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Composable logical gate error in approximate quantum error correction: reexamining gate implementations in Gottesman-Kitaev-Preskill codes

Lukas Brenner, Beatriz Dias, and Robert Koenig

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AbstractQuantifying the accuracy of logical gates is paramount in approximate error correction, where perfect implementations are often unachievable with the available set of physical operations. To this end, we introduce a single scalar quantity we call the (composable) logical gate error. It captures both the deviation of the logical action from the desired target gate as well as leakage out of the code space. It is subadditive under successive application of gates, providing a simple means for analyzing circuits. We show how to bound the composable logical gate error in terms of matrix elements of physical unitaries between (approximate) logical computational basis states. In the continuous-variable context, this sidesteps the need for computing energy-bounded norms. As an example, we study the composable logical gate error for linear optics implementations of Paulis and Cliffords in approximate Gottesman-Kitaev-Preskill (GKP) codes. We find that the logical gate error for implementations of Paulis depends linearly on the squeezing parameter. This implies that their accuracy improves monotonically with the amount of squeezing. For some Cliffords, however, linear optics implementations which are exact for ideal GKP codes fail in the approximate case: they have a constant logical gate error even in the limit of infinite squeezing. This is consistent with previous results about the limitations of certain gate implementations for approximate GKP codes. It shows that findings applicable to ideal GKP codes do not always translate to the realm of physically realizable approximate GKP codes.► BibTeX data@article{Brenner2026composablelogical, doi = {10.22331/q-2026-10-08-2230}, url = {https://doi.org/10.22331/q-2026-10-08-2230}, title = {Composable logical gate error in approximate quantum error correction: reexamining gate implementations in {G}ottesman-{K}itaev-{P}reskill codes}, author = {Brenner, Lukas and Dias, Beatriz and Koenig, Robert}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2230}, month = oct, year = {2026} }► References [1] Ivan Rojkov, Paul Moser Röggla, Martin Wagener, Moritz Fontboté-Schmidt, Stephan Welte, Jonathan Home, and Florentin Reiter. Two-qubit operations for finite-energy Gottesman-Kitaev-Preskill encodings. Phys. Rev. Lett., 133:100601, Sep 2024. doi:10.1103/PhysRevLett.133.100601. https://doi.org/10.1103/PhysRevLett.133.100601 [2] Jacob Hastrup, Mikkel V. Larsen, Jonas S. Neergaard-Nielsen, Nicolas C. Menicucci, and Ulrik L. Andersen. Unsuitability of cubic phase gates for non-Clifford operations on Gottesman-Kitaev-Preskill states. Phys. Rev. A, 103:032409, Mar 2021. doi:10.1103/PhysRevA.103.032409. https://doi.org/10.1103/PhysRevA.103.032409 [3] Hector Bombin and Miguel A. Martin-Delgado. Topological quantum distillation. Phys. Rev. Lett., 97:180501, Oct 2006. doi:10.1103/PhysRevLett.97.180501. https://doi.org/10.1103/PhysRevLett.97.180501 [4] Marcel Bergmann and Peter van Loock. Quantum error correction against photon loss using multicomponent cat states. Phys. Rev. A, 94:042332, Oct 2016. doi:10.1103/PhysRevA.94.042332. https://doi.org/10.1103/PhysRevA.94.042332 [5] Linshu Li, Chang-Ling Zou, Victor V. Albert, Sreraman Muralidharan, S. M. Girvin, and Liang Jiang. Cat codes with optimal decoherence suppression for a lossy bosonic channel. Phys. Rev. Lett., 119:030502, Jul 2017. doi:10.1103/PhysRevLett.119.030502. https://doi.org/10.1103/PhysRevLett.119.030502 [6] Zaki Leghtas, Gerhard Kirchmair, Brian Vlastakis, Robert J. Schoelkopf, Michel H. Devoret, and Mazyar Mirrahimi. Hardware-efficient autonomous quantum memory protection. Phys. Rev. Lett., 111:120501, Sep 2013. doi:10.1103/PhysRevLett.111.120501. https://doi.org/10.1103/PhysRevLett.111.120501 [7] Jacob Hastrup and Ulrik Lund Andersen. All-optical cat-code quantum error correction. Phys. Rev. Res., 4:043065, Oct 2022. doi:10.1103/PhysRevResearch.4.043065. https://doi.org/10.1103/PhysRevResearch.4.043065 [8] Timothy C. Ralph, Alexei Gilchrist, Gerard J. Milburn, William J. Munro, and Scott Glancy. Quantum computation with optical coherent states. Phys. Rev. A, 68:042319, Oct 2003. doi:10.1103/PhysRevA.68.042319. https://doi.org/10.1103/PhysRevA.68.042319 [9] Andreas Winter. Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities, Dec 2017. doi:10.48550/arXiv.1712.10267. https://doi.org/10.48550/arXiv.1712.10267 [10] Maksim E. Shirokov. On the energy-constrained diamond norm and its application in quantum information theory. Probl. Inf. Transm., 54(1):20–33, Jan 2018. doi:10.1134/s0032946018010027. https://doi.org/10.1134/s0032946018010027 [11] Robert Koenig and Cambyse Rouzé. Limitations of local update recovery in stabilizer-GKP codes: a quantum optimal transport approach, Sep 2023. doi:10.48550/arXiv.2309.16241. https://doi.org/10.48550/arXiv.2309.16241 [12] Takaya Matsuura, Nicolas C. Menicucci, and Hayata Yamasaki. Continuous-variable fault-tolerant quantum computation under general noise. Nature Communications, 17:1709, Feb 2026. doi:10.1038/s41467-026-69036-5. https://doi.org/10.1038/s41467-026-69036-5 [13] Daniel Gottesman, Alexei Kitaev, and John Preskill. 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Schoelkopf, Luigi Frunzio, Mazyar Mirrahimi, and Michel H. Devoret. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584:368–372, Aug 2020. doi:10.1038/s41586-020-2603-3. https://doi.org/10.1038/s41586-020-2603-3 [17] Maxime Boissonneault, Jay M. Gambetta, and Alexandre Blais. Dispersive regime of circuit QED: Photon-dependent qubit dephasing and relaxation rates. Phys. Rev. A, 79:013819, Jan 2009. doi:10.1103/PhysRevA.79.013819. https://doi.org/10.1103/PhysRevA.79.013819 [18] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin. Hybrid oscillator-qubit quantum processors: Instruction set architectures, abstract machine models, and applications. PRX Quantum, 7:010201, Jan 2026. doi:10.1103/4rf7-9tfx. https://doi.org/10.1103/4rf7-9tfx [19] John Watrous. The Theory of Quantum Information.

Cambridge University Press, Apr 2018. doi:https://doi.org/10.1017/9781316848142. https://doi.org/10.1017/9781316848142 [20] Daniel Kressner, Ding Lu, and Bart Vandereycken. Subspace acceleration for the crawford number and related eigenvalue optimization problems. SIAM Journal on Matrix Analysis and Applications, 39(2):961–982, 2018. doi:10.1137/17M1127545. https://doi.org/10.1137/17M1127545 [21] Lukas Brenner, Libor Caha, Xavier Coiteux-Roy, and Robert Koenig. Complexity of gottesman-kitaev-preskill states. Physical Review X, 15(3), 2025. doi:10.1103/4ww5-4yww. https://doi.org/10.1103/4ww5-4yww [22] Barbara M. Terhal and Daniel J. Weigand. Encoding a qubit into a cavity mode in circuit QED using phase estimation. Phys. Rev. A, 93(1), Jan 2016. doi:10.1103/physreva.93.012315. https://doi.org/10.1103/physreva.93.012315 [23] Kasper Duivenvoorden, Barbara M. Terhal, and Daniel Weigand. Single-mode displacement sensor. Phys. Rev. A, 95:012305, Jan 2017. doi:10.1103/PhysRevA.95.012305. https://doi.org/10.1103/PhysRevA.95.012305 [24] Daniel J. Weigand and Barbara M. Terhal. Generating grid states from Schrödinger-cat states without postselection. Phys. Rev. A, 97(2), Feb 2018. doi:10.1103/physreva.97.022341. https://doi.org/10.1103/physreva.97.022341 [25] Jacob Hastrup, Kimin Park, Jonatan Bohr Brask, Radim Filip, and Ulrik Lund Andersen. Measurement-free preparation of grid states. npj Quantum Inf., 7(1), Jan 2021. doi:10.1038/s41534-020-00353-3. https://doi.org/10.1038/s41534-020-00353-3 [26] Lukas Brenner, Libor Caha, Xavier Coiteux-Roy, and Robert Koenig. Factoring an integer with three oscillators and a qubit. Nature Communications, 17(1), December 2025. doi:10.1038/s41467-025-67694-5. https://doi.org/10.1038/s41467-025-67694-5 [27] Takaya Matsuura, Hayata Yamasaki, and Masato Koashi. Equivalence of approximate Gottesman-Kitaev-Preskill codes. Physical Review A, 102(3), Sept 2020. doi:10.1103/physreva.102.032408. https://doi.org/10.1103/physreva.102.032408 [28] Nicolas C. Menicucci. Fault-tolerant measurement-based quantum computing with continuous-variable cluster states. Phys. Rev. Lett., 112:120504, Mar 2014. doi:10.1103/PhysRevLett.112.120504. https://doi.org/10.1103/PhysRevLett.112.120504 [29] Alexei Yu Kitaev, Aleksandr H. Shen, and Mikhail N. Vyalyi. Classical and Quantum Computation.

