption scheme is essentially a single-rail photonic qubit, and might be extendable to multi-photon or time-bin encoded wavepackets while preserving the deterministic nature and increasing robustness to loss.
  • The gate-based swapping approach could be adapted to other bosonic codes, such as cat or binomial codes, if a controlled-Z and homodyne-based logical measurement exist, potentially transferring the advantage beyond GKP states.
  • The assumption of error-free transmon operations might be relaxed by incorporating gate errors into the secret-key calculation, giving a more complete comparison with beamsplitter-based repeaters.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a gate-based microwave quantum repeater (GBMQR) using GKP grid states in superconducting resonators, with sequential single-photon absorption for entanglement generation and an all-bosonic controlled-Z plus homodyne X-basis measurements for entanglement swapping. The authors claim that at a stationary damping time of 40 ms the entanglement generation and swapping success probabilities are approximately 0.75 and 0.58, respectively, surpassing the 1/2 success probability of ideal beamsplitter-based Bell-state measurements, and they compare the resulting secret key rate with a beamsplitter-based microwave quantum repeater.

Significance. If the reported success probabilities were correct, the protocol would be a notable advance: it would provide a deterministic microwave repeater architecture that avoids beamsplitter interference, exploits autonomously error-corrected GKP memories, and uses only currently available cQED components. The paper contains useful analytical material, including explicit finite-energy GKP overlap expressions, a POVM treatment of homodyne detection, and a comparison with a beamsplitter-based alternative. However, the central quantitative claim is not supported as stated: the 'success probability' of entanglement generation is defined in Appendix E as a fidelity, and the tabulated values are inconsistent with the displayed fidelity expression. The headline comparison with the 1/2 beamsplitter ceiling therefore currently rests on a metric conflation.

major comments (3)
  1. [Appendix E, Eqs. (E4)-(E5) and Table I] Appendix E explicitly states: 'The success probability of the entanglement generation protocol is quantified by calculating the fidelity of the previous state against the target finite-energy Bell state' and then defines F via Eq. (E4). The quantity P_succ_EG in Table I is thus a fidelity, not a probability that an entanglement-generation attempt succeeds. This conflates two different objects: in Eq. (15), the key rate multiplies P_succ_EG by R_raw, and R_raw is itself derived from the final-state fidelity F; using F as P_succ_EG double-counts errors. In addition, the numerical values do not follow from Eq. (E5) with the stated parameters: for α=4e-4, β=0.90, η=0.6, Eq. (E5) gives F ≈ 0.43, not 0.75; for α=1e-3, β=0.80, η=0.6 it gives F ≈ 0.26, not 0.60. Finally, a true success probability for a single-photon wavepacket that must survive a channel of transmissivity η=0.6 is at most η (or
  2. [Section IV and Appendix F, P_succ_SW] The text states that the controlled-Z gate has a success probability approximately β^2 and that the homodyne measurement fidelity is 0.95, but the reported values P_succ_SW ≈ 0.58 (κ^-1=40 ms, β=0.90) and ≈ 0.36 (κ^-1=25 ms, β=0.80) correspond numerically to β^4 × (0.95)^2, not β^2 × (0.95)^2. No derivation of the β^4 factorization is provided. Furthermore, in a feedforward scheme all homodyne outcomes are accepted; the projective-measurement 'success probability' is a confidence or fidelity, not an erasure probability. Multiplying such a quantity into a rate is unjustified. The swapping success probability therefore has the same fidelity-versus-probability problem as entanglement generation.
  3. [Section I and Conclusion] The authors explicitly state that 'we assume error-free transmon operations' in both the introduction and the conclusion. This is an important limitation: the remote controlled-Z gate and the transmon measurements are the operative mechanism for entanglement generation, so the quoted P_succ_EG ≈ 0.75 is an idealized bound that excludes transmon gate and readout errors. Given that this number is the basis of the headline comparison, the idealization should be quantified (e.g., with typical gate-error rates) or clearly excluded from any claim about actual 'success probability' in the abstract.
minor comments (5)
  1. [Appendix E, Eq. (E4)] The display after the Bell state '|¯Φ−⟩Δ,AB' is missing a closing parenthesis or an 'and' before 'F=...'; as typeset, the equation is ambiguous.
  2. [Appendix F, Eq. (F1)] The term '− ¯0¯0¯1¯1⟩Δ,ABCD' is missing the opening '|' before '¯0'.
  3. [Appendix B, Eq. (B2)] The 2×2 matrix is written with an equality between the vector of output operators and the matrix, which is not the intended operation. The formatting should be corrected.
  4. [References] Reference [53] is listed as 'Szeg'; the correct spelling is 'Szegő'.
  5. [Section III.C] The sentence introducing P_succ_{¯0/¯1} ≈ 0.95 calls it a 'success probability' of detecting a logical codeword. This is the same fidelity-versus-probability ambiguity as in the main text and should be clarified.

