Fig. 1: All most expensive 6-qubit Clifford unitaries requiring 15 entangling gates (up to left and right multiplication by the single-qubit gates and qubit relabeling).

a left: a compact representation in the form (U ⊗ U)SWAP, right: its optimal implementation. b left: a compact representation in the form \((U^{\prime} \otimes V^{\prime} )\,{{\mbox{SWAP}}}\,\), right: its optimal implementation. Not illustrated is the cyclic SWAP of all 6 qubits, that also requires 15 entangling gates.