Determine the depth required for generic three-qubit logical unitary compilation

Determine the depth at which a bubble-sort exchange network can exactly or approximately compile generic elements of $SU(8)$ on three exchange-only logical qubits while maintaining leakage-free operation.

Background

For two three-spin logical qubits, the authors successfully compile the tested real orthogonal targets at depth $R=4N$. For three logical qubits, however, compilation of Haar-random and GPT-2-derived polar factors remains imperfect even at the largest depths examined, while specially planted reachable targets can be recovered.

The authors caution that the larger-depth results become optimization-limited and therefore cannot distinguish insufficient circuit expressivity from unsuccessful optimization. The depth at which generic three-qubit logical unitaries become reachable is consequently unresolved.