Novel Encodings of Homology, Cohomology, and Characteristic Classes

TL;DR

This work develops a unified framework to encode topological invariants inside quantum error-correcting codes by extending toric/code-like constructions to higher homology and cohomology with finitely generated abelian coefficients. It introduces multiple families of stabilizers that preserve commutativity and enables both homology and cohomology encodings, then analyzes the structure and dynamics of errors as geometric submanifolds (erbs) whose boundaries reveal stabilizer violations. The main contributions include constructing Hamiltonians that encode obstruction classes of fiber bundles, exemplified by the Hairy Ball Theorem for the Euler class, and detailing how to realize and gap these obstructions via $P$-type and $V$-type operators across general bundles. The results provide a pathway to encode characteristic classes such as Chern and Pontryagin classes in topological QECCs, with a concrete, gapped Hamiltonian framework and explicit strategies for handling integer-coefficient and torsion components, potentially impacting fault-tolerant implementations of geometric/topological quantum information tasks.

Abstract

Topological quantum error-correcting codes (QECC) encode a variety of topological invariants in their code space. A classic structure that has not been encoded directly is that of obstruction classes of a fiber bundle, such as the Chern or Euler class. Here, we construct and analyze extensions of toric codes. We then analyze the topological structure of their errors and finally construct a novel code using these errors to encode the obstruction class to a fiber bundle. In so doing, we construct an encoding of characteristic classes such as the Chern and Pontryagin class in topological QECC. An example of the Euler class of $S^2$ is constructed explicitly.

Figures (14)

• Figure 1: The left side is a snapshot of the triangulation of two $3$-cells on the boundary of a $3$-dimensional triangulation (with the boundary facing the viewer), and the right side is the dual of this snapshot. The red cells on the right are in the closed dual triangulation but not the open dual triangulation. Each $3$-cell on the left goes to a corresponding shaded vertex on the right. Similarly, the shaded face on the left goes to the corresponding shaded face on the right. On the left, the two $2$-cells on the boundary go to the corresponding edges from the shaded vertex to the red vertex on the right.
• Figure 2: A visual representation of two $P$-errors in the normal triangulation on the left and in the dual triangulation on the right.
• Figure 3: This is a visual representation of error sub-pseudo manifolds over $\mathbb{Z}_3$, where red indicates a degree-$(1)$ error and blue indicates a degree-$(2)$ error
• Figure 4: A visual representation of erbs of degrees of $-p,-q$, and $p+q$ in the toric code
• Figure 5: The erb in blue wiggles after a $P$ operation is applied to the two 2-cells marked in red.

Theorems & Definitions (11)

• Lemma 1.1: Higher Homologies over $\mathbb{Z}_2$
• proof
• Lemma 1.2: Higher Homologies over Finitely Generated Abelian Groups
• proof
• Definition 2.1: Oriented $m$-pseudo-manifold
• Definition 2.2: $V$-error sub-pseudo-manifold
• Definition 2.3: $P$-error sub-pseudo-manifold on the dual triangulation
• Lemma 2.4
• proof
• Definition 2.5: Error Boundary/Erb