Coined Quantum Walks on Complex Networks for Quantum Computers

We propose a quantum circuit design for implementing coined quantum walks on complex networks. In complex networks, the coin and shift operators depend on the varying degrees of the nodes, which makes circuit construction more challenging than for regular graphs. To address this issue, we use a dual-register encoding. This approach enables a simplified shift operator and reduces the resource overhead compared to previous methods. We implement the circuit using Qmod, a high-level quantum programming language, and evaluated the performance through numerical simulations on Erd\H{o}s–R\'enyi, Watts–Strogatz, and Barab\'asi–Albert models. The results show that the circuit depth scales as approximately $N^{1.9}$ regardless of the network topology. Furthermore, we execute the proposed circuits on the \texttt{ibm\_torino} superconducting quantum processor for Watts–Strogatz models with $N=4$ and $N=8$. The experiments show that hardware-aware optimization slightly improved the $L_1$ distance for the larger graph, whereas connectivity constraints imposed overhead for the smaller one. These results indicate that while current NISQ devices are limited to small-scale validations, the polynomial scaling of our framework makes it suitable for larger-scale implementations in the early fault-tolerant quantum computing era.