Surjectivity of the fifteen-rotation two-qubit gate parametrization

Prove that the fixed fifteen-Pauli-rotation word map used for each trainable two-qubit gate is onto $\mathrm{SU}(4)$, thereby establishing the equality case of the first-rung geometric-entanglement bound.

Background

At peaking depth τp=1\tau_p=1, the variational probe states are product states across the brick partition, so the peakedness is bounded by the geometric measure of entanglement of the scrambled state. Equality requires each two-qubit gate parametrization to reach every element of SU(4)\mathrm{SU}(4), or equivalently every two-qubit unit vector up to phase.

The paper reports numerical verification of surjectivity for the chosen fifteen-rotation word order but does not provide a proof. Thus, the unconditional upper bound is established, while the exact equality statement depends on an unresolved parametrization property.

References

Surjectivity of the gate map onto $SU(4)$, which acts transitively on unit vectors of $\mathbb{C}4$, turns that inclusion into an equality of sets and the bound into an equality; it is verified numerically for the fixed word order (Sec.~\ref{sec:gauge}) and not proved, a modeling hypothesis of the equality as Gaussianity is one of Lemma~\ref{lem:gaussian}.

— The optimization landscape of peaked-circuit generation  (2608.11890 - Jamoussi, 12 Aug 2026) in Section 4.2, The atom and the first-rung theorem; Appendix A, proof of Theorem First rung