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The optimal cost of a three-qubit Fredkin gate is 5 two-qubit entanglinggates, and the overhead climbs to 8 when restricted to controlled-not (CNOT)gates. By harnessing higher-dimensional Hilbert spaces, we reduce the cost of athree-qubit Fredkin gate from 8 CNOTs to 5 nearest-neighbor CNOTs. We alsopresent construction of an n-control-qubit Fredkin gate with 2n+3 CNOTs and 2nsingle-qudit operations. Finally, we design deterministic and nondeterministicthree-qubit Fredkin gates in photonic architectures. The cost of anondeterministic three-qubit Fredkin gate is further reduced to 4nearest-neighbor CNOTs, and the success of such a gate is heralded by asingle-photon detector. Our insights bridge the gap between the theoreticallower bound and the current best result for the n-qubit quantum computation.

Optimal synthesis of the Fredkin gate in a multilevel system