Circularity Check

2 steps flagged

Headline 'success probabilities' are fidelities by definition; the comparison to the 1/2 beamsplitter bound is constructed.

specific steps
  1. self definitional [Appendix E, Eqs. (E4)-(E5); used in Sec. IV-A and Table I]
    "The success probability of the entanglement generation protocol is quantified by calculating the fidelity of the previous state against the target finite-energy Bell state | ¯Φ−⟩∆,AB = 1√2 (|¯0¯0⟩∆,AB − |¯1¯1⟩∆,AB F= Tr{ ∆,AB⟨ ¯Φ−|ρS∆,AB| ¯Φ−⟩∆,AB}. Consequently, the fidelity becomes ..."

    The paper defines P_succ_EG to be the fidelity F of Eq. (E5), not the probability that an entanglement-generation attempt succeeds. The quoted Table I values 0.60 and 0.75 are therefore fidelity values, not success probabilities. The abstract and Sec. IV-A then compare these F-values with BSMQR success probabilities (0.17, 0.32) and with the beamsplitter bound 1/2. The headline 'success probability approx. 0.75, surpassing 1/2' is thus true by construction only if 'success probability' is redefined as fidelity; it is not a comparison of like quantities.

  2. renaming known result [Sec. IV-A and Appendix F]
    "The controlled-Z gate has a success probability that approximately scales as the squared overlap β2 (see Appendix F). When κ−1damp = 25 ms, the overlap is β≈0.80, whereas when κ−1damp = 40 ms, β≈0.90 ... Accordingly, entanglement swapping success probability is PsuccSW ≈ 0.36, and ≈ 0.58 respectively."

    The controlled-Z gate is a deterministic unitary operation; its 'success probability' is not a probability of the swapping attempt succeeding but is set equal to β^2, which is a squared overlap of the finite-energy GKP codewords. This is multiplied by the homodyne measurement fidelity to obtain P_succ_SW. With feedforward, all four homodyne outcomes are accepted, so measurement imperfections should reduce the fidelity of the output state, not the probability of performing the swap. Thus the quoted 0.58 is a product of fidelities repackaged as a success probability, and the comparison to 1/2 is again a comparison of a fidelity-like quantity with a probability.

full rationale

The underlying numerical work is not circular: α, β, and η are taken from external GKP overlap formulas and reported experimental parameters, and the fidelity F(E5) is an independent calculation from those inputs. The circularity is in the central labelling: Appendix E explicitly defines the entanglement-generation 'success probability' as fidelity, and Sec. IV-A defines the swapping 'success probability' as β^2 times homodyne fidelity. Consequently, the abstract's headline claim that GBMQR reaches 'entanglement generation and swapping success probabilities approx. 0.75 and 0.58 respectively, surpassing ... 1/2' is a constructed comparison of redefined fidelities with a genuine success probability. This is a central reduction-by-definition, not merely a terminology slip, because the same P_succ_EG is then inserted into Eq. (15) and multiplied by R_raw, which already depends on F, double-counting the same fidelity. No load-bearing self-citation chain is present; [25,26] are not used to force the headline result. Score 6 reflects that the central claim reduces by construction, while the rest of the repeater analysis remains an independent performance estimate.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central numbers depend on experimental inputs (κ, η, homodyne fidelity), on the finite-energy GKP model, and on asserted success-probability factorizations. No machine-checked math or code is provided.

