Fiber Bundle Fault Tolerance of GKP Codes
1Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
2Department of Computer Science, Virginia Tech, Alexandria, USA
3Phasecraft Inc., Washington DC, USA
4Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
| Published: | 2025-10-29, volume 9, page 1899 |
| Editor: | Ulysse Chabaud |
| Eprint: | arXiv:2410.07332v3 |
| Doi: | https://doi.org/10.22331/q-2025-10-29-1899 |
| Citation: | Quantum 9, 1899 (2025). |
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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.
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 summary
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.
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Cited by
[1] Jonathan Conrad, Jens Eisert, and Steven T. Flammia, "Chasing shadows with Gottesman-Kitaev-Preskill codes", Quantum 10, 1973 (2026).
[2] Sayan Chakraborty and Victor V. Albert, "Hybrid Oscillator-Qudit Quantum Processors: Stabilizer States, Stabilizer Codes, Symplectic Operations, and Noncommutative Geometry", PRX Quantum 7 2, 020320 (2026).
[3] Jonathan Conrad, Jens Eisert, and Steven T. Flammia, "Chasing shadows with Gottesman-Kitaev-Preskill codes", arXiv:2411.00235, (2024).
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