Figure 4.

Numerical results on the weighted sum of the RQCs utilizing the weighting PQCs. (a) A typical five-qubit RQC organization with interlacing two-qubit layer and randomly set single-qubit gate. (b) Scalability tests from two-qubit to eight-qubit quantum processes. The left-axis (blue circle) shows the maximum $F_{\mathrm{avg}}$ reached during repetitive trials, and the logarithmic right-axis represents the corresponding number of input states N needed, while the red rectangular denotes the input state in our PQC, which is at least two orders of magnitude fewer than that in SQPT. It is noted that the data presented is valid for both fixed and variational weighting parameter conditions, since we utilized the same PQC configuration and achieved the same results. (c) VQPT on the weighted sum of the randomly generated quantum circuit on n-qubit, d-depth, and N-input configurations (denoted as $(n,d,N)$). The weighting probability $p=0.1$ is a fixed and known parameter. (d) VQPT on the weighted sum of the randomly generated quantum circuit on $(6,7,35)$ cases with unknown weighting parameters p. We initialized the parameter $p_{\mathrm{init}}=0.1$, and we presented two trials with different target values $p_{\mathrm{opt}}=0.05$ and $p_{\mathrm{opt}}=0.20$ (red dotted lines). Results demonstrated convergence to the optimal values, and the maximum $F_{\mathrm{avg}}$ is over $99\%$.