When is the Clifford hierarchy generalized semi-Clifford?

We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level k and qubit number n. We show that C k(n) is semi-Clifford if and only if k 2, n 2, or k=3 and n 6. We also establish that C k(n) is generalized semi-Clifford when k 4 or n 3 and construct counterexamples when k,n at least 5. For n=4 and k 5, we show that every gate defined over Q(i) up to a scalar is generalized semi-Clifford, reduce the general case to a single level, and propose a strategy to prove our conjecture that C k(n) is generalized semi-Clifford if and only if k 4 or n 4. Lastly, we go beyond this structure by introducing n c-generalized semi-Clifford gates, which we show are exactly the gates of the form sum xπ(x) x V x up to Clifford gates on both sides, where π permutes the computational basis states of n c control qubits. We conjecture that for k 3 every gate in C k(n) has this form for some n c 1 with every V x in C k-3(n-n c), refining the conjecture of Zeng, Chen, and Chuang.