Cubical sheaf complexes enable new efficient quantum error correcting codes
Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs
Information TheoryComputational Complexity
Summary
Quantum computers need special codes to protect information from errors while still allowing computations. This paper shows how to build certain multi-dimensional structures called cubical sheaf complexes that lead to quantum codes with good properties: they have a high rate of information, can detect errors reliably, and are efficiently constructed. The authors use advanced tools from coding theory and geometry to build these codes explicitly, improving on previous designs. This helps pave the way for better quantum error correction methods.
What this means in practice
- •For quantum hardware engineers: Construct quantum memory systems with high error detection and correction efficiency using these codes for more reliable quantum computations.
- •For quantum software developers: Implement fault-tolerant quantum algorithms that use explicitly defined, efficiently decodable quantum codes with good error-resilience.
Authors
Yeyuan Chen, Miryam Mi-Ying Huang, Yinchen Liu, Er-Cheng Tang
Abstract
For every fixed integers $r \ge 4$ and $2 \le k \le r-2$, we construct $r$-dimensional cubical sheaf complexes whose degree-$k$ CSS codes have positive constant rate, linear distance, and constant soundness, with bounded row and column weights. Taking $r=4$ and $k=2$ gives a family of asymptotically good binary qLTCs. At the core of our construction is a uniform product-expansion theorem for explicit Reed-Solomon codes on norm-one evaluation sets. The key point is that the expansion constant stays bounded away from zero as the local code lengths grow. We place these codes on arithmetic cubical complexes, obtaining constant local expansion for both the resulting sheaf and its dual. Together with the local-to-global framework of Dinur, Lin, and Vidick (FOCS 2024) and sheaf duality, this gives linear distance and constant soundness, while an asymmetric choice of local code dimensions gives positive rate. The resulting codes are explicit and polynomial-time computable.