  Quantum computing and quantum simulation can be implemented by concatenation
of one- and two-qubit gates and interactions. For most physical
implementations, however, it may be advantageous to explore state components
and interactions that depart from this universal paradigm and offer faster or
more robust access to more advanced operations on the system. In this article,
we show that adiabatic passage along the dark eigenstate of excitation exchange
interactions can be used to implement fast multi-qubit Toffoli (C$_k$-NOT) and
fan-out (C-NOT$^k$) gates. This mechanism can be realized by simultaneous
excitation of atoms to Rydberg levels, featuring resonant exchange interaction.
Our theoretical estimates and numerical simulations show that these multi-qubit
Rydberg gates are possible with errors below 1% for up to 20 qubits. The
excitation exchange mechanism is ubiquitous across experimental platforms and
we show that similar multi-qubit gates can be implemented in superconducting
circuits.