free parameters (4)
  • GKP envelope variance Δ = 0.3
    Assumed in the main text; sets the finite-energy overlap α, β that determine all reported success probabilities.
  • Channel transmissivity η = 0.60
    Chosen as 2 dB/km cryogenic transmission loss; directly enters P_succ_EG and the BSMQR comparison.
  • Stationary damping rate κ^{-1}_damp = 25–40 ms
    Taken from experimental demonstrations of autonomous GKP error correction; determines β and the generation/swapping success values.
  • Homodyne measurement success probability = 0.95
    Assumed from Ref. [24]; used in the P_succ_SW calculation as the measurement fidelity factor.
axioms (4)
  • domain assumption Error-free transmon operations and measurements
    Stated in Sec. I and the Conclusion; all quoted success probabilities assume ideal transmon gates and readout.
  • domain assumption Deterministic absorption/reflection of the single-photon wavepacket
    The remote controlled-Z gate is conditioned on successful round-trip absorption; loss branches are handled only through reduced-state expressions, not as a separately modeled failure probability.
  • ad hoc to paper Swapping success factorizes as β^4 × P_meas^2
    Sec. III.C and Appendix F state that the controlled-Z gate succeeds as β² and homodyne measurements as 0.95; the combined swapping success is asserted rather than derived step-by-step.
  • standard math Mehler-formula evaluation of GKP overlaps
    Appendix D uses Mehler's formula and Hermite-polynomial representation for finite-energy GKP codewords; this is standard, though the resulting expressions are long.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Gate-Based Microwave Quantum Repeater Via Grid-State Encoding." pith.science (2026). https://pith.science/paper/45VOFCRY

@misc{pith2026251219896,
  author       = {Pith},
  title        = {Pith review of: Gate-Based Microwave Quantum Repeater Via Grid-State Encoding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/45VOFCRY}},
  note         = {Machine review of arXiv:2512.19896}
}
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read the original abstract

In autonomous quantum error correction the lifetime of a logical bosonic qubit can be extended beyond its physical constituents without feedback measurements. Leveraging autonomous error correction, we propose a gate-based microwave quantum repeater (GBMQR) with encoded bosonic grid states. Each repeater station comprises a transmon and two bosonic resonators: one resonator serving as a stationary quantum memory utilizing autonomous error correction, and the other as an information bus for entanglement generation. Entanglement is generated sequentially through the successful absorption of a microwave photon wavepacket. This method enables deterministic entanglement generation, in contrast to a probabilistic mixing of two heralding signals on a balanced beamsplitter. Furthermore, our GBMQR employs an all-bosonic entanglement swapping Bell-state measurement. This is implemented via a bosonic controlled-Z gate and two separate X-basis projective homodyne measurements on the stationary stored codewords. Our approach circumvents mode-mismatch losses associated with routing and interfering of heralding modes on a beamsplitter, and confines losses to those arising from stationary storage. We evaluate the performance of the proposed quantum repeater by calculating its secret key rate under realistic lab environments. Moreover, we explicitly demonstrate that at stationary damping rate of $\kappa^{-1}_{\text{damp}}=$~\SI{40}{\milli\second}, GBMQR can achieve entanglement generation and swapping success probabilities approx.~$0.75$, and $0.58$ respectively, surpassing the hallmark success probability of $1/2$ set by ideal linear beamsplitter-based Bell-state measurements. The proposed device can be implemented using currently available superconducting microwave technology and is suited for secure chip-to-chip communication and distributed quantum computing.

Figures

Figures reproduced from arXiv: 2512.19896 by Hany Khalifa, Matti Silveri.

Figure 1
Figure 1. Figure 1: FIG. 1. A schematic of the proposed GBMQR. The figure depicts two repeater segments, where nodes A and D are the end [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Overlap between two logical [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. A comparison between GBMQR and BSMQR secret [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

Works this paper leans on

59 extracted references · 1 linked inside Pith

  1. [1]

    A. K. Ekert, Phys. Rev. Lett.67, 661 (1991)

    1991

  2. [2]

    C. H. Bennett, G. Brassard, C. Cr´ epeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett.70, 1895 (1993)

    1993

  3. [3]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys.89, 035002 (2017)

    2017

  4. [4]

    H. J. Kimble, Nature453, 1023 (2008)

    2008

  5. [5]

