A combinatorial optimization problem is minimizing or maximizing their cost or objective function among many variables that take discrete values. In general, it takes time to solve the combinatorial optimization problem. To deal with many combinatorial optimization problems, we utilize generic solvers to solve them efficiently. Quantum annealing (QA) is one of the generic solvers for solving combinatorial optimization problems (Kadowaki and Nishimori, 1998) using the quantum tunneling effect. Quantum annealing is a computational technique to search for good solutions to combinatorial optimization problems by expressing the objective function and constraint time requirements of the combinatorial optimization problem by quantum annealing in terms of the energy function of the Ising model or its equivalent QUBO (Quadratic Unconstrained Binary Optimization) and manipulating the Ising model and QUBO to search for low energy states (Shu Tanaka and Seki, 2022). Various applications of QA are proposed in traffic flow optimization (Neukart et al., 2017; Hussain et al., 2020; Inoue et al., 2021), finance (Rosenberg et al., 2016; Orús et al., 2019; Venturelli and Kondratyev, 2019), logistics (Feld et al., 2019; Ding et al., 2021), manufacturing (Venturelli et al., 2016; Haba et al., 2022; Yonaga et al., 2022), preprocessing in material experiments (Tanaka et al., 2023), marketing (Nishimura et al., 2019), steel manufacturing (Yonaga et al., 2022), and decoding problems (Ide et al., 2020; Arai et al., 2021a). The model-based Bayesian optimization is also proposed in the literature (Koshikawa et al., 2021). A comparative study of quantum annealer was performed for benchmark tests to solve optimization problems (Oshiyama and Ohzeki, 2022). The quantum effect on the case with multiple optimal solutions has also been discussed (Yamamoto et al., 2020; Maruyama et al., 2021). As the environmental effect cannot be avoided, the quantum annealer is sometimes regarded as a simulator for quantum many-body dynamics (Bando et al., 2020; Bando and Nishimori, 2021; King et al., 2022). Furthermore, applications of quantum annealing as an optimization algorithm in machine learning have also been reported (Neven et al., 2012; Amin et al., 2018; Khoshaman et al., 2018; Kumar et al., 2018; OḾalley et al., 2018; Arai et al., 2021b; Sato et al., 2021; Urushibata et al., 2022; Goto and Ohzeki, 2023; Hasegawa et al., 2023). In this sense, developing the power of quantum annealing by considering hybrid use with various techniques is important, as mentioned in several previous studies (Hirama and Ohzeki, 2023; Takabayashi and Ohzeki, 2023).

In this study, we propose the generation of the maze by quantum annealing. In the application of quantum annealing to mazes, algorithms for finding the shortest path through a maze have been studied (Pakin, 2017). Automatic map generation is an indispensable technique for game production, including rogue-like games. Maze generation has been used to construct random dungeons in rogue-like games by assembling mazes (mok Bae et al., 2015). Therefore, considering maze generation as one of the rudiments of this technology, we studied maze generation using a quantum annealing machine. Several algorithms for the generation of the maze have been proposed. In this study, we focused on maze-generating algorithms. One can take the bar-tipping algorithm (Alg, 2023a), the wall-extending algorithm (Alg, 2023b), and the hunt-and-kill algorithm (Alg, 2023c).

The bar-tipping algorithm is an algorithm that generates a maze by extending evenly spaced bars one by one. For the sake of explanation, we will explain the terminology here. A path represents an empty traversable part of the maze and a bar a filled non-traversable part. Figure 1 shows where the outer wall, bars, and coordinate (i, j) are in a 3 × 3 maze. The maze is surrounded by an outer wall, as shown in Figure 1. It requires the following three constraints. First, each bar can be extended by one cell only in one direction. Second, the first column can be extended in four directions: up, down, left, and right, while the second and subsequent columns can be extended only in three directions: up, down, and right. Third, adjacent bars cannot overlap each other. We explain the detailed process of the bar-tipping algorithm using the 3 × 3 size maze. In this study, a maze generated by extending the N × N bars is called N × N size maze. First, standing bars are placed in every two cells in a field surrounded by an outer wall, as shown in Figure 1. Second, Figure 2 shows each step of bar-tipping algorithm. Figure 2A shows the first column of bars extended. The bars in the first column are randomly extended in only one direction with no overlaps, as shown in Figure 2A. The bars can be extended in four directions (up, down, right, and left) at this time. Figure 2B shows the second column of bars being extended. Third, the bars in the second column are randomly extended in one direction without overlap, as shown in Figure 2B. The bars can be extended in three directions (up, down, and right) at this time. Figure 2C shows the state in which the bars after the second column are extended. Fourth, the bars in subsequent columns are randomly extended in one direction, likewise the bars in the second column, as shown in Figure 2. Figure 2D shows the complete maze in its finished state. Following the process, we can generate a maze, as shown in Figure 2D.

