Fig. 3

The second program to quantitatively study FGR. a Brief sketch of logic diagram. In this program, 26 temporal modes are created. 13 of them are still performed with \(U\) (i.e., the program in Fig. 2), and the other 13 are performed with \({U}_{P}\). This structure constitutes a big controlled gate by an ancilla qubit. \(n\) = 5 or 6 periods of evolutions are compiled in this program. The \({R}_{x}\) operation belongs to the state preparation and detection, and is performed by the transformations between temporal to polarization modes (See Supplementary Information). Using this algorithm, the average fidelity \(\bar{F}\) can be directly readout. b Fidelity decay under \(k=1\) and \(k=12\). When \(k=1\), fidelity fluctuates as the period index \(i\) grows; when \(k=12\), fidelity decays following FGR, and the fitted slope of F denoted as \(=c(j)\triangle\) is just the decay rate, where \(c(j)\) is a negative constant related to \(j\). c Fidelity decay rates represented by \(\triangle\) with various \(\delta\) under \(k=12\). It demonstrates that the decay rate is proportional to \({\delta }^{2}\) within the errors of Trotter expansion, and the fidelity decay rate coincidences with FGR. d shows that when the evolution is located in Fermi golden region, a constant decay rate will be observed under various k