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Non-Abelian quantum LDPC codes
Sheaf codes gauging
Long-range magic
Knill-Laflamme condition
Logical Clifford measurements
Non-Abelian sheaf quantum LDPC codes: good and magical
🤖 AI Summary
This study addresses the challenges of logical constraints and long-range magic state preparation in non-Abelian quantum LDPC codes. To overcome these limitations, it proposes a general framework built upon cup-product gauge homological codes, integrating Knill–Laflamme conditions with extended cleaning techniques to transcend Pauli stabilizer restrictions and achieve protection against arbitrary low-weight errors. The primary contribution is the construction of a family of asymptotically good codes exhibiting constant encoding rate, linear distance, and long-range magic throughout the entire code space, thereby enabling the preparation of encoded magic states via logical Clifford measurements. Furthermore, this work establishes a theoretical foundation for non-Abelian phases beyond geometric locality and advances the investigation of the no low-energy trivial magic conjecture.
📝 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.
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