Fig. 2: Decomposition of the double-controlled composite gates in the LCU circuits.
A Example of the decomposition of a double-controlled −ZZ gate, which is controlled on \(\left\vert 1\right\rangle\) of a0 and \(\left\vert 0\right\rangle\) of a1, into CCZ (blue) along with other single- and two-qubit gates. The X gates (green) are used to adjust the control states; the CZ gate on a0 and a1 (purple) is used to adjust the overall multiplicative factor, which is −1 in this case; the CNOT gate equivalents (orange) are used to extend the weights of the Pauli string as in ref. 52. B Decomposition of the CCZ gates with the iToffoli gate, which is a \({\rm{CC}}{\hbox{-}}i{\rm{X}}\) gate with both control qubits controlled on \(\left\vert 0\right\rangle\). The decomposition includes the equivalent of a \({\rm{CC}}{\hbox{-}}i{\rm{Z}}\) gate (light blue) and the equivalent of a long-range CS†gate (yellow). The SWAP gates are simplified in the transpilation stage or further decomposed with CZ gates according to ref. 53.
