Fiber Bundle Fault Tolerance of GKP Codes
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AbstractWe investigate multi-mode GKP (Gottesman–Kitaev–Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman–Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.Featured image: Illustration of fault tolerant Gaussian operations on GKP codes as topologically non-trivial paths on the space of GKP codes. Top Left: A GKP stabilizer is generated by a set of $2N$ displacement operators with phases. Bottom Left: The stabilizer gives rise to a lattice in phase space. Pictured is the 2-dimensional hexagonal lattice. The unit cell area of this lattice must be a multiple of $2\pi$. The shown area of $4 \pi$ corresponds to an encoded qubit. Top Right: The GKP moduli space is a union of tori, each corresponding to a choice of phases for a given point in the lattice moduli space. a) Two paths on the code manifold are illustrated, one wrapping nontrivially around a fiber. Bottom Right: In the single mode case (N=1) the space of lattices is given by a three sphere $S^3$ with an embedded trefoil knot (b) removed. The three-sphere is pictured as two filled balls with surfaces identified. c) GKP code paths whose projections interlink with the trefoil knot can give rise to nontrivial logical Clifford action (up to Paulis). A contractible path necessarily produces trivial Clifford logical action. The Pauli part of the logical action is specified by how a path wraps around the fibers over the projected path. Popular summaryFault tolerance is a notion of fundamental importance to the field of quantum information processing. It is one of the central properties a quantum computer must posses in order to enable the achievement of large scale practical quantum computation. Intuitively it is clear what it means for a computer to be fault tolerant: in the presence of sufficiently weak noise with sufficiently low correlations, arbitrarily long and precise computations are possible. While a widely used, general, and intuitive concept, within the literature the term fault tolerant is often applied to specific procedures in an ad-hoc fashion tailored to details of the context or platform under discussion. On the contrary in previous work Gottesman and Zhang conjectured that all types of fault-tolerant gates can in fact be regarded as topological, so that a unifying definition of fault tolerance becomes possible. The main contribution of this work is to prove the Gottesman–Zhang conjecture for logical Clifford operations on arbitrary multi-mode GKP codes, a set of gates physically implemented by Gaussian unitary operations. This constitutes the first class of continuous-variable codes for which the conjecture has been shown to be true, highlighting the topological nature of fault-tolerant operations. The paper also contributes to the theory of multi-mode GKP codes and more generally the theory of fault tolerance in bosonic codes, a class of technologically promising codes, by providing a canonical form for GKP codes that avoids ambiguities present in prior work. Both of these contributions combine in a natural way resulting in a three-dimensional visualization of fault-tolerant logical gates as paths that interlink nontrivially with a trefoil knot.► BibTeX data@article{Burchards2025fiberbundlefault, doi = {10.22331/q-2025-10-29-1899}, url = {https://doi.org/10.22331/q-2025-10-29-1899}, title = {Fiber {B}undle {F}ault {T}olerance of {GKP} {C}odes}, author = {Burchards, Ansgar G. and Flammia, Steven T. and Conrad, Jonathan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1899}, month = oct, year = {2025} }► References [1] Peter W. Shor ``Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer'' SIAM Journal on Computing 26, 1484-1509 (1997). https://doi.org/10.1137/s0097539795293172 [2] Dorit Aharonovand Michael Ben-Or ``Fault-Tolerant Quantum Computation with Constant Error Rate'' SIAM Journal on Computing 38, 1207–1282 (2008). https://doi.org/10.1137/S0097539799359385 [3] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek, ``Resilient quantum computation: error models and thresholds'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365–384 (1998). https://doi.org/10.1098/rspa.1998.0166 [4] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' (1997). https://doi.org/10.48550/arXiv.quant-ph/9705052 https://arxiv.org/abs/quant-ph/9705052 [5] A.Yu. Kitaev ``Fault-tolerant quantum computation by anyons'' Annals of Physics 