Non-Abelian quantum codes achieve high error protection and new magic states
Non-Abelian sheaf quantum LDPC codes: good and magical
Information Theory
Summary
Quantum computers need special codes to protect information from errors, and most codes studied so far use simpler mathematical structures called Abelian groups. This work by the authors develops a new way to build quantum error-correcting codes using more complex non-Abelian structures, which can protect data better and create special quantum states called magic states. These codes have good properties like keeping a constant rate of encoding and detecting many errors, which is important for reliable quantum computing. They also introduce methods for measuring and handling quantum information in these advanced codes.
What this means in practice
- •For quantum hardware engineers: Implement error-correcting codes that maintain performance with complex non-Abelian structures, improving fault-tolerance in future quantum processors.
- •For quantum algorithm developers: Prepare encoded magic states through logical Clifford measurements enabled by these codes for more efficient quantum computation protocols.
Authors
Zimu Li, Fuchuan Wei, Zhengyi Han, Zi-Wen Liu
Abstract
Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to obtain families with constant encoding rate and linear distance. We resolve the coupled logical constraints using explicit representatives to characterize the full gauged code space. We provide a fundamental treatment of code distance based on the general Knill--Laflamme condition and combine expansion with cleaning to establish protection against arbitrary low-weight errors. We further construct an almost-good family whose entire code space exhibits long-range magic. Gauging and ungauging also enable logical Clifford measurements that prepare encoded magic states. These results extend good qLDPC codes beyond the Pauli stabilizer setting and provide a concrete foundation for exploring non-Abelian phases beyond geometric locality and pursuing the no low-energy trivial magic conjecture.