    Van Meter and S

    R. Van Meter and S. J. Devitt, Computer49, 31 (2016)

    2016

  6. [6]

    Sangouard, C

    N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, Rev. Mod. Phys.83, 33 (2011)

    2011

  7. [7]

    J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett.80, 3891 (1998)

    1998

  8. [8]

    L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature414, 413 (2001)

    2001

  9. [9]

    Azuma, K

    K. Azuma, K. Tamaki, and H.-K. Lo, Nat. Commun.6, 6787 (2015)

    2015

  10. [10]

    Jiang, J

    L. Jiang, J. M. Taylor, K. Nemoto, W. J. Munro, R. Van Meter, and M. D. Lukin, Phys. Rev. A79, 032325 (2009)

    2009

  11. [11]

    S. S. Chelluri, S. Sharma, F. Schmidt, S. V. Kusminskiy, and P. van Loock, arXiv:2503.21569 (2025)

    arXiv 2025

  12. [12]

    K. Noh, S. Girvin, and L. Jiang, Phys. Rev. Lett.125, 080503 (2020)

    2020

  13. [13]

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsiout- sios, S. Ganjam, A. Miano, B. Brock, A. Ding, L. Frunzio, et al., Nature616, 50 (2023)

    2023

  14. [14]

    B. L. Brock, S. Singh, A. Eickbusch, V. V. Sivak, A. Z. Ding, L. Frunzio, S. M. Girvin, and M. H. Devoret, Nature 641, 612 (2025)

    2025

  15. [15]

    Royer, S

    B. Royer, S. Singh, and S. Girvin, Phys. Rev. Lett.125, 260509 (2020)

    2020

  16. [16]

    Campagne-Ibarcq, E

    P. Campagne-Ibarcq, E. Zalys-Geller, A. Narla, S. Shankar, P. Reinhold, L. Burkhart, C. Axline, W. Pfaff, L. Frunzio, R. Schoelkopf,et al., Phys. Rev. Lett.120, 200501 (2018)

    2018

  17. [17]

    Narla, S

    A. Narla, S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, E. Zalys-Geller, S. O. Mundhada, W. Pfaff, L. Frun- zio, R. J. Schoelkopf,et al., Phys. Rev. X6, 031036 (2016)

    2016

  18. [18]

    Casariego, E

    M. Casariego, E. Z. Cruzeiro, S. Gherardini, T. Gonzalez- Raya, R. Andr´ e, G. Fraz˜ ao, G. Catto, M. M¨ ott¨ onen, D. Datta, K. Viisanen,et al., Quantum Sci. Technol.8, 023001 (2023)

    2023

  19. [19]

    E. J. Zhang, S. Srinivasan, N. Sundaresan, D. F. Bogorin, Y. Martin, J. B. Hertzberg, J. Timmerwilke, E. J. Pritch- ett, J.-B. Yau, C. Wang,et al., Sci. Adv.8, eabi6690 (2022)

    2022

  20. [20]

    W.-J. Lin, H. Cho, Y. Chen, M. G. Vavilov, C. Wang, and V. E. Manucharyan, PRX Quantum6, 010349 (2025)

    2025

  21. [21]

    Marxer, J

    F. Marxer, J. Mro˙ zek, J. Andersson, L. Abdurakhimov, J. Adam, V. Bergholm, R. Beriwal, C. F. Chan, S. Dahl, S. R. Das,et al., arXiv:2508.16437 (2025)

    Pith/arXiv arXiv 2025

  22. [22]

    E. T. Holland, B. Vlastakis, R. W. Heeres, M. J. Reagor, U. Vool, Z. Leghtas, L. Frunzio, G. Kirchmair, M. H. Devoret, M. Mirrahimi,et al., Phys. Rev. Lett.115, 180501 (2015)

    2015

  23. [23]

    Baskov, D

    R. Baskov, D. K. Weiss, and S. M. Girvin, arXiv:2501.07784 (2025)

    arXiv 2025

  24. [24]

    Strandberg, A

    I. Strandberg, A. M. Eriksson, B. Royer, M. Kervinen, and S. Gasparinetti, Phys. Rev. Lett.133, 063601 (2024)

    2024

  25. [25]

