Quantum error correction beyond $SU(2)$ spin, bosonic, and permutation-invariant codes from convex geometry

2025-09-24

AI Summary

This work addresses quantum error correction for three classes of symmetric quantum systems: composite permutation-invariant subspaces, constant-excitation Fock-state subspaces, and single-block nuclear spin subspaces of atoms/ions/molecules. Methodologically, it maps each state space onto a discrete simplex and an irreducible representation of $SU(q)$, and—novelty—integrates Tverberg’s theorem from convex geometry, Sidon set constructions, and $ell_1$-norm-based classical coding techniques. Key contributions are: (1) achieving asymptotic code distance $Omega(N)$, surpassing the linear-distance barrier inherent in prior symmetric codes; (2) constructing highly efficient encodings with shorter physical length or lower excitation number; and (3) presenting the first bosonic code enabling non-Gaussian logical operations. The framework unifies code construction and fault-tolerant logical gate design across diverse physical platforms, advancing high-fidelity, platform-agnostic quantum information processing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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 inter-converted between the three state spaces. We construct new code instances for all three spaces using classical $ell_1$ codes and Tverberg's theorem, a classic result from convex geometry. We obtain new families of quantum codes with distance that scales almost linearly with the code length $N$ by constructing $ell_1$ codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This improves upon 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, new bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.

Problem

• Develops quantum error-correcting codes for composite permutation-invariant spaces
• Constructs codes for bosonic modes and nuclear state spaces of atoms
• Creates new code families with distance scaling linearly with length

Innovation

• Framework converts SU(q) codes between three quantum spaces
• New codes constructed using classical l1 codes and Tverberg theorem
• Codes achieve near-linear distance scaling via Sidon sets