Finding diagonal logical gates in CSS codes and circuits
arXiv:2607.26477
The paper introduces efficient algorithms to identify all diagonal logical gates that preserve the code space of CSS codes or circuits, enabling the systematic discovery of fault‑tolerant non‑Clifford operations.
Abstract
Finding efficient schemes for non-Clifford logic or magic state preparation is one of the central challenges on the way to fault-tolerant quantum computation. Many of the proposed schemes rely on diagonal non-Clifford logical gates acting on CSS codes in space or decorating CSS-type syndrome-extraction circuits in spacetime. Here we propose and implement efficient algorithms to find all (spacetime) logical gates of a given CSS code (circuit) composed from a prescribed set of ansatz gates. Depending on the choice of ansatz gates, this means finding transversal gates, more general locality-preserving logical circuits, folding gates, or similar. While we focus on qubit diagonal gates in the Clifford hierarchy, we also discuss the generalization to arbitrary diagonal non-hierarchy gates, certain non-diagonal gates, as well as prime and composite-dimensional qudits. Our method works by rephrasing code-space preserving gates as the kernel of the ``pullback'' of the check matrix onto phase functions, which maps between finite abelian 2-groups. We implement a fast ``filtration'' method to find this kernel. The runtime for finding fault-tolerant logical gates in a qLDPC code with qubits or a circuit with gates in a naive dense implementation is , with potential for improvement making use of sparsity.