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Physical Review A

Proposal for implementing quantum walks on multidimensional networks

Qi-Ping Su, Rui-Qi Shen, Li Zhi Yu, Yu Wang, Yi‐Hao Kang, Lijiong Shen, Wei Feng, Guo-Qiang Zhang, Chui‐Ping Yang

Published Aug 15, 2025• 0 citations
Publisher

Abstract

Discrete-time quantum walks (DTQWs) have broad applications in quantum computation and simulation, particularly in universal quantum computing. Prior research in circuit quantum electrodynamics (QED) has focused on implementing DTQWs in one-dimensional chains. However, to achieve practical quantum tasks, it is crucial to extend DTQWs to multidimensional networks with diverse structures. Here, we propose a protocol for realizing DTQWs in multidimensional networks, which accommodate nodes with different numbers of neighbors. Using this protocol, we numerically simulate a Grover walk search algorithm in a cubic network with circuit QED. Our simulations, which consider decoherence, align well with theoretical predictions. Our universal protocol can be adapted for various quantum systems and paves the way for DTQWs in multidimensional networks, which has potential applications in quantum computing and simulation.

DOI
10.1103/f3kt-qf1d

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Discrete-time quantum walks (DTQWs) have broad applications in quantum computation and simulation, particularly in universal quantum computing. Prior research in circuit quantum electrodynamics (QED) has focused on implementing DTQWs in one-dimensional chains. However, to achieve practical quantum tasks, it is crucial to extend DTQWs to multidimensional networks with diverse structures. Here, we propose a protocol for realizing DTQWs in multidimensional networks, which accommodate nodes with different numbers of neighbors. Using this protocol, we numerically simulate a Grover walk search algorithm in a cubic network with circuit QED. Our simulations, which consider decoherence, align well with theoretical predictions. Our universal protocol can be adapted for various quantum systems and paves the way for DTQWs in multidimensional networks, which has potential applications in quantum computing and simulation.
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Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-qubit processor and address scalability to higher dimensions as well as larger quantum processors.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. Quantum walks represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves onto the multi-qubit states in a quantum processor, we present different configurations of quantum circuits for the implementation of discrete-time quantum walks in one-dimensional position space. With example circuits for a five qubit machine we address scalability to higher dimensions and larger quantum processors.
Research has shown that quantum walks can accelerate certain quantum algorithms and act as a universal paradigm for quantum processing. The discrete-time quantum walk (DTQW) model, owing to its discrete nature, stands out as one of the most suitable choices for circuit implementation. Nevertheless, most current implementations are characterized by extensive, multi-layered quantum circuits, leading to higher computational expenses and a notable decrease in the number of confidently executable time steps on current quantum computers. Since quantum computers are not scalable enough in this NISQ era, we also must confine ourselves to the ancilla-free frontier zone. Therefore, in this paper, we have successfully cut down the circuit cost concerning gate count and circuit depth by half through our proposed methodology in qubit systems as compared to the state-of-the-art increment–decrement approach. Furthermore, for the engineering excellence of our proposed approach, we implement DTQW in any finite-dimensional quantum system with akin efficiency. To ensure an efficient implementation of quantum walks without requiring ancilla, we have incorporated an intermediate qudit technique for decomposing multi-qubit gates. Experimental outcomes hold significance far beyond the realm of just a few time steps, laying the groundwork for dependable implementation and utilization on quantum computers.