    Khalifa,Microwave quantum communications: new ap- proaches to sensing and mitigation of the bosonic pure-loss channel, Ph.D

    H. Khalifa,Microwave quantum communications: new ap- proaches to sensing and mitigation of the bosonic pure-loss channel, Ph.D. thesis, Aalto University, Espoo, Finland (2024)

    2024

  26. [26]

    Khalifa, R

    H. Khalifa, R. J¨ antti, and G. S. Paraoanu, IEEE Trans. Quantum Eng.5, 4100917 (2024)

    2024

  27. [27]

    H¨ aussler and P

    S. H¨ aussler and P. van Loock, Phys. Rev. A111, 062611 (2025)

    2025

  28. [28]

    Kurpiers, P

    P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal, J. Heinsoo, Y. Salath´ e, A. Akin, S. Storz, J.-C. Besse, et al., Nature558, 264 (2018)

    2018

  29. [29]

    Kn¨ orzer, X

    J. Kn¨ orzer, X. Liu, B. F. Schiffer, and J. Tura, arXiv:2510.15630 (2025)

    arXiv 2025

  30. [30]

    LaRacuente, K

    N. LaRacuente, K. N. Smith, P. Imany, K. L. Silverman, and F. T. Chong, Quantum9, 1581 (2025)

    2025

  31. [31]

    A. P. Babu, O. Kerppo, A. Mu˜ noz-Moller, M. Haghparast, and M. Silveri, Quantum Sci. Technol.10, 035004 (2025)

    2025

  32. [32]

    A. L. Grimsmo and S. Puri, PRX Quantum2, 020101 (2021)

    2021

  33. [33]

    C. C. Gerry and P. L. Knight,Introductory quantum optics (Cambridge university press, Cambridge, UK, 2023)

    2023

  34. [34]

    Walls and G

    D. Walls and G. J. Milburn,Quantum Optics(Springer, Berlin, 2025)

    2025

  35. [35]

    Y. Liu, S. Singh, K. C. Smith, E. Crane, J. M. Martyn, A. Eickbusch, A. Schuckert, R. D. Li, J. Sinanan-Singh, M. B. Soley, T. Tsunoda, I. L. Chuang, N. Wiebe, and S. M. Girvin, PRX Quantum 10.1103/4rf7-9tfx (2025)

    doi:10.1103/4rf7-9tfx 2025

  36. [36]

    Kurpiers, M

    P. Kurpiers, M. Pechal, B. Royer, P. Magnard, T. Walter, J. Heinsoo, Y. Salath´ e, A. Akin, S. Storz, J.-C. Besse, et al., Phys. Rev. Appl.12, 044067 (2019)

    2019

  37. [37]

    Ilves, S

    J. Ilves, S. Kono, Y. Sunada, S. Yamazaki, M. Kim, K. Koshino, and Y. Nakamura, npj Quantum Inf.6, 34 (2020). ix

    2020

  38. [38]

    McIntyre and W

    Z. McIntyre and W. Coish, Phys. Rev. Research7, 023255 (2025)

    2025

  39. [39]

    Furusawa and P

    A. Furusawa and P. Van Loock,Quantum teleportation and entanglement: a hybrid approach to optical quantum information processing(John Wiley & Sons, Weinheim, 2011)

    2011

  40. [40]

    Gisin, G

    N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys.74, 145 (2002)

    2002

  41. [41]

    Scarani, H

    V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Duˇ sek, N. L¨ utkenhaus, and M. Peev, Rev. Mod. Phys. 81, 1301 (2009)

    2009

  42. [42]

    Pirandola, U

    S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Vil- loresi, and P. Wallden, Adv. Opt. Photon.12, 1012 (2020)

    2020

  43. [43]

    Hong, Z.-Y

    C.-K. Hong, Z.-Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987)

    2044

  44. [44]

    Weihs and A

    G. Weihs and A. Zeilinger, inCoherence and Statistics of Photons and Atoms, edited by J. Peˇ rina (Wiley, New Jersey, 2001) pp. 262–288

    2001

  45. [45]

    Bouchard, A

    F. Bouchard, A. Sit, Y. Zhang, R. Fickler, F. M. Miatto, Y. Yao, F. Sciarrino, and E. Karimi, Rep. Prog. Phys.84, 012402 (2020)

