Selecting self-dual bases to minimize CCZ count in the 2^s-to-s qubit gate decomposition

Identify analytically the isomorphism between F_{2^s} and F_2^s (specified by a self-dual basis of F_{2^s} over F_2) that yields, for s=10, the decomposition of the induced 10-qubit non-Clifford gate corresponding to U_1^{(7)} with the smallest number of CCZ gates, and develop an efficient method to find the optimal self-dual basis for general s.

Background

The paper’s qudit-to-qubit mapping depends on a choice of self-dual basis for F_{2^{10}} over F_2, and the resulting 10-qubit diagonal third-level gate can be decomposed into Z, CZ, and CCZ gates whose count depends on this choice.

Minimizing the number of CCZ gates directly reduces the overhead for converting between the distilled 10‑qubit magic state and standard CCZ magic states, motivating an analytic and scalable selection criterion for the basis.