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Accurate, nontrivial quantum operations on many qubits are experimentallychallenging. As opposed to the standard approach of compiling larger unitariesinto sequences of 2-qubit gates, we propose a protocol on Hamiltonian controlfields which implements highly selective multi-qubit gates in astrongly-coupled many-body quantum system. We exploit the selectiveness ofresonant driving to exchange only 2 out of $2^N$ eigenstates of some backgroundHamiltonian, and discuss a basis transformation, the eigengate, that makes thisoperation relevant to the computational basis. The latter has a second use as aHahn echo which undoes the dynamical phases due to the background Hamiltonian.We find that the error of such protocols scales favourably with the gate timeas $t^{-2}$, but the protocol becomes inefficient with a growing number ofqubits N. The framework is numerically tested in the context of a spin chainmodel first described by Polychronakos, for which we show that an earliersolution method naturally gives rise to an eigengate. Our techniques could beof independent interest for the theory of driven many-body systems.

Quantum gates by resonantly driving many-body eigenstates, with a focus on Polychronakos' model