If multiple maze solutions are possible, the maze solution is not unique, simplifying the time and difficulty of reaching the maze goal. These constraints must be followed for the reasons described below. The first constraint prevents a maze from generating a maze with multiple maze solutions and closed circuits. Figure 3A shows a maze state that violates the first constraint. The step violating the first constraint because one bar in the upper right corner is extended in two directions, as shown Figure 3A. The second constraint prevents generating a maze from a maze with closed circuits and multiple maze solutions. Figure 3B shows a state that violates the second constraint. The second constraint is violated, it has a closed circuit and multiple maze solutions, as shown in Figure 3B. The third constraint prevents maze generation from a maze with multiple maze solutions. Figure 3C shows a state that violates the third constraint. The bars overlap in the upper right corner, making it the third constraint as Figure 3C.

Next, we describe the wall-extending algorithm. It is an algorithm that generates a maze by extending walls. Figure 4 shows the extension starting coordinates of the wall-extending algorithm. Figure 5A shows the initial state of the wall-extending algorithm. First, as an initial condition, the outer perimeter of the maze is assumed to be the outer wall, and the rest of the maze is assumed to be the path, as shown in Figure 5A. Coordinate system is different from the bar-tipping algorithm, and all cells are labeled coordinates. As shown in Figure 4, the coordinates where both x and y are even and not walls are listed as starting coordinates for wall extending. The following process is repeated until all starting coordinates change to walls, as shown in Figure 5C. The coordinates were randomly chosen from the non-wall extension start coordinates. The next extending direction is randomly determined from which the adjacent cell is a path. Figure 5B shows how the path is extended. The extension will be repeated while two cells ahead of the extending direction to be extended are a path, as shown in Figure 5B. Figure 5C shows all starting coordinates changed to walls. These processes are repeated until all the starting coordinates change to walls, as shown in Figure 5C. Figure 5D shows a maze created by wall-extending. Following the process, we can generate a maze, as shown in Figure 5D.

As a third, the hunt-and-kill algorithm is explained below. It is an algorithm that generates a maze by extending paths. Figure 6 shows the extension starting coordinates of the hunt-and-kill algorithm. Figure 7A shows the initial state of the hunt-and-kill algorithm. The entire surface is initially walled off, as shown in Figure 7A. Coordinates, where both x and y are odd, are listed as starting coordinates for path extension, as shown in Figure 6. As with the wall-extending algorithm, all cells are set to coordinates. Figure 7B shows the state in which the path is extended. A coordinate is chosen randomly from the starting coordinates, and the path is extended from there, as shown in Figure 7B. Figure 7C shows the coordinate selection and re-extension after the path can no longer be extended. If the path can no longer be extended, a coordinate is randomly selected from the starting coordinates, which are already paths, and extension starts again from it, as shown in Figure 7C. This process is repeated until all the starting coordinates turn into paths to generate the maze. Figure 7D shows the complete maze with the hunt-and-kill algorithm. Following the process, we can generate a maze, as shown in Figure 7D.

Of the three maze generation algorithms mentioned above, the bar-tipping algorithm is relevant to the combinatorial optimization problem. In addition, unlike other maze generation algorithms, the bar-tipping algorithm is easy to apply because it only requires the consideration of adjacent elements. Thus, we have chosen to deal with this algorithm. Other maze generation algorithms could be generalized by reformulating them as combinatorial optimization problems. The wall-extending and hunt-and-kill algorithms will be implemented in future studies, considering the following factors. The former algorithm introduces the rule that adjacent walls are extended and so are their walls. The number of connected components will be computed for the latter, and the result will be included in the optimization.

