Poincar\'e Duality and Multiplicative Structures on Quantum Codes

TL;DR

This paper extends Poincaré duality to quantum codes via sheaf theory, revealing new multiplicative structures and logical gate constructions that enhance fault-tolerant quantum computing capabilities.

Contribution

It introduces a generalized Poincaré duality framework for quantum codes using sheaf theory, and develops multiplicative structures leading to new logical gate methods.

Findings

Established duality between chain and cochain complexes of sheaf codes.

Constructed explicit isomorphisms between (co)homology groups via cap products.

Proposed transversal logical gates with linear scaling on quantum LDPC codes.

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 Poincar\'e 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 Poincar\'e 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 $k_{\mathrm{C}Z}=\Theta(n)$ 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, \v{C}ech cohomology, and the cohomology of sheaf codes, thereby introducing new mathematical tools into quantum coding theory.