Homological origin of transversal implementability of logical diagonal gates in quantum CSS codes

TL;DR

This paper introduces a homological framework to understand the limitations and conditions for implementing transversal logical diagonal gates in quantum CSS codes, linking algebraic conditions to topological obstructions.

Contribution

It develops a homological classification of transversal diagonal gates and formulates obstruction maps that determine their implementability in quantum error correction.

Findings

Homological data classifies logical actions of transversal gates.

Obstruction maps identify when finer angle gates can be implemented.

Algebraic conditions like divisibility are reinterpreted as homological obstructions.

Abstract

Transversal Pauli $Z$ rotations provide a natural route to fault-tolerant logical diagonal gates in quantum CSS codes, but their capability is inherently constrained. We develop a homological framework that organizes transversal diagonal gates in terms of their logical action and physical implementation, revealing two layers of structure that govern their behavior. At a fixed level, we establish that their logical action admits a classification in terms of homological data of the underlying chain complex, extending the standard description of logical operators. We then formulate the refinement to finer angles as a lifting problem and derive two Bockstein-type obstruction maps, whose vanishing is a necessary and sufficient condition for the existence of a transversal logical diagonal gate at the next level. Within this framework, known algebraic conditions such as divisibility and triorthogonality are reinterpreted as necessary conditions for the existence of transversal logical diagonal gates with uniform rotation angles. Our results identify homological obstructions governing transversal implementability and provide a conceptual foundation for a formal theory of transversal structures in quantum error correction.