Table 1 Comparison of distribution protocols for a fully non-locally N-controlled unitary.

Existing Algorithms^8,9,12 (Fig. 1)

2N

\(N+M\)

N

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

Sequential Cascade (Fig. 4)

2N

\(\max \left\{ 3,\,M+1\right\}\)

N

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

Binary Parallel Cascades (Fig. 3)

2N

\(\max \left\{ 4,\,M+2\right\}\)

N

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

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

2N

\(\lceil \sqrt{N}+M\rceil\)

N

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