Fiber Bundle Fault Tolerance of GKP Codes

TL;DR

This work develops a geometric framework for multi-mode GKP quantum error-correcting codes by constructing the moduli space $\mathcal{M}_{\textnormal{GKP}}$ as a fiber bundle over the lattice moduli space $\mathcal{M}_{\textnormal{lat}}$ and establishing a flat projective connection induced by Gaussian unitaries. The Gottesman--Zhang conjecture is proven for logical Clifford gates: such gates arise from parallel transport along paths in $\mathcal{M}_{\textnormal{GKP}}$, with nontrivial Clifford actions linked to non-contractible loops in the moduli space. The moduli-space structure is enriched by a Frobenius-standard form-based, unique Gram-matrix normalization and a mapping-torus description of restricted bundles, clarifying how Pauli and Clifford information is encoded and manipulated geometrically. The results provide a universal, topology-driven view of fault tolerance for multi-mode bosonic codes and open avenues for extending the framework to finite-energy codes and non-Clifford operations.

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.

Figures (2)

• Figure 1: Illustration of the GKP moduli space. Top Left: A GKP stabilizer is generated by a set of $2n$ displacement operators with phases. Bottom Left: The stabilizer gives rise to a symplectic 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 the set of possible phases $\{\phi_{i}\}$ for a given point in the lattice moduli space. a) A path on the GKP moduli space wrapping around a fiber implements a logical Pauli gate. Bottom Right: In the single mode case ($n=1$) each connected component of the space of lattices is topologically equivalent to $S^{3}- \gamma$, the 3-sphere with a trefoil knot removed, the 3-sphere is pictured as two filled balls with surfaces identified. b) Paths on $\mathcal{M}_{\textnormal{GKP}}$ whose projections interlink with the trefoil knot can give rise to nontrivial logical Clifford action (up to Paulis). Contractible paths necessarily produce trivial logical Clifford action (up to Paulis). The Pauli part of the logical action is specified by how the path wraps around the fibers over its projection.
• Figure 2: Commutative diagram illustrating the proof of Theorem \ref{['thm:restriction_bundles']}. Objects on the outer commutative square are fiber bundles, their projection maps pointing to the base spaces on the inner commutative square. We first construct the isomorphism $\varphi$ between two bundles over the unit interval and then construct the sought-after isomorphism $\theta$ through the universal product of the quotient topology.

Theorems & Definitions (33)

• Theorem 1.1
• Definition 2.1: Lattice
• Definition 2.2: Dual Lattice
• Definition 2.3: Gram Matrix
• Definition 2.4
• Definition 2.5: Stabilizer Code
• Definition 2.6
• Lemma 3.1
• proof
• Corollary 3.2