Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.07332

Fiber Bundle Fault Tolerance of GKP Codes

Ansgar G. Burchards, Steven T. Flammia, Jonathan Conrad

Published Oct 9, 2024• 0 citations
arXivHTMLPDF

Abstract

We 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.

DOI
10.48550/arxiv.2410.07332

Related papers

Fiber Bundle Fault Tolerance of GKP Codes

Burchards, Ansgar G., Steven T. Flammia, Conrad, Jonathan

We 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.

•Jan 1, 2025•0 citations

Geometric aspects of fault tolerance

Jonathan Conrad

We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a $\rm CCZ$-type magic state arising this way.

quant-pharXiv:2610.35513•Oct 27, 2026•0 citations

Fiber Bundle Fault Tolerance of GKP Codes

Ansgar G. Burchards, Steven T. Flammia, Jonathan Conrad

We 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.

Quantum•Oct 29, 2025•1 citation

Complex abelian varieties and quantum error correction: a mathematical framework for GKP codes

Maxence Mayrand, Baptiste Royer

We study a class of quantum error-correcting codes through the geometry of complex abelian varieties. These codes, introduced by Gottesman--Kitaev--Preskill, are built from symplectically integral lattices and therefore naturally define polarized complex abelian varieties. We give a precise mathematical formulation of this relationship and extend it to a dictionary between the main structures of GKP code theory and classical objects in the theory of abelian varieties. For instance, under this dictionary, the finite-dimensional code space becomes the space of theta functions $H^0(X, L)$, logical Pauli gates arise from the theta group, passive logical Clifford gates correspond to automorphisms of the polarized abelian variety, and concatenation with stabilizer codes corresponds to isogeny. We also prove several key results that give precise mathematical formulations of statements about these codes that often appear in heuristic form in the physics literature. In particular, we prove that the encoding is asymptotically isometric, that every logical Clifford gate is realized by a Gaussian unitary, and that, for noise of small variance, the failure probability is governed to first order by the shortest nontrivial displacement in the kernel of the polarization isogeny, a systolic invariant of the underlying polarization. This leads naturally to optimization problems on the moduli space of polarized abelian varieties.

math.AGCross-listedarXiv:2605.28784•May 28, 2026•0 citations

Probabilistic Error Cancellation for Single-Mode Gottesman-Kitaev-Preskill Codes

Victoria Wadewitz, Alessandro Ciani

In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.

quant-pharXiv:2609.17095•Sep 16, 2026•0 citations

Lattices, Gates, and Curves: GKP codes as a Rosetta stone

Jonathan Conrad, Ansgar G. Burchards, Steven T. Flammia

We explain how GKP Clifford gates arise as symplectic automorphisms of the corresponding GKP lattice and show that, for scaled codes of type $D=dI_n$, their integral action agrees with the action of the mapping class group of a genus-$n$ surface on first homology. This correspondence introduces a topological interpretation of fault tolerance for GKP codes and motivates the connection between GKP codes (lattices), their Clifford gates, and algebraic curves, which we explore in depth. For a single-mode GKP code, we identify the space of oriented fixed-covolume lattice realizations with $S^3 - K$, the three sphere $S^3$ with a trefoil knot $K$ removed, and explain how logical degrees of freedom arise from the choice of a level structure on the corresponding curves. Specified Clifford implementations define loops in the space of lattices whose trefoil linking number is a topological invariant. Finally, we relate our observations to the idea of fiber bundle fault tolerance as proposed by Gottesman and Zhang for the GKP code, where logical Clifford and Pauli operations of the single-mode GKP code arise as the monodromy representation of a finite covering of the space of nonzero-distance GKP codes.

arXiv:2407.03270•Jul 3, 2024•1 citation
We 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.
We 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.
We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a CCZ\rm CCZCCZ-type magic state arising this way.
We 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.
We study a class of quantum error-correcting codes through the geometry of complex abelian varieties. These codes, introduced by Gottesman--Kitaev--Preskill, are built from symplectically integral lattices and therefore naturally define polarized complex abelian varieties. We give a precise mathematical formulation of this relationship and extend it to a dictionary between the main structures of GKP code theory and classical objects in the theory of abelian varieties. For instance, under this dictionary, the finite-dimensional code space becomes the space of theta functions H0(X,L)H^0(X, L)H0(X,L), logical Pauli gates arise from the theta group, passive logical Clifford gates correspond to automorphisms of the polarized abelian variety, and concatenation with stabilizer codes corresponds to isogeny. We also prove several key results that give precise mathematical formulations of statements about these codes that often appear in heuristic form in the physics literature. In particular, we prove that the encoding is asymptotically isometric, that every logical Clifford gate is realized by a Gaussian unitary, and that, for noise of small variance, the failure probability is governed to first order by the shortest nontrivial displacement in the kernel of the polarization isogeny, a systolic invariant of the underlying polarization. This leads naturally to optimization problems on the moduli space of polarized abelian varieties.
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.
We explain how GKP Clifford gates arise as symplectic automorphisms of the corresponding GKP lattice and show that, for scaled codes of type D=dInD=dI_nD=dIn​, their integral action agrees with the action of the mapping class group of a genus-nnn surface on first homology. This correspondence introduces a topological interpretation of fault tolerance for GKP codes and motivates the connection between GKP codes (lattices), their Clifford gates, and algebraic curves, which we explore in depth. For a single-mode GKP code, we identify the space of oriented fixed-covolume lattice realizations with S3−KS^3 - KS3−K, the three sphere S3S^3S3 with a trefoil knot KKK removed, and explain how logical degrees of freedom arise from the choice of a level structure on the corresponding curves. Specified Clifford implementations define loops in the space of lattices whose trefoil linking number is a topological invariant. Finally, we relate our observations to the idea of fiber bundle fault tolerance as proposed by Gottesman and Zhang for the GKP code, where logical Clifford and Pauli operations of the single-mode GKP code arise as the monodromy representation of a finite covering of the space of nonzero-distance GKP codes.