Extended Data Fig. 7: Process χ matrix block for 25 μs of PReSPA.

The matrix converts elements of input density matrices (top axis) to output density matrix elements (left axis) expressed in the Fock state basis. a, Amplitude values of the χ matrix elements. The upper left block (\({\chi }_{nn,mm}\)) describes the conversion of diagonal elements of the input and output density matrices, which is associated with transfer of photon occupation probabilities calculated from transmon spectroscopy experiments (as in Fig. 2d, e). The lower right block (\({\chi }_{nm,kl}\)) describes the conversion of relevant off-diagonal elements of density matrices, which is calculated from Wigner tomography and density matrix reconstruction⁵⁹ (as in Fig. 3). The greyed blocks are assumed to be zero owing to the absence of interference between the four conversion paths in PReSPA. b, Phase values of the χ matrix elements for the lower right block in a. For best illustration of the PReSPA process, the phases are reported in the full rotating frame where all Fock states have zero energy. Values in grey are measured but not statistically significant because the corresponding amplitude value is not large enough. In this frame, as prescribed by equation (2), PReSPA requires zero phase for the six \({\chi }_{nm,(n+1)(m+1)}\) elements representing the coherence of the even-to-odd conversion process, which is accomplished by our PReSPA calibration. The diagonal elements \({\chi }_{nm,nm}\) representing the preservation of odd-state superpositions should have zero phase by definition. Their systematic deviation from zero was caused by parameter drift in the experiment as that block of data was acquired at a later time than the earlier rotating frame calibration.