Using the bar-tipping algorithm, we reformulated it to solve a combinatorial optimization problem that generates a maze with a longer solving time and optimizes it using quantum annealing. Quantum annealing (DW_2000Q_6 from D-Wave), classical computing (simulated annealing, simulated quantum annealing, and algorithmic solution of the bar-tipping algorithm), and hybrid computing were compared with each other according to the generation time of mazes, and their performance was evaluated. The solver used in this experiment is as follows: DW_2000Q_6 from D-Wave, simulated annealer called SASampler and simulated quantum annealer called SQASampler from OpenJij (Ope, 2023), D-Wave's quantum-classical hybrid solver called hybrid_binary_quadratic_model_version2 (BQM), and classical computer [MacBook Pro(14-inch, 2021), OS: macOS Monterey Version 12.5, Chip: Apple M1 Pro, Memory: 16 GB]. This comparison showed that quantum annealing was faster. This may be because the direction of the bars is determined at once using quantum annealing, which is several times faster than the classical algorithm. We do not use an exact solver to solve the combinatorial optimization problem. We expect some diversity in the optimal solution and not only focus on the optimal solution in maze generation. Thus, we compare three solvers, which generate various optimal solutions.

In addition, we generate mazes that reflect individual characteristics, whereas existing maze generation algorithms rely on randomness and fail to incorporate other factors. In this case, we incorporated the maze solution time as one of the other factors to solve the maze. The maze solving time was defined as the time (in seconds) from the start of solving the maze to the end of solving the maze.

The study is organized as follows. In the next Section, we explain the methods of our experiments. In Section 3, we describe the results of our experiments. In Section 4, we summarize this study.

In this study, we show that generating difficult (longer the maze solving time) mazes using the bar-tipping algorithm is also possible with quantum annealing. By reformulating the bar-tipping algorithm as the combinatorial optimization problem, we generalize it more flexibly to generate mazes. In particular, our approach is simple but can adjust the difficulty in solving mazes by quantum annealing.

In Section 3.2, regarding comparing computational costs to solve our approach to generating mazes using TTS, DW_2000Q_6 has a smaller coefficient of N² than the classical counterpart. Therefore, as N increases, the computational cost of DW_2000Q_6 can be expected to be lower than that of the classical simulated annealing for a certain time. Unfortunately, since the number of qubits in the D-Wave quantum annealer is finite, the potential power of generating mazes by quantum annealing is limited. However, our insight demonstrates some advantages of quantum annealing against its classical counterpart. In addition, we observed that the hybrid solver's computational cost was constant up to N = 18. This indicates that hybrid solvers will be potentially effective if they are developed to deal with many variables in future.

In Section 3.3, we proposed Q_update to increase the solving time using quantum annealing. We demonstrated that introducing Q_update increased the time to solve the maze and changed the difficulty compared with the case where Q_update was not introduced. At this time, the parameters (λ_update1, λ_update2, and a) were fixed. Difficult maze generation for everyone may be possible by adjusting the parameters individually.

One of the directions in the future study is in applications of our cost function in various realms. We should emphasize that Q_update proposed in this study also has the potential use in various fields related to training and education. The powerful computation of quantum annealing and its variants open the way to such realms with high-speed computation and various solutions.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Ethical approval was not required for the studies involving humans because the data was obtained through collaboration among the participants of a group workshop developing an application using quantum annealing. The studies were conducted in accordance with the local legislation and institutional requirements. Written informed consent for participation was not required from the participants or the participants' legal guardians/next of kin in accordance with the national legislation and institutional requirements because the data was obtained through collaboration among the participants of a group workshop developing an application using quantum annealing.

YI: Writing – original draft, Writing – review & editing, Conceptualization, Data curation, Formal analysis, Investigation, Software, Validation, Visualization, Methodology. TY: Writing – original draft, Writing – review & editing, Conceptualization, Validation, Methodology. KO: Writing – original draft, Writing – review & editing, Conceptualization, Validation, Methodology. MO: Supervision, Writing – review & editing, Funding acquisition, Project administration.