Prove the conjectured Lie saturation depth pattern

Prove that, for the ring-CX fixed-block ansatz with one parameterized R_Y rotation per qubit, the Lie saturation depth is 4 when the number of qubits n is a multiple of 3 and 2 otherwise.

Background

The paper studies a parameterized quantum circuit whose trainable block consists of single-qubit R_Y rotations and whose repeated fixed entangling block is a ring of controlled-X gates. The effective generators at successive layers are obtained by repeatedly conjugating the initial Pauli-Y generators by the fixed Clifford block.

The conjugation period of the generators can be substantially larger than the depth required for Lie saturation, because the initial effective generators may generate later conjugates through Lie brackets. Numerical results summarized for n=2 through n=8 suggest a simple parity/divisibility pattern for the Lie saturation depth: depth 4 when n is divisible by 3 and depth 2 otherwise. The paper leaves proving this pattern unresolved.

References

Besides, we indicate the Lie closure depth $L_{\mathrm{Lie}}$ which is actually much smaller; actually it seems (and it should not be too difficult to prove) that $L_{\mathrm{Lie}}$ is $4$ if $n$ is a multiple of $3$, and $2$ otherwise.

— Dynamical Lie Algebras for Fixed-Gate Quantum Ansätze  (2610.13709 - Zang et al., 9 Oct 2026) in Section IV.E, subsection “Specific cases for differents number of qubits,” paragraph immediately preceding Table 1