Figure 3

Experimental results (dots) for 2-time correlation functions. In this case, only two controlled quantum gates \({U}_{\alpha }^{0}\) and \({U}_{\beta }^{1}\) are applied with an interval of \(t\). For example, \({U}_{\alpha }^{0}\) and \({U}_{\beta }^{1}\) should be chosen as \(C-i{R}_{x}^{2}(\pi )\) and \(C-{R}_{y}^{2}(\pi )\), respectively, to measure the 2-time correlation function \(\langle {\sigma }_{y}(t){\sigma }_{x}\rangle \). t is swept from 0.5 ms to 10 ms with a 0.5 ms increment. The input state of ¹H nuclei \({\rho }_{{\rm{in}}}=|\phi \rangle \langle \phi |\) is shown on each diagram. All experimental results are directly obtained from measurements of the expectation values of \(\langle {\sigma }_{x}\rangle \) and \(\langle {\sigma }_{y}\rangle \) of the ancillary qubit. The orange and blue results respectively mean the real and imagine part of the observed 2-time correlation functions.