Finding diagonal logical gates in CSS codes and circuits

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 $X$ 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 $O(n)$ qubits or a circuit with $O(n)$ gates in a naive dense implementation is $O(n^3)$, with potential for improvement making use of sparsity.

arXiv:2607.26477 [quant-ph]

(or arXiv:2607.26477v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.26477 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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