Establish whether additional two-qubit tangent directions enable conditioning recovery

Establish whether the additional tangent directions available in the two-qubit ancilla and parallel circuits, compared with the single-qubit serial circuit, are what enables their Jacobian conditioning to recover along the optimisation path.

Background

The experiments find that the serial circuit remains poorly conditioned more often during training, while the two-qubit control circuits more reliably recover well-conditioned Jacobians and reach the optimum. The paper presents the larger tangent spaces of the two-qubit circuits as a candidate explanation, but explicitly leaves unresolved whether those additional directions cause the observed recovery of conditioning along optimisation trajectories.