### NOR and NAND operation features

NOR is known to be a versatile computing unit. In our method, the Hamiltonian is designed to reduce the power of the NOR logic components. The superconducting quantum circuit is constructed by direct implementation of the Hamiltonian circuit shown in Supplementary Fig. 1(a). Two types of samples consisting of three qubits corresponding to . are prepared *a* And the *B* for input and *s* For the boolean result, with critical currents (*I*_{c}) from 6.25 A (NOR1) and 3.75 A (NOR2). The configuration model is described in the “Methods” section. Boolean components of NOR, corresponding to four groups of (*a*And the *B*And the *s*) with minimum power, appear at dissolution point after quality assurance. Theoretically, the dissolution point is expressed as.

$$ I_{h2} = frac {{M_{23}}}{{M_{31}}}cdot I_{h1}$$

And the

$$ I_{h3} = frac {{M_{1}}}{{M_{3}}}cdotfrac {{M_{23}}}{{M_{12}}}cdot I_{h1 } $$

(1)

where *I*_{hI} (*I* = 1–3) is the external bias of the qubit *I*(corresponding to the labels *a* And the *B,*And the *s* ), *M*_{I} (*I* = 1–3) is the mutual inductance between qubits *I*and the external bias line, and *M*_{IJ} (*I*= 1–3, *y*= 1–3) is the mutual induction between the qubits *I*And the *y*. Equation derivation process. (1) Described in Supplementary Methods. The inductances of qubits and their mutual inductors are extracted from the circuit diagram (see Methods). The theoretical decay points of NOR1 and NOR2 are estimated as (*I*_{h1}And the *I*_{h2}And the *I*_{h3}) = (1.4, 1.4, 3.1) and (1.3, 1.3, 2.8) [µA], Straight. Figure 1a-c respectively show the state diagrams obtained from the theory, from simulations using the Josephson Integrated Circuit Simulator (JSIM)^{27} with *I*_{h3}= 2.0 µA, and from experience with *I*_{h3}= 2.0 µA carried out at 10 mK. The detailed JSIM and experimental methods are presented in the Methods section and in the “Experimental Configuration” section of the Supplementary Methods, respectively. In JSIM analysis, the thermal noise current is neglected in order to emphasize the direction of the boundary state in each logic component. A breakpoint is found, where each boolean component of a NOR appears, about the current state of (*I*_{h1}And the *I*_{h2} And the *I*_{h3} ) = (1.8, 1.8, 2.0) [µA] Both in experiments and in JSIM analysis in NOR1. Supplementary Figure 3 shows the iterative distribution of the logical components in the experiments performed at the dissolution point. Boolean components corresponding to Hamiltonian minimum energy are selectively generated. In the case diagram, the boundary is found along the diagonal direction, which we call a “ladder” for convenience. When the ladder goes up diagonally in the left direction *I*_{h3}It decreases from the point of dissolution (Fig. 1d-f). On the other hand, a ladder that rises diagonally in the right direction is created when *I*_{h3}It increases the point of dissolution (Fig. 1g-i). These trends are qualitatively consistent with theory, JSIM analysis, and experiments. At the experimentally obtained decay point, the logical components of the NOR occur randomly (see Supplementary Fig. 4 and “Detailed characteristics of the NOR process” section in the Supplementary Note). Note that we can produce a desired logic component by applying an appropriate compensation current (α) against the decay point. For example, the logical component of (*a*And the *B*) = (0, 1) can be considered by applying the outflow bias of (*I*_{h1} ‘, *I*_{h2} ′) = (*I*_{h1}– a, *I*_{h2}+ α). This corresponds to the dependence of α along the diagonal direction of the decay point. By applying an appropriate value of α, the NOR logic can be reproduced with high accuracy (see Supplementary Fig. 5). We confirm that injecting flux into one of the qubits by adopting α into the initial state restricts the state of the other qubits because the qubits interact with each other to reduce energy after QA. Moreover, this quantum circuit behaves like a NAND when *I*_{h}It is supplied with a negative sign. In the NAND state plot, the absolute value of the degeneration point is almost the same as the value of NOR. The boundaries of each logical component are modified by *I*_{h3,} The same is the case for NOR (see Supplementary Fig. 6 and the “NAND Operation” section in the Supplementary Note). Each logic component of a NAND is reproduced with a success probability of 100% by adopting an appropriate value of α (see Supplementary Fig. 7). QA in NOR1 shows a high probability of success in NOR and NAND operations, but its breaking point varies between theory, JSIM analysis and experiments.

### Gray Zone Rating

In NOR2, the barrier height in the energy potential per qubit is reduced compared to that in NOR1 due to the reduction of *I*_{c}. Supplementary Figure 8 shows the frequency distribution for each logic component with the current state of (*I*_{h1}And the *I*_{h2} ) = (1.6, 1.6) [µA] and modify *I*_{h3}Between 0 and 9 A. About *I*_{h3}From 2.8 µA, all candidate logic components occur in the NOR. Figure 2a,b show case diagrams with 2D and 3D images in *I*_{h3}From 2.8 the experimental decay point is close to the theoretical point. Note that the boundaries of each logical component change drastically around this point. The empirical decay point of NOR2 is (*I*_{h1} And the *I*_{h2}And the *I*_{h3} ) = (1.6, 1.6, 2.8) [µA]. For convenience, we define the transient width between two different logical regions as the ‘gray region’. There are two types of gray areas: the first type is created between adjacent areas, such as “100” – “001” and “001” – “010”, and the second type occurs in the same diagonal direction as the ladder. Theoretically, the width of the ladder decreases monotonously with the external bias *I*_{h3}before the degeneration point. Later, it increases monotonously with *I*_{h3}. The first and second types can be evaluated from four types of line profiles (L._{1} -to_{4}) and L profiles._{5} and me_{6} , respectively (Fig. 2c and f). in L._{5} and me_{6} , four logical components of NOR have been identified. Type I gray areas depend on the annealing time (*T*_{a} ) (See Supplementary Figure 9 and the “Type I gray area feature” section of the Supplementary Note). as such *T*_{a} decreases, the spread of the gray area becomes wider. with a longer period *T*_{a} , the effect of noise can be time-averaged. This contributes to the reduction of the gray area, which leads to the use of the quantum annealing effect. These gray areas are illustrated in the case of JSIM analysis with thermal noise current (see Supplementary Fig. 10). Figure 3a,b show the type I gray areas assessed in the experiments and in the JSIM analysis, respectively. The minimum width of the gray area varies between experiments and JSIM analysis. The effect of the flow due to the surrounding circles appears differently between the JSIM and the experiments, resulting in a difference in the minimum width of the gray area. However, consideration of the current equidistant step in assessing the gray area contributes to suppressing the effect of generating a secondary logic component. The gray areas between ‘100’ and ‘001’ and between ‘001’ and ‘010’ tend to be significant. These trends are consistent with the fact that the border situation is likely to change due to *I*_{h3} In Figure 1, indicating the ease of changing the energy state. On the other hand, the values are small in the boundary cases between “110” and “010” and between “100” and “110”. These trends are consistent with the fact that the values of *I*_{h1} And the *I*_{h2}Do not change with modification *I*_{h3}In Figure 1, indicating the difficulty of changing the energy state. These relationships are also confirmed regardless of the value of *T*_{a}(See Supplementary Figure 9). Figure 3c,d show a type II gray area with two orientations in experiments and JSIM analysis, respectively. The first is a monotonous response against the absolute value of *I*_{h3}Starting at the degeneration point. This trend is consistent with the predictions of the theory. The second is the gray area that spreads a little wider with lower *I*_{h3}Before the dissolution point of increase *I*_{h3}beyond the point of dissolution. These trends are consistent with the result shown in Fig. 1, where the occupation of the ‘001’ region is broadly modified with decreasing *I*_{h3}Compared with the case of increased *I*_{h3}. JSIM analysis also reproduces the same trends seen in experiments. Note that the trends change for a thermal noise current above 2.5 pA/Hz in the JSIM analysis. Below 2.0 pA/Hz, trapping to the local minimum state occurs (see the “JSIM gray area analysis” section in the Supplementary Methods). The logic in NOR and NAND can be achieved with high precision by tuning the current state with α values above 1 µA along a diagonal direction from the decay point, which contributes to avoiding the gray area.