Compare Papers

Paper 1

MacWilliams Identities for Intrinsic Quantum Codes

Eric Kubischta, Ian Teixeira

Year

2026

Journal

arXiv preprint

DOI

arXiv:2604.16023

arXiv

2604.16023

We develop an intrinsic enumerator framework for quantum error correction in unitary representations of symmetry groups. An intrinsic quantum code is a subspace of a representation $V$ of a group $G$, and errors are organized by the decomposition of the conjugation representation on $\mathcal{L}(V)$ into isotypic subspaces. Associated with any orthogonal decomposition of $\mathcal{L}(V)$ we introduce two families of quadratic enumerators, called projector and twirl enumerators, which satisfy positivity, normalization, and Knill--Laflamme type inequalities. When the conjugation representation is multiplicity--free, these enumerators are related by a linear transform that we interpret as an intrinsic MacWilliams identity. For $G=\mathrm{SU}(2)$, we compute this transform explicitly in terms of Wigner $6j$-symbols. Applied to symmetric-power representations, this gives linear programming bounds for permutation-invariant qubit and qudit codes, including extremality results for the four-qubit, seven-qubit, and three-qutrit examples treated here. We also develop the general equivariant theory in the presence of multiplicities, where the enumerators become matrix-valued, the MacWilliams transform becomes block unitary, and the resulting feasibility problem becomes semidefinite; we illustrate this theory in a first non-multiplicity-free $\mathrm{SU}(3)$ example.

Open paper

Paper 2

Qubit-oscillator concatenated codes: decoding formalism & code comparison

Yijia Xu, Yixu Wang, En-Jui Kuo, Victor V. Albert

Year

2022

Journal

arXiv preprint

DOI

arXiv:2209.04573

arXiv

2209.04573

Concatenating bosonic error-correcting codes with qubit codes can substantially boost the error-correcting power of the original qubit codes. It is not clear how to concatenate optimally, given there are several bosonic codes and concatenation schemes to choose from, including the recently discovered GKP-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020)}] that allow protection of a logical bosonic mode from fluctuations of the mode's conjugate variables. We develop efficient maximum-likelihood decoders for and analyze the performance of three different concatenations of codes taken from the following set: qubit stabilizer codes, analog/Gaussian stabilizer codes, GKP codes, and GKP-stabilizer codes. We benchmark decoder performance against additive Gaussian white noise, corroborating our numerics with analytical calculations. We observe that the concatenation involving GKP-stabilizer codes outperforms the more conventional concatenation of a qubit stabilizer code with a GKP code in some cases. We also propose a GKP-stabilizer code that suppresses fluctuations in both conjugate variables without extra quadrature squeezing, and formulate qudit versions of GKP-stabilizer codes.

Open paper