In this paper, we give a generalization of Kitaev’s stabilizer code based on chain complex theory of bicommutative Hopf algebras. Due to the bicommutativity, the Kitaev’s stabilizer code extends to a broader class of spaces, e.g. finite CW-complexes ; more generally short abstract complex over a commutative unital ring R which is introduced in this paper. Given a finite-dimensional bisemisimple bicommutative Hopf algebra with an R-action, we introduce some analogues of \(\mathbb {A}\)-stabilizers, \(\mathbb {B}\)-stabilizers and the local Hamiltonian, which we call by the \((+)\)-stabilizers, the \((-)\)-stabilizers and the elementary operator respectively. We prove that the eigenspaces of the elementary operator give an orthogonal decomposition and the ground-state space is isomorphic to the homology Hopf algebra. In application to topology, we propose a formulation of topological local stabilizer models in a functorial way. It is known that the ground-state spaces of Kitaev’s stabilizer code extends to Turaev-Viro TQFT. We prove that the 0-eigenspaces of a topological local stabilizer model extends to a projective TQFT which is improved to a TQFT in typical examples. Furthermore, we give a generalization of the duality in the literature based on the Poincaré-Lefschetz duality of R-oriented manifolds.

在本文中，我们基于双交换Hopf代数的链复杂理论对Kitaev的稳定器代码进行了概括。由于具有双交换性，因此Kitaev的稳定器代码扩展到了更广泛的空间类别，例如，有限的CW复数；本文介绍了一个更一般的交换单环R上的短抽象复合体。给定有限维bisemisimple bicommutative Hopf代数与ř -action，我们介绍的一些类似物\（\ mathbb {A} \） -stabilizers，\（\ mathbb {B} \） -stabilizers和本地哈密顿量，我们由\（（+）\）-稳定器调用，\（（-）\）-稳定剂和基本运算符。我们证明了基本算子的本征空间给出了一个正交分解，并且基态空间对于同构Hopf代数是同构的。在拓扑结构中，我们以函数形式提出了拓扑局部稳定器模型的提法。众所周知，Kitaev稳定器代码的基态空间扩展到Turaev-Viro TQFT。我们证明了拓扑局部稳定器模型的0本征空间扩展到射影TQFT，在典型示例中，该射影TQFT被改进为TQFT。此外，我们基于R定向流形的Poincaré-Lefschetz对偶性对文献中的对偶性进行了概括。