Stabilizer Codes with Exotic Local-dimensions

TL;DR

The paper generalizes stabilizer codes to exotic local-dimensions via the local-dimension-invariant (LDI) framework, enabling importing codes across local-dimension choices using the $\phi_\infty$ symplectic representation and a canonical form. It proves two main results: (i) any stabilizer code over a finite field or prime dimension can be transformed into an LDI code over a ring or integral domain with preserved or improved distance; (ii) any stabilizer code can be used to construct analog continuous-variable (CV) codes with distance at least the original code's in the infinite/distance sense, $d' \ge d$ and $d' \ge d^*$. The work further analyzes restricted cases such as bounded phase space (continuous but wrapping) and discrete/integer precision, showing distance promises under large enough base $p$ or lattice discretization, linking to GKP-like encodings. These results broaden the applicability of stabilizer codes to systems described by rings, infinite dimensions, or rotor-like degrees of freedom, with potential implications for topological codes and rotor/GKP constructions.

Abstract

Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same $n$, $k$, and $d$ parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.

Figures (3)

• Figure 1: Above is a schematic of an integer code, assuming a code with lattice points only differing by $X$ operators. The lattice would formally continue infinitely, but we only highlight this sublattice. Two codewords are placed on the blue diamonds. Their distance is $\sqrt{14}$, so their spacing is scaled by this factor and the inherent lattice spacing. This acts as an encoding of a discrete logical subspace in a possibly continuous phase space, or explicitly here a discretized phase space.
• Figure 2: This figure shows the transformations on the stabilizing plaquettes for the qubit toric code to generate a toric code that works regardless of the local-dimension of the system. This transforms the code from a $[[2N^2,2,N]]_2$ code into a $[[2N^2,2,N]]$ code for any choice of local-dimension with additive inverses.
• Figure 3: This figure illustrates the distance argument, showing that the distance of the code is preserved upon changing the local-dimension of the code. Notice that bridges generated are only existent in the non-qubit case, as modulo $2$ they disappear and a mere distortion of the logical string operator occurs

Theorems & Definitions (31)

• Definition 1
• Definition 2: $\phi$ representation of a qudit operator
• Definition 3
• Definition 4
• Definition 5
• Theorem 6
• proof
• Theorem 7
• Definition 8
• Definition 9