    2020

  46. [46]

    Y. Y. Gao, B. J. Lester, Y. Zhang, C. Wang, S. Rosenblum, L. Frunzio, L. Jiang, S. Girvin, and R. J. Schoelkopf, Physical Review X8, 021073 (2018)

    2018

  47. [47]

    Storz, J

    S. Storz, J. Sch¨ ar, A. Kulikov, P. Magnard, P. Kurpiers, J. L¨ utolf, T. Walter, A. Copetudo, K. Reuer, A. Akin, et al., Nature617, 265 (2023)

    2023

  48. [48]

    Takeoka, S

    M. Takeoka, S. Guha, and M. M. Wilde, Nat. Commun. 5, 5235 (2014)

    2014

  49. [49]

    Pirandola, R

    S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, Nat. Commun.8, 15043 (2017)

    2017

  50. [50]

    J. Y. Qiu, A. Grimsmo, K. Peng, B. Kannan, B. Lienhard, Y. Sung, P. Krantz, V. Bolkhovsky, G. Calusine, D. Kim, et al., Nature Physics19, 706 (2023)

    2023

  51. [51]

    Brecht, D

    B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, Phys. Rev. X5, 041017 (2015)

    2015

  52. [52]

    Pechal, L

    M. Pechal, L. Huthmacher, C. Eichler, S. Zeytino˘ glu, A. Abdumalikov Jr, S. Berger, A. Wallraff, and S. Filipp, Phys. Rev. X4, 041010 (2014)

    2014

  53. [53]

    Szeg,Orthogonal polynomials, Vol

    G. Szeg,Orthogonal polynomials, Vol. 23 (American Math- ematical Soc., 1939)

    1939

  54. [54]

    Gottesman,Surviving as a quantum computer in a classical world(Textbook manuscript preprint, 2024)

    D. Gottesman,Surviving as a quantum computer in a classical world(Textbook manuscript preprint, 2024)

    2024

  55. [55]

    J. H. Shapiro, Quantum optical communication, MIT OpenCourseWare lecture notes (2016)

    2016

  56. [56]

    Glancy and E

    S. Glancy and E. Knill, Phys. Rev. A73, 012325 (2006). Appendix A: Photon temporal basis Temporal modes (TMs) form a complete orthonormal basis for single-photon states [ 51]. Although the concept of TMs can be extended to more complex photonic states, this study focuses primarily on single-photon temporal modes. Given a temporal mode shape, the wavepacke...

    2006

  57. [57]

    Controlled-Z joint operation A bosonic controlled-Z operation is defined as e−iQcQt . The action of the gate on logical states is shown as e−iQcQt |¯0⟩c|¯0⟩t =e −iQcQt X k∈Z |2k√π⟩qc ⊗ X k′∈Z |2k′√π⟩qt = X k∈Z X k′∈Z (e−i4π)kk′ |2k√π⟩qc ⊗ |2k′√π⟩qt =| ¯0⟩|¯0⟩, e−iQcQt |¯0⟩|¯1⟩ =e −iQcQt X k∈Z |2k√π⟩qc ⊗ X k′∈Z |(2k′ + 1)√π⟩qt = X k∈Z X k′∈Z (e−i2π)k(e−i4π...

  58. [58]

    However, losses due to couplings, detection inefficiency, and mode mismatch decrease the indistin- guishability between the two interfering codewords

    Coupling and mode-mismatch errors In a beamsplitter-based Bell-state measurement, two identical GKP codewords are routed to a coupling balanced beamsplitter. However, losses due to couplings, detection inefficiency, and mode mismatch decrease the indistin- guishability between the two interfering codewords. In order to quantify this error, the overlap bet...

  59. [59]

    For projective homodyne detection, an operator description of this process is captured by the continuous variable positive operator-valued measure (POVM) for- malism

    Imperfect projective measurement inGKPbasis The next step in the swapping process is to perform two projective homodyne measurements on the middle modes. For projective homodyne detection, an operator description of this process is captured by the continuous variable positive operator-valued measure (POVM) for- malism. As defined earlier, the GKP logical ...

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.