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The controlled-SWAP and controlled-controlled-NOT gates are at the heart ofthe original proposal of reversible classical computation by Fredkin andToffoli. Their widespread use in quantum computation, both in theimplementation of classical logic subroutines of quantum algorithms and inquantum schemes with no direct classical counterparts, has made it imperativeearly on to pursue their efficient decomposition in terms of the lower-levelgate sets native to different physical platforms. Here, we add to this body ofliterature by providing several logically equivalent circuits for the Toffoliand Fredkin gates under all-to-all and linear qubit connectivity, the latterwith two different routings for control and target qubits. Besides achievingthe lowest CNOT counts in the literature for all these configurations, we alsodemonstrate the remarkable effectiveness of the obtained decompositions atmitigating coherent errors on near-term quantum computers via equivalentcircuit averaging. We first quantify the performance of the method in silicowith a coherent-noise model before validating it experimentally on asuperconducting quantum processor. In addition, we consider the case where thethree qubits on which the Toffoli or Fredkin gates act nontrivially are notadjacent, proposing a novel scheme to reorder them that saves one CNOT forevery SWAP. This scheme also finds use in the shallow implementation oflong-range CNOTs. Our results highlight the importance of considering differententanglement structures and connectivity constraints when designing efficientquantum circuits.

Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging