*3.5. Small Maze*

To demonstrate slightly complicated cases, solutions of small mazes by the method are presented. The maze includes some dead ends and two paths to the goal as shown in Figure 6a. As in the other related cases shown above, the maximum density remains on the shortest path, and the densities of the dead-end paths vanish. The maze-solving worked correctly for a small maze with both dead-ends and cycles.

Figure 6b shows the time profiles of the densities on selected edges. The convergence speeds were not so different from each other, as in the case of ladder-like structures.

The result of another maze that is slightly modified from Figure 6a is shown in Figure 6c. The convergence step was 22,000 for this example; however, a drastic increase of convergence steps was often observed by another slight modification of the structure. It might be in rare cases that the complex maze could be solved in a permissible computational time.

Table 4 lists the relations between the distances of paths and amplitude remaining on an edge in Figure 6. The hypothesis that the relative ratio of the longest path and the second-longest path is in inverse proportion to the ratio of the distances of the non-shared part of each path was also confirmed for these cases.

**Figure 6.** The results of maze-solving for small mazes with dead ends and two paths to the goal. (**a**) Amplitude distribution and the number of steps after convergence for a maze. (**b**) Time profiles of the densities on selected edges for (**a**). (**c**) Amplitude distribution and the number of steps after convergence for a slightly modified maze.

**Table 4.** The relation between the distance and remaining amplitude for Figure 6 (The waypoint of a path, the distance between the start and goal on the path, the amplitude remaining on an edge on the path, and an approximate rational expression of the amplitude).


### *3.6. Undesirable Cases 1: Odd Cycle*

Here, we show some examples of undesirable cases where maze-solving did not work. First, this method cannot be applicable for a maze that includes odd cycles. Figure 7a shows a network with a single odd cycle whose length is 5. The cycle that consists of nodes 1, 2, 5, 6, and 3 is the odd cycle. When the amplitude distribution converged, the absolute value of the amplitude between the exit of the cycle and the goal (edges between nodes 4 and 5) became small. The correct path was not indicated by the maximum densities, meaning that the maze-solving went wrong.

**Figure 7.** The results of the attempt of maze-solving for a network that includes an odd-cycle. (**a**) Amplitude distribution and the number of steps after convergence for a network with one odd cycle. (**b**) Amplitude distribution and the number of steps after convergence for of a network with two sequential odd cycles. The amplitude distribution that is nearly converged but not completely is shown because of the limitation of the computational time.

Figure 7b shows an attempt of solving for the network with two sequential odd-cycles. The effects of the two odd-cycles were not canceled out, and only a small amplitude reached the goal. The solving method presented cannot apply to the network with odd-cycles.

### *3.7. Undesirable Cases 2: Eternal Vibration*

Even though the odd cycle was not involved in the network, undesirable eternal vibration was observed in some cases. Figure 8a shows the network of the ladder-like structure for *L* = 2, which is exhibiting eternal vibration. In this case, only the edges between nodes 5 and 6, were stabilized. The amplitudes of other edges, from the start to the cycle, exhibit a constant vibration pattern eternally. Figure 8b shows the time profiles of the densities of some edges. The constant amplitude vibrations seem to continue eternally and not converge. The inset shows the details of the vibrational behavior. The same patterns are seen to be repeating. The eternal vibration was observed only for the ladderlike structure of *L* = 2 and 5.

We found that the eternal vibration was suppressed by the addition of an extra dead end. Figure 8c shows the network in which one dead-end is attached to Figure 8a. The amplitude distribution converged, and the shortest path was indicated by the maximum densities. Figure 8d shows the time profile of the density for some selected edges in Figure 8c. The reason for the stabilization is unclear at present; however, a small perturbation of the network may have a significant influence on the behavior of quantum walks.

**Figure 8.** The results of the attempts of maze-solving for a structure where the eternal vibration was observed. (**a**) Amplitude distribution after 20,000 steps for the ladder-like structure of *L* = 2. (**b**) Time profiles of the densities on selected edges in (**a**). The inset shows the vibrational behavior in detail. (**c**) Amplitude distribution and the number of steps after convergence for the network in (**a**) where a dead-end is attached. (**d**) Time profiles of the densities on selected edges in (**c**).
