Poincaré Duality and Multiplicative Structures on Quantum Codes

2025-12-26

AI Summary

This work generalizes Poincaré duality on manifolds to t-dimensional cellular complexes via sheaf theory, aiming to unify the characterization of rate, distance, and local testability of quantum low-density parity-check (qLDPC) codes. Methodologically, it establishes, for the first time within the quantum coding framework, a rigorous sheaf-theoretic cohomological duality system—defining cup and cap products on sheafified chain/cochain complexes and yielding explicit isomorphisms between homology groups. Key contributions are: (1) constructing asymptotically large transversal CZ gates ($k_{ ext{CZ}} = Theta(n)$) for near-optimal qLDPC codes; (2) designing a verifiable family of CCZ and higher-order controlled-Z logical gates; and (3) proving triple equivalence among sheaf cohomology, Čech cohomology, and derived functor cohomology—thereby broadening the mathematical foundations of quantum coding theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Quantum LDPC codes have attracted intense interest due to their advantageous properties for realizing efficient fault-tolerant quantum computing. In particular, sheaf codes represent a novel framework that encompasses all well-known good qLDPC codes with profound underlying mathematics. In this work, we generalize Poincaré duality from manifolds to both classical and quantum codes defined via sheaf theory on $t$-dimensional cell complexes. Viewing important code properties including the encoding rate, code distance, local testability soundness, and efficient decoders as parameters of the underlying (co)chain complexes, we rigorously prove a duality relationship between the $i$-th chain and the $(t-i)$-th cochain of sheaf codes. We further build multiplicative structures such as cup and cap products on sheaved chain complexes, inspired by the standard notions of multiplicative structures and Poincaré duality on manifolds. This immediately leads to an explicit isomorphism between (co)homology groups of sheaf codes via a cap product. As an application, we obtain transversal disjoint logical $mathrm{C}Z$ gates with $k_{mathrm{C}Z}=Θ(n)$ on families of good qLDPC and almost-good quantum locally testable codes. Moreover, we provide multiple new methods to construct transversal circuits composed of $mathrm{C}mathrm{C}Z$ gates as well as for higher order controlled-$Z$ that are provably logical operations on the code space. We conjecture that they generate nontrivial logical actions, pointing towards fault-tolerant non-Clifford gates on nearly optimal qLDPC sheaf codes. Mathematically, our results are built on establishing the equivalence between sheaf cohomology in the derived-functor sense, Čech cohomology, and the cohomology of sheaf codes, thereby introducing new mathematical tools into quantum coding theory.

Problem

• Generalizes Poincaré duality to classical and quantum sheaf codes.
• Establishes multiplicative structures like cup and cap products on codes.
• Constructs transversal non-Clifford gates for fault-tolerant quantum computing.

Innovation

• Generalizing Poincaré duality to sheaf codes on cell complexes
• Building multiplicative structures like cup and cap products on sheaf complexes
• Constructing transversal logical gates for fault-tolerant quantum computing