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arXiv:2512.16400 · Quantum Information Processing

Coined Quantum Walks on Complex Networks for Quantum Computers

Rei Sato

quant-phSubmitted Dec 18, 2025• Announced Dec 19, 2025• 0 citations
v1 · Dec 18, 2025v2 · Apr 23, 2026
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Abstract

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 Erdős-Rényi, Watts-Strogatz, and Barabási-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 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.

DOI
10.1007/s11128-026-05203-3
Comments
6 figures, 11 pages

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In this paper, we propose a circuit design for implementing quantum walks on complex networks. Quantum walks are powerful tools for various graph-based applications such as spatial search, community detection, and node classification. Although many quantum-walk-based graph algorithms have been extensively studied, specific quantum circuits for implementing these algorithms have not yet been provided. To address this issue, we present a circuit design for implementing the discrete-time quantum walk on complex networks. We investigate the functionality of our circuit using the small-sized Watts-and-Strogatz model as the complex network model, comparing it with theoretical calculations. This work offers a new approach to constructing quantum circuits for implementing quantum walks on arbitrary complex networks.
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Quantum random walks, which drew much attention over the past few decades for their distinctly nonclassical behavior, are a promising subfield within quantum computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing coin quantum random walks using a Quantum Adder-based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit depth, gate count, connectivity requirements and scalability. We propose and analyze a novel approach to implementing boundary conditions for these walks, demonstrating the technique explicitly in one and two dimensions. Finally, we present several fidelity results from running our circuits on IBM’s quantum volume 32 ‘Toronto’ chip, showcasing the extent to which these noisy intermediate-scale quantum devices can currently handle quantum walks.
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