Constant-cost implementations of Clifford operations and multiply controlled gates using global interactions
A frequent physical-level requirement for the ability to implement quantum operations simultaneously is their commutativity. In this work, we asked if quantum computation by commuting Ising-type entangling operations and ``free'' single-qubit gates can be advantageous compared to quantum computation by the two-qubit gates and ``free'' single-qubit gates. We focused on the elements of the Clifford group and the multiply controlled gates. It turned out that such circuits and composite gates can be implemented with very little effort -- using constantly or effectively constantly many blocks of commuting sets of gates, in all cases except some with the most severe restrictions on the number of ancillary qubits available. Specifically, we show constant-cost implementations of Clifford operations with and without ancilla, constant-cost implementation of the multiply controlled gates with linearly many ancillae, and an $O(\log^*(n))$ implementation of the $n$-controlled single-target gates using logarithmically many ancillae. This shows significant (asymptotic) improvement of circuits enabled by the global gates vs those over single- and two-qubit gates.