303, 2–30 (2003). https://doi.org/10.1016/s0003-4916(02)00018-0 [6] Daniel Gottesman, Alexei Kitaev, and John Preskill, ``Encoding a qubit in an oscillator'' Physical Review A 64 (2001). https://doi.org/10.1103/physreva.64.012310 [7] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd, ``Gaussian quantum information'' Reviews of Modern Physics 84, 621–669 (2012). https://doi.org/10.1103/revmodphys.84.621 [8] Daniel Gottesmanand Lucy Liuxuan Zhang ``Fibre bundle framework for unitary quantum fault tolerance''. https://doi.org/10.48550/arXiv.1309.7062 arXiv:1309.7062 [9] Jonathan Conrad, Ansgar G. Burchards, and Steven T. Flammia, ``Lattices, Gates, and Curves: GKP codes as a Rosetta stone'' (2024). https://doi.org/10.48550/arXiv.2407.03270 arXiv:2407.03270 [10] Christopher Gerryand Peter Knight ``Introductory Quantum Optics'' Cambridge University Press (2004). https://doi.org/10.1017/CBO9780511791239 [11] J. Conwayand N. Sloane ``Sphere packings, lattices and groups'' Springer, New York, NY (1988). https://doi.org/10.1007/978-1-4757-6568-7 [12] John C. Baezand Javier P. Muniain ``Gauge Fields, Knots And Gravity'' World Scientific Publishing Company (1994). https://doi.org/10.1142/2324 [13] Theodore Frankel ``The Geometry of Physics: An Introduction'' Cambridge University Press (2011). https://doi.org/10.1017/CBO9781139061377 https://www.cambridge.org/core/books/geometry-of-physics/94894F70DB22055BD7BC2B84C135ABAF [14] Tommaso Guaita, Lucas Hackl, and Thomas Quella, ``Representation theory of Gaussian unitary transformations for bosonic and fermionic systems'' (2024). https://doi.org/10.48550/arXiv.2409.11628 arXiv:2409.11628 [15] Jonathan Conrad, Jens Eisert, and Francesco Arzani, ``Gottesman-Kitaev-Preskill codes: A lattice perspective'' Quantum 6, 648 (2022). https://doi.org/10.22331/q-2022-02-10-648 [16] Emanuel Knill, Raymond Laflamme, and Lorenza Viola, ``Theory of Quantum Error Correction for General Noise'' Physical Review Letters 84, 2525–2528 (2000). https://doi.org/10.1103/physrevlett.84.2525 [17] Baptiste Royer, Shraddha Singh, and S.M. Girvin, ``Encoding Qubits in Multimode Grid States'' PRX Quantum 3 (2022). https://doi.org/10.1103/prxquantum.3.010335 [18] J. W. Harrington ``Analysis of quantum error-correcting codes: Symplectic lattice codes and toric codes'' thesis (2004). https://doi.org/10.7907/AHMQ-EG82 https://resolver.caltech.edu/CaltechETD:etd-05122004-113132 [19] Jim Harringtonand John Preskill ``Achievable rates for the Gaussian quantum channel'' Physical Review A 64 (2001). https://doi.org/10.1103/physreva.64.062301 [20] Christina Birkenhakeand Herbert Lange ``Complex Abelian Varieties'' Springer Berlin Heidelberg (2004). https://doi.org/10.1007/978-3-662-06307-1 [21] John Milnor ``Introduction to Algebraic K-Theory. (AM-72), Volume 72'' Princeton University Press (2016). https://doi.org/10.1515/9781400881796 [22] Etienne Ghys ``Lorenz and Modular Flows: A Visual Introduction'' (2006). http://www.ams.org/publicoutreach/feature-column/fcarc-lorenz [23] Baptiste Royer, Shraddha Singh, and S.M. Girvin, ``Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States'' Physical Review Letters 125 (2020). https://doi.org/10.1103/physrevlett.125.260509 [24] V. G. Matsos, C. H. Valahu, M. J. Millican, T. Navickas, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, ``Universal Quantum Gate Set for Gottesman-Kitaev-Preskill Logical Qubits'' (2024). https://doi.org/10.48550/arXiv.2409.05455 arXiv:2409.05455 [25] Shubham P. Jain, Joseph T. Iosue, Alexander Barg, and Victor V. Albert, ``Quantum spherical codes'' Nature Physics 20, 1300–1305 (2024). https://doi.org/10.1038/s41567-024-02496-y [26] Aurélie Denysand Anthony Leverrier ``Quantum Error-Correcting Codes with a Covariant Encoding'' Physical Review Letters 133 (2024). https://doi.org/10.1103/physrevlett.133.240603 [27] N. Bourbaki ``Algébre'' Springer Berlin Heidelberg (2007). https://doi.org/10.1007/978-3-540-35339-3 [28] G. Frobenius ``Theorie der linearen Formen mit ganzen Coefficienten.'' Journal für die reine und angewandte Mathematik 86, 146–208 (1879). https://doi.org/10.1515/crll.1879.86.146 http://eudml.org/doc/148392Cited byCould not fetch Crossref cited-by data during last attempt 2025-10-29 13:13:02: Could not fetch cited-by data for 10.22331/q-2025-10-29-1899 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-10-29 13:13:08: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractWe investigate multi-mode GKP (Gottesman–Kitaev–Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman–Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.Featured image: Illustration of fault tolerant Gaussian operations on GKP codes as topologically non-trivial paths on the space of GKP codes. Top Left: A GKP stabilizer is generated by a set of $2N$ displacement operators with phases. Bottom Left: The stabilizer gives rise to a lattice in phase space. Pictured is the 2-dimensional hexagonal lattice. The unit cell area of this lattice must be a multiple of $2\pi$. The shown area of $4 \pi$ corresponds to an encoded qubit. Top Right: The GKP moduli space is a union of tori, each corresponding to a choice of phases for a given point in the lattice moduli space. a) Two paths on the code manifold are illustrated, one wrapping nontrivially around a fiber. Bottom Right: In the single mode case (N=1) the space of lattices is given by a three sphere $S^3$ with an embedded trefoil knot (b) removed. The three-sphere is pictured as two filled balls with surfaces identified. c) GKP code paths whose projections interlink with the trefoil knot can give rise to nontrivial logical Clifford action (up to Paulis). A contractible path necessarily produces trivial Clifford logical action. The Pauli part of the logical action is specified by how a path wraps around the fibers over the projected path. Popular summaryFault tolerance is a notion of fundamental importance to the field of quantum information processing. It is one of the central properties a quantum computer must posses in order to enable the achievement of large scale practical quantum computation. Intuitively it is clear what it means for a computer to be fault tolerant: in the presence of sufficiently weak noise with sufficiently low correlations, arbitrarily long and precise computations are possible. While a widely used, general, and intuitive concept, within the literature the term fault tolerant is often applied to specific procedures in an ad-hoc fashion tailored to details of the context or platform under discussion. On the contrary in previous work Gottesman and Zhang conjectured that all types of fault-tolerant gates can in fact be regarded as topological, so that a unifying definition of fault tolerance becomes possible. The main contribution of this work is to prove the Gottesman–Zhang conjecture for logical Clifford operations on arbitrary multi-mode GKP codes, a set of gates physically implemented by Gaussian unitary operations. This constitutes the first class of continuous-variable codes for which the conjecture has been shown to be true, highlighting the topological nature of fault-tolerant operations. The paper also contributes to the theory of multi-mode GKP codes and more generally the theory of fault tolerance in bosonic codes, a class of technologically promising codes, by providing a canonical form for GKP codes that avoids ambiguities present in prior work. Both of these contributions combine in a natural way resulting in a three-dimensional visualization of fault-tolerant logical gates as paths that interlink nontrivially with a trefoil knot.► BibTeX data@article{Burchards2025fiberbundlefault, doi = {10.22331/q-2025-10-29-1899}, url = {https://doi.org/10.22331/q-2025-10-29-1899}, title = {Fiber {B}undle {F}ault {T}olerance of {GKP} {C}odes}, author = {Burchards, Ansgar G. and Flammia, Steven T. and Conrad, Jonathan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1899}, month = oct, year = {2025} }► References [1] Peter W. Shor ``Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer'' SIAM Journal on Computing 26, 1484-1509 (1997). https://doi.org/10.1137/s0097539795293172 [2] Dorit Aharonovand Michael Ben-Or ``Fault-Tolerant Quantum Computation with Constant Error Rate'' SIAM Journal on Computing 38, 1207–1282 (2008). https://doi.org/10.1137/S0097539799359385 [3] Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek, ``Resilient quantum computation: error models and thresholds'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 365–384 (1998). https://doi.org/10.1098/rspa.1998.0166 [4] Daniel Gottesman ``Stabilizer Codes and Quantum Error Correction'' (1997). https://doi.org/10.48550/arXiv.quant-ph/9705052 https://arxiv.org/abs/quant-ph/9705052 [5] A.Yu. Kitaev ``Fault-tolerant quantum computation by anyons'' Annals of Physics 303, 2–30 (2003). https://doi.org/10.1016/s0003-4916(02)00018-0 [6] Daniel Gottesman, Alexei Kitaev, and John Preskill, ``Encoding