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The Journal of Chemical Physics, 18(3):365–375, 1950. doi:10.1063/1.1747632. https://doi.org/10.1063/1.1747632 [40] Istvan Mayer. Simple Theorems, Proofs, and Derivations in Quantum Chemistry. Kluwer Academic / Plenum Publishers, New York, NY, 2003. doi:10.1007/978-1-4757-6519-9. https://doi.org/10.1007/978-1-4757-6519-9Cited by[1] Lukas Brenner, Beatriz Dias, and Robert Koenig, "Trading modes against energy", arXiv:2509.18854, (2025). [2] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026). [3] Lukas Brenner, Beatriz Dias, and Robert Koenig, "Qubit-oscillator-based gate implementations for approximate Gottesman-Kitaev-Preskill codes", Physical Review A 113 4, 042447 (2026). [4] Yixu Wang, Yijia Xu, and Zi-Wen Liu, "Approximate quantum error correction theory of non-isometric codes", arXiv:2606.13559, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-10-11 14:11:02). The list may be incomplete as not all publishers provide suitable and complete citation data.On Crossref's cited-by service no data on citing works was found (last attempt 2026-10-11 14:10:58).This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractQuantifying the accuracy of logical gates is paramount in approximate error correction, where perfect implementations are often unachievable with the available set of physical operations. To this end, we introduce a single scalar quantity we call the (composable) logical gate error. It captures both the deviation of the logical action from the desired target gate as well as leakage out of the code space. It is subadditive under successive application of gates, providing a simple means for analyzing circuits. We show how to bound the composable logical gate error in terms of matrix elements of physical unitaries between (approximate) logical computational basis states. In the continuous-variable context, this sidesteps the need for computing energy-bounded norms. As an example, we study the composable logical gate error for linear optics implementations of Paulis and Cliffords in approximate Gottesman-Kitaev-Preskill (GKP) codes. We find that the logical gate error for implementations of Paulis depends linearly on the squeezing parameter. This implies that their accuracy improves monotonically with the amount of squeezing. For some Cliffords, however, linear optics implementations which are exact for ideal GKP codes fail in the approximate case: they have a constant logical gate error even in the limit of infinite squeezing. This is consistent with previous results about the limitations of certain gate implementations for approximate GKP codes. It shows that findings applicable to ideal GKP codes do not always translate to the realm of physically realizable approximate GKP codes.► BibTeX data@article{Brenner2026composablelogical, doi = {10.22331/q-2026-10-08-2230}, url = {https://doi.org/10.22331/q-2026-10-08-2230}, title = {Composable logical gate error in approximate quantum error correction: reexamining gate implementations in {G}ottesman-{K}itaev-{P}reskill codes}, author = {Brenner, Lukas and Dias, Beatriz and Koenig, Robert}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2230}, month = oct, year = {2026} }► References [1] Ivan Rojkov, Paul Moser Röggla, Martin Wagener, Moritz Fontboté-Schmidt, Stephan Welte, Jonathan Home, and Florentin Reiter. Two-qubit operations for finite-energy Gottesman-Kitaev-Preskill encodings. Phys. Rev. Lett., 133:100601, Sep 2024. doi:10.1103/PhysRevLett.133.100601. https://doi.org/10.1103/PhysRevLett.133.100601 [2] Jacob Hastrup, Mikkel V. Larsen, Jonas S. Neergaard-Nielsen, Nicolas C. Menicucci, and Ulrik L. Andersen. Unsuitability of cubic phase gates for non-Clifford operations on Gottesman-Kitaev-Preskill states. Phys. Rev. A, 103:032409, Mar 2021. doi:10.1103/PhysRevA.103.032409. https://doi.org/10.1103/PhysRevA.103.032409 [3] Hector Bombin and Miguel A. Martin-Delgado. Topological quantum distillation. Phys. Rev. Lett., 97:180501, Oct 2006. doi:10.1103/PhysRevLett.97.180501. https://doi.org/10.1103/PhysRevLett.97.180501 [4] Marcel Bergmann and Peter van