Figure 1

Two-qubit quantum circuit for measuring general \(n\)-time correlation functions. The first line is the ancilla (held by the nuclear spin of ¹³C), and second line is the system qubit (held by the nuclear spin of ¹H). The blue zone between the different controlled gates \({U}_{\alpha }^{k}\) on the line of qubit A represents the decoupling of the ¹³C nucleus from the nuclear spin of ¹H, while the latter evolves according to \(U({t}_{k+1};{t}_{k})\). The measurement of the quantities \(\langle {\sigma }_{x}\rangle \) and \(\langle {\sigma }_{y}\rangle \) of the ancillary qubit at the end of the circuit will directly provide the real and imaginary values of the \(n\)-time correlation function for the initial state \({\rho }_{{\rm{in}}}=|\phi \rangle \langle \phi |\).