Establish the full orthogonal dynamical Lie algebra for all qubit numbers

Prove that, for the ring-CX fixed-block ansatz with parameterized R_Y rotations on n qubits, the dynamical Lie algebra is \(\mathfrak{so}(2^n)\) for every integer n, rather than only for the cases checked through n=8.

Background

For the ring-CX ansatz, the paper computes the effective-generator orbits and Lie closures for qubit numbers from 2 through 8. In each of these cases, the resulting dynamical Lie algebra is reported to be the special orthogonal algebra so(2n)\mathfrak{so}(2^n), whose dimension is 2n−1(2n−1)2^{n-1}(2^n-1).

The observed exponential growth of the DLA dimension is subsequently used to discuss overparameterization and barren-plateau susceptibility. Extending the finite verification to a proof for arbitrary n would establish the general algebraic structure of this ansatz.

References

Finally, we have checked up to $n=8$ that the DLA $\mathfrak g_n$ is $\mathfrak{so}(2n)$, of dimension $2{n-1}(2n - 1)$. It is very likely to be true for all $n$.

— 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