Finding diagonal logical gates in CSS codes and circuits

Overview

Finding efficient, fault-tolerant non-Clifford logical gates is a primary hurdle in achieving universal quantum computation. While Clifford gates are well-understood within the stabilizer formalism, non-Clifford gates—essential for magic state preparation—are significantly harder to implement. This paper introduces a systematic, efficient algorithmic framework to identify all diagonal logical gates for a given CSS code, provided a set of ansatz gates (e.g., T, CS, or CCZ gates) is specified.

The Algorithmic Approach

The author frames the problem of finding code-space preserving gates as a linear algebra task over finite abelian 2-groups. By defining a pullback of the X-check matrix onto phase functions, the author shows that any diagonal logical gate must lie in the kernel of this pullback map. This transforms the search for logical gates into a kernel computation, which is solved using a fast filtration method. The resulting algorithm has a runtime of O(n^3) for qLDPC codes with O(n) qubits, with potential for further optimization using sparsity.

Versatility and Applications

A key strength of this method is its flexibility. It does not require translation invariance, allowing it to discover gates that arise from higher-order symmetries or complex folding/stacking geometries. Beyond standard transversal gates, the framework extends to:

• Spacetime local logical gates for decorating syndrome-extraction circuits.
• Generalizations to higher levels of the Clifford hierarchy and arbitrary qudit dimensions.
• Non-diagonal gates, by interpreting them as phase terms coupling qubits across time steps.

This approach is particularly useful for extending known logical gates from a code's bulk to its boundaries or for blending different logical gates across domain walls in interfaced codes.