Toffoli gates are key building blocks in quantum programs, and on most current quantum computers, they must be implemented with smaller gates. Such an implementation requires five 2-qubit gates if we assume that each gate can operate on any two qubits. However, many current quantum computers have only 2-qubit gates that operate on neighboring qubits; we call them neighbor gates. How many neighbor gates are required to implement a Toffoli gate? In this article, we show that six neighbor gates are necessary and sufficient, and we generalize to a characterization of all 3-qubit diagonal gates.

We present a compilation scheme for conditional quantumgates. Our scheme compiles amulti-qubit conditional to a linear number of two-qubit conditionals. This can be done straightforwardly with helper qubits, but we show how to do it without using helper qubits and with much fewer gates than in previous work. Specifically, our scheme requires 1/3 as many gates as the previous best scheme without using helper qubits, which is essential for practical use. Our experiments show that several quantum-circuit optimizers have little impact on the compiled code from the previous best scheme, confirming the need for our new scheme. Our experiments with Grover’s algorithm and quantum walk also show that our scheme has a major impact on the reliability of the compiled code.