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  • Open Access

Quantum Error Correction beyond SU(2): Spin, Bosonic, and Permutation-Invariant Codes from Convex Geometry

Arda Aydin1,*, Victor V. Albert2,†, and Alexander Barg1,2,‡

  • *Contact author: aaydin@umd.edu
  • †Contact author: vva@umd.edu
  • ‡Contact author: abarg@umd.edu
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PRX Quantum 7, 010341 – Published 27 February, 2026

DOI: https://doi.org/10.1103/kx3b-4nrp

Abstract

We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces—composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group SU(q), we prove that many codes and their gates in SU(q) can be interconverted between the three state spaces. We construct code instances for all three spaces using classical ℓ1 codes and Tverberg’s theorem, a classic result from convex geometry. We obtain families of quantum codes with distance that scales almost linearly with the code length N by constructing ℓ1 codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This compares favorably with the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

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Popular Summary

Quantum platforms come in many shapes and sizes, and there is no one-size-fits-all way of robustly processing quantum information on all of them. This work takes steps to solve this problem by showing that the geometry, physical noise, and certain operations of two seemingly different platforms are actually interconvertible. This finding allows for the application of a single, unified strategy for encoding and processing quantum information across both platforms. This work provides one such strategy by building quantum error-correcting codes out of classical ones, extending the well-known Calderbank-Shor-Steane (CSS) construction to more exotic state spaces. Several new examples as well as an infinite family of high-performing codes compatible with both platforms are provided. The research presents several new examples as well as an infinite family of high-performing codes that are compatible with both quantum platforms.

Article Text

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PRX Quantum 7, 010341– Published 27 February, 2026
  • Received 6 October 2025
  • Revised 19 December 2025
  • Accepted 14 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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