Figure 2

NMR sequence to realize the quantum algorithm for measuring \(n\)-time correlation functions. The black line and blue line mean the ancillary qubit (marked by ¹³C) and the system qubit (marked by ¹H). All the controlled quantum gates \({U}_{\alpha }^{k}\) are decomposed into the following sequence in the bottom of the plot. Gz means a \(z\)-gradient pulse which is used to cancel the polarization in \(x-y\) plane. (a) NMR sequence for measuring the 2-time correlation function \(\langle {\sigma }_{y}(t){\sigma }_{x}\rangle \). Other 2-time correlation functions can be similarly measured. (b) NMR sequence for measuring the 3-time correlation function \(\langle {\sigma }_{y}({t}_{2}){\sigma }_{y}({t}_{1}){\sigma }_{z}\rangle \). (c) NMR sequence for measuring the 2-time correlation function \(\langle {\sigma }_{x}(t){\sigma }_{x}\rangle \) with a time-dependent Hamiltonian \( {\mathcal H} ^{\prime} (t)=500{e}^{-300t}\pi {\sigma }_{y}\). The method to decouple the interaction between ¹³C and ¹H nuclei is Waltz-4 sequence.