Loock. Quantum error correction against photon loss using multicomponent cat states. Phys. Rev. A, 94:042332, Oct 2016. doi:10.1103/PhysRevA.94.042332. https://doi.org/10.1103/PhysRevA.94.042332 [5] Linshu Li, Chang-Ling Zou, Victor V. Albert, Sreraman Muralidharan, S. M. Girvin, and Liang Jiang. Cat codes with optimal decoherence suppression for a lossy bosonic channel. Phys. Rev. Lett., 119:030502, Jul 2017. doi:10.1103/PhysRevLett.119.030502. https://doi.org/10.1103/PhysRevLett.119.030502 [6] Zaki Leghtas, Gerhard Kirchmair, Brian Vlastakis, Robert J. Schoelkopf, Michel H. Devoret, and Mazyar Mirrahimi. Hardware-efficient autonomous quantum memory protection. Phys. Rev. Lett., 111:120501, Sep 2013. doi:10.1103/PhysRevLett.111.120501. https://doi.org/10.1103/PhysRevLett.111.120501 [7] Jacob Hastrup and Ulrik Lund Andersen. All-optical cat-code quantum error correction. Phys. Rev. Res., 4:043065, Oct 2022. doi:10.1103/PhysRevResearch.4.043065. https://doi.org/10.1103/PhysRevResearch.4.043065 [8] Timothy C. Ralph, Alexei Gilchrist, Gerard J. Milburn, William J. Munro, and Scott Glancy. Quantum computation with optical coherent states. Phys. Rev. A, 68:042319, Oct 2003. doi:10.1103/PhysRevA.68.042319. https://doi.org/10.1103/PhysRevA.68.042319 [9] Andreas Winter. Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities, Dec 2017. doi:10.48550/arXiv.1712.10267. https://doi.org/10.48550/arXiv.1712.10267 [10] Maksim E. Shirokov. On the energy-constrained diamond norm and its application in quantum information theory. Probl. Inf. Transm., 54(1):20–33, Jan 2018. doi:10.1134/s0032946018010027. https://doi.org/10.1134/s0032946018010027 [11] Robert Koenig and Cambyse Rouzé. Limitations of local update recovery in stabilizer-GKP codes: a quantum optimal transport approach, Sep 2023. doi:10.48550/arXiv.2309.16241. https://doi.org/10.48550/arXiv.2309.16241 [12] Takaya Matsuura, Nicolas C. Menicucci, and Hayata Yamasaki. Continuous-variable fault-tolerant quantum computation under general noise. Nature Communications, 17:1709, Feb 2026. doi:10.1038/s41467-026-69036-5. https://doi.org/10.1038/s41467-026-69036-5 [13] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A, 64:012310, Jun 2001. doi:10.1103/PhysRevA.64.012310. https://doi.org/10.1103/PhysRevA.64.012310 [14] Lukas Brenner, Beatriz Dias, and Robert Koenig. Qubit-oscillator-based gate implementations for approximate Gottesman-Kitaev-Preskill codes. Phys. Rev. A, 113:042447, Apr 2026. doi:10.1103/x758-5lc2. https://doi.org/10.1103/x758-5lc2 [15] Alec Eickbusch, Volodymyr Sivak, Andy Z. Ding, Salvatore S. Elder, Shantanu R. Jha, Jayameenakshi Venkatraman, Baptiste Royer, S. M. Girvin, Robert J. Schoelkopf, and Michel H. Devoret. Fast universal control of an oscillator with weak dispersive coupling to a qubit. Nat. Phys., 18(12):1464–1469, Oct 2022. doi:10.1038/s41567-022-01776-9. https://doi.org/10.1038/s41567-022-01776-9 [16] Philippe Campagne-Ibarcq, Alec Eickbusch, Steven Touzard, Evan Zalys-Geller, Nicholas E. Frattini, Volodymyr V. Sivak, Philip Reinhold, Shruti Puri, Shyam Shankar, Robert J. Schoelkopf, Luigi Frunzio, Mazyar Mirrahimi, and Michel H. Devoret. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584:368–372, Aug 2020. doi:10.1038/s41586-020-2603-3. https://doi.org/10.1038/s41586-020-2603-3 [17] Maxime Boissonneault, Jay M. Gambetta, and Alexandre Blais. Dispersive regime of circuit QED: Photon-dependent qubit dephasing and relaxation rates. Phys. Rev. A, 79:013819, Jan 2009. doi:10.1103/PhysRevA.79.013819. https://doi.org/10.1103/PhysRevA.79.013819 [18] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin. Hybrid oscillator-qubit quantum processors: Instruction set architectures, abstract machine models, and applications. PRX Quantum, 7:010201, Jan 2026. doi:10.1103/4rf7-9tfx. https://doi.org/10.1103/4rf7-9tfx [19] John Watrous. The Theory of Quantum Information.

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Composable logical gate error in approximate quantum error correction: reexamining gate implementations in Gottesman-Kitaev-Preskill codes

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