SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Here, we compare the performance of the baselines (NFT-sequential and NFT-random) to our NFT-with-EMICoRe under different settings of  $Q$ ,  $L$ , and  $N_{\mathrm{shots}}$ . All figures in this appendix (Figures 9-17) are shown in the same format as Figure 3: for each figure, the energy (left column) and the fidelity (right column) are shown for the Ising (top row) and the Heisenberg (bottom row) Hamiltonians. In each panel, the left plot shows the optimization progress with the median (solid) and the 25- and 75-th percentiles (shadow) over the 50 seeded trials, as described in Appendix H, as a function of the observation costs, i.e., the number of observed points. The portrait square on the right shows the distribution of the final solution after 200, 600, and 1000 observations have been performed, respectively, for  $Q = 3$ , 5, and 7 qubit cases. As mentioned earlier, the prior standard deviation  $\sigma_0$  is set roughly proportional to  $Q$ . Specifically we use  $\sigma_0 = 4$ , 6, 9 for  $Q = 3$ , 5, 7, respectively.
3-qubit setup: Figures 9-11 show results for  $Q = 3$ ,  $L = 3$ , and  $N_{\mathrm{shots}} = 256$ , 512, 1024. Given the relatively low-dimensional problem with only  $D = 24$ , the convergence rate is comparable for both baselines and our approach. However, the red sharp peak in the density plot of the final solutions
(right portrait square) in each panel implies that the robustness against initialization is improved by our NFT-with-EMICoRe approach, thus highlighting its enhanced stability and noise-resiliency.
5-qubit setup: Figures 12-14 show results for  $Q = 5$ ,  $L = 3$ , and  $N_{\mathrm{shots}} = 256$ , 512, 1024. The case for  $N_{\mathrm{shots}} = 1024$  is identical to Figure 3 in the main text. For all noise levels ( $N_{\mathrm{shots}} = 256$ , 512, 1024), our EMICoRe (red) consistently achieves lower energy and higher fidelity compared to the baselines, thus demonstrating the superiority of our NFT-with-EMICoRe over NFT [11]. Remarkably, we observe that in high-noise-level cases, such as for the Heisenberg Hamiltonian for  $N_{\mathrm{shots}} = 256$ , 512 (bottom rows of Figure 12 and Figure 13), the achieved energy by the state-of-the-art baselines (purple and green) fail to even surpass the energy level of the first excited state for the 600 observed data points, whereas EMICoRe successfully accomplishes this task.
7-qubit setup: Figures 15-17 present results for  $Q = 7$ ,  $L = 5$ , and  $N_{\mathrm{shots}} = 256, 512, 1024$ . Again, NFT-with-EMICoRe consistently outperforms the baselines in all experimental setups. Given the increased complexity associated with  $Q = 7$  and  $D = 84$ , the optimization process becomes more challenging, necessitating a greater number of observed points to approach the ground-state energy. Nonetheless, even with just 1000 observed points, NFT-with-EMICoRe already exhibits significant superiority over NFT [11], particularly in high-noise scenarios such as  $N_{\mathrm{shots}} = 256$ . We also observed that the optimization process faces difficulties with the Heisenberg Hamiltonian case. We attribute this behavior to the greater complexity of the latter task and defer further analysis of this regime to future studies.
Figure 9: Comparison (in the same format as Figure 3) between our NFT-with-EMICoRe (red) and the NFT baselines (green and purple) in VQE for the Ising (top row) and Heisenberg (bottom row) Hamiltonians with the  $(L = 3)$ -layered  $(Q = 3)$ -qubit quantum circuit (thus,  $D = 24$ ) and  $N_{\mathrm{shots}} = 256$ .
Figure 10: Same comparison as in Fig. 9, with the  $(L = 3)$ -layered  $(Q = 3)$ -qubit quantum circuit (thus,  $D = 24$ ) and  $N_{\mathrm{shots}} = 512$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
Figure 11: Same comparison as in Fig. 9, with the  $(L = 3)$ -layered  $(Q = 3)$ -qubit quantum circuit (thus,  $D = 24$ ) and  $N_{\mathrm{shots}} = 1024$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
Figure 12: Same comparison as in Fig. 9, with the  $(L = 3)$ -layered  $(Q = 5)$ -qubit quantum circuit (thus,  $D = 40$ ) and  $N_{\mathrm{shots}} = 256$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
Figure 13: Same comparison as in Fig. 9, with the  $(L = 3)$ -layered  $(Q = 5)$ -qubit quantum circuit (thus,  $D = 40$ ) and  $N_{\mathrm{shots}} = 512$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
Figure 14: Same comparison as in Fig. 9, with the  $(L = 3)$ -layered  $(Q = 5)$ -qubit quantum circuit (thus,  $D = 40$ ) and  $N_{\mathrm{shots}} = 1024$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
Figure 15: Same comparison as in Fig. 9, with the  $(L = 5)$ -layered  $(Q = 7)$ -qubit quantum circuit (thus,  $D = 84$ ) and  $N_{\mathrm{shots}} = 256$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
EMiCoRe (ours) NFT-Sequential NFT-Random
Figure 16: Same comparison as in Fig. 9, with the  $(L = 5)$ -layered  $(Q = 7)$ -qubit quantum circuit (thus,  $D = 84$ ) and  $N_{\mathrm{shots}} = 512$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.
Figure 17: Same comparison as in Fig. 9, with the  $(L = 5)$ -layered  $(Q = 7)$ -qubit quantum circuit (thus,  $D = 84$ ) and  $N_{\mathrm{shots}} = 1024$ . The NFT-with-EMICoRe (red) and NFT baselines (green and purple) are shown for both Ising (top row) and Heisenberg (bottom row) Hamiltonians.