Table 2 Comparison of distribution protocols for a fully non-local N-controlled unitary.

Full Clustering^8,9 (Fig. 1)

2N

\(N+M\)

N

\({\mathcal {O}}\left( 1\right)\)

Local Clustering¹² (Fig. 2)

2K

\(\max \left\{ n+1,\,K+M\right\}\)

K

\({\mathcal {O}}\left( 1\right)\)

Sequential Cascade (Fig. 4)

2K

\(\max \left\{ n+2,\,M+1\right\}\)

K

\({\mathcal {O}}\left( K\right)\)

Binary Parallel Cascades (Fig. 3)

2K

\(\max \left\{ n+3,\,M+2\right\}\)

K

\({\mathcal {O}}\left( \log K\right)\)

\(\sqrt{N}\) Parallel Cascades (Fig. 3)

2K

\(\max \left\{ \lceil n+\sqrt{K}\rceil ,\,\lceil M+\sqrt{K}\rceil \right\}\)

K

\({\mathcal {O}}\left( 1\right)\)