a qubit in an oscillator'' Physical Review A 64 (2001). https://doi.org/10.1103/physreva.64.012310 [7] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd, ``Gaussian quantum information'' Reviews of Modern Physics 84, 621–669 (2012). https://doi.org/10.1103/revmodphys.84.621 [8] Daniel Gottesmanand Lucy Liuxuan Zhang ``Fibre bundle framework for unitary quantum fault tolerance''. https://doi.org/10.48550/arXiv.1309.7062 arXiv:1309.7062 [9] Jonathan Conrad, Ansgar G. Burchards, and Steven T. Flammia, ``Lattices, Gates, and Curves: GKP codes as a Rosetta stone'' (2024). https://doi.org/10.48550/arXiv.2407.03270 arXiv:2407.03270 [10] Christopher Gerryand Peter Knight ``Introductory Quantum Optics'' Cambridge University Press (2004). https://doi.org/10.1017/CBO9780511791239 [11] J. Conwayand N. Sloane ``Sphere packings, lattices and groups'' Springer, New York, NY (1988). https://doi.org/10.1007/978-1-4757-6568-7 [12] John C. Baezand Javier P. Muniain ``Gauge Fields, Knots And Gravity'' World Scientific Publishing Company (1994). https://doi.org/10.1142/2324 [13] Theodore Frankel ``The Geometry of Physics: An Introduction'' Cambridge University Press (2011). https://doi.org/10.1017/CBO9781139061377 https://www.cambridge.org/core/books/geometry-of-physics/94894F70DB22055BD7BC2B84C135ABAF [14] Tommaso Guaita, Lucas Hackl, and Thomas Quella, ``Representation theory of Gaussian unitary transformations for bosonic and fermionic systems'' (2024). https://doi.org/10.48550/arXiv.2409.11628 arXiv:2409.11628 [15] Jonathan Conrad, Jens Eisert, and Francesco Arzani, ``Gottesman-Kitaev-Preskill codes: A lattice perspective'' Quantum 6, 648 (2022). https://doi.org/10.22331/q-2022-02-10-648 [16] Emanuel Knill, Raymond Laflamme, and Lorenza Viola, ``Theory of Quantum Error Correction for General Noise'' Physical Review Letters 84, 2525–2528 (2000). https://doi.org/10.1103/physrevlett.84.2525 [17] Baptiste Royer, Shraddha Singh, and S.M. Girvin, ``Encoding Qubits in Multimode Grid States'' PRX Quantum 3 (2022). https://doi.org/10.1103/prxquantum.3.010335 [18] J. W. Harrington ``Analysis of quantum error-correcting codes: Symplectic lattice codes and toric codes'' thesis (2004). https://doi.org/10.7907/AHMQ-EG82 https://resolver.caltech.edu/CaltechETD:etd-05122004-113132 [19] Jim Harringtonand John Preskill ``Achievable rates for the Gaussian quantum channel'' Physical Review A 64 (2001). https://doi.org/10.1103/physreva.64.062301 [20] Christina Birkenhakeand Herbert Lange ``Complex Abelian Varieties'' Springer Berlin Heidelberg (2004). https://doi.org/10.1007/978-3-662-06307-1 [21] John Milnor ``Introduction to Algebraic K-Theory. (AM-72), Volume 72'' Princeton University Press (2016). https://doi.org/10.1515/9781400881796 [22] Etienne Ghys ``Lorenz and Modular Flows: A Visual Introduction'' (2006). http://www.ams.org/publicoutreach/feature-column/fcarc-lorenz [23] Baptiste Royer, Shraddha Singh, and S.M. Girvin, ``Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States'' Physical Review Letters 125 (2020). https://doi.org/10.1103/physrevlett.125.260509 [24] V. G. Matsos, C. H. Valahu, M. J. Millican, T. Navickas, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, ``Universal Quantum Gate Set for Gottesman-Kitaev-Preskill Logical Qubits'' (2024). https://doi.org/10.48550/arXiv.2409.05455 arXiv:2409.05455 [25] Shubham P. Jain, Joseph T. Iosue, Alexander Barg, and Victor V. Albert, ``Quantum spherical codes'' Nature Physics 20, 1300–1305 (2024). https://doi.org/10.1038/s41567-024-02496-y [26] Aurélie Denysand Anthony Leverrier ``Quantum Error-Correcting Codes with a Covariant Encoding'' Physical Review Letters 133 (2024). https://doi.org/10.1103/physrevlett.133.240603 [27] N. Bourbaki ``Algébre'' Springer Berlin Heidelberg (2007). https://doi.org/10.1007/978-3-540-35339-3 [28] G. Frobenius ``Theorie der linearen Formen mit ganzen Coefficienten.'' Journal für die reine und angewandte Mathematik 86, 146–208 (1879). https://doi.org/10.1515/crll.1879.86.146 http://eudml.org/doc/148392Cited byCould not fetch Crossref cited-by data during last attempt 2025-10-29 13:13:02: Could not fetch cited-by data for 10.22331/q-2025-10-29-1899 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-10-29 13:13:08: No response from ADS or unable to decode the received json data when getting the list of citing works.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.
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