Fig. 1: Illustration of the pulse inverse used in the KIK method.

a Quantum gates are executed via classical control signals, or pulses. The left panel shows a pulse schedule used for a CNOT gate in the IBM quantum computing platform. The pulse schedule in the right panel performs the inverse of the CNOT through the inverse driving \({{{{\mathcal{H}}}}}_{{{{\rm{I}}}}}(t)\). It is constructed from the original pulse schedule \({{{\mathcal{H}}}}(t)\), by inverting the amplitudes of the pulses (black curved arrow) and their time ordering (red curved arrow). b Instead of the pulse inverse, circuit folding and other variants of unitary folding^13,34 use the CNOT as its own inverse. Therefore, the pulse schedule for the inverse evolution is not modified. c Noisy implementations of \({{{\mathcal{K}}}}\) and \({{{{\mathcal{K}}}}}_{{{{\rm{I}}}}}\). We assume that during the executions of \({{{\mathcal{K}}}}\) and \({{{{\mathcal{K}}}}}_{{{{\rm{I}}}}}\) temporal variations of the noise due to external factors (e.g. temperature variations) are negligible. Thus, any time dependence in \({{{\mathcal{L}}}}(t)\) is induced by the time dependence of \({{{\mathcal{H}}}}(t)\). (Top) This leads to gate dependent noise depicted by different border colors in the gates U_a, U_b, and U_c. (Bottom) Since \({{{{\mathcal{H}}}}}_{{{{\rm{I}}}}}(t)\) reverses the time ordering of \({{{\mathcal{H}}}}(t)\), the time ordering of \({{{\mathcal{L}}}}(t)\) is also reversed. However, the sign of \({{{\mathcal{L}}}}(t)\) does not change because otherwise the inverse evolution would undo the noise.