Searching via quantum walk is a topic that has been extensively studied. Most previous results provide approximate solutions, while in this paper we prove an algorithm that can find a marked vertex certainly. We adopt the coined discrete-time quantum walk (DTQW) model with adjusted operators and prove that, on complete bipartite graphs, when parameters are set properly, coined DTQW can deterministically find a marked vertex, i.e., the success probability is exactly 1. Before this paper there have been results of an alternating continuous-time quantum walk method that achieve deterministic spatial search, but this paper provides a deterministic quantum spatial search result via DTQW, while maintaining a quadratic speedup compared to classical algorithms. We also provide the quantum circuit implementation. Published by the American Physical Society 2024Searching via quantum walk is a topic that has been extensively studied. Most previous results provide approximate solutions, while in this paper we prove an algorithm that can find a marked vertex certainly. We adopt the coined discrete-time quantum walk (DTQW) model with adjusted operators and prove that, on complete bipartite graphs, when parameters are set properly, coined DTQW can deterministically find a marked vertex, i.e., the success probability is exactly 1. Before this paper there have been results of an alternating continuous-time quantum walk method that achieve deterministic spatial search, but this paper provides a deterministic quantum spatial search result via DTQW, while maintaining a quadratic speedup compared to classical algorithms. We also provide the quantum circuit implementation. Published by the American Physical Society 2024

Physical Review ResearchJul 9, 20245 citations

Decoherent quantum walks driven by a generic coin operation

Gonzalo Abal, R. Donangelo, F. Severo +1 more

arXiv:0708.1297Physica A Statistical Mechanics and its ApplicationsOct 1, 200720 citations

Discrete-time quantum walks with energy-dependent coins

Alana Spak dos Santos, Thiago T. Tsutsui, Fabiano M. Andrade

In this work, we extend the scattering quantum walk (SQW) framework to a lattice of energy-dependent point interactions. This yields, within the coined quantum walk (CQW) formalism, a coin operator that is directly related to the scattering matrix of zero-range potentials. The model thus provides a discrete-time quantum-walk (DTQW) analog of a periodic array of point interactions of the Kronig-Penney type, where the walker's wavenumber serves as a continuous, physically transparent control parameter for the coin operation. We analyze the spectra and the dynamics of position probability and entanglement, yielding distinct results for specific energies and point interactions. We relate the spectral structure to the spatial probability distribution and explicitly characterize the long-time entanglement behavior for each point interaction. The transmission modulus determines the quasienergy gap, bandwidth, and maximum group velocity, while also controlling the long-time coin-position entanglement for the initial state considered. The four families of one-dimensional point interactions ( $δ$ , $δ'$ , crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.In this work, we extend the scattering quantum walk (SQW) framework to a lattice of energy-dependent point interactions. This yields, within the coined quantum walk (CQW) formalism, a coin operator that is directly related to the scattering matrix of zero-range potentials. The model thus provides a discrete-time quantum-walk (DTQW) analog of a periodic array of point interactions of the Kronig-Penney type, where the walker's wavenumber serves as a continuous, physically transparent control parameter for the coin operation. We analyze the spectra and the dynamics of position probability and entanglement, yielding distinct results for specific energies and point interactions. We relate the spectral structure to the spatial probability distribution and explicitly characterize the long-time entanglement behavior for each point interaction. The transmission modulus determines the quasienergy gap, bandwidth, and maximum group velocity, while also controlling the long-time coin-position entanglement for the initial state considered. The four families of one-dimensional point interactions ($δ$, $δ'$, crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.

quant-pharXiv:2609.03272Sep 4, 20260 citations

Discrete-Time Quantum Walks with Even-Dimensional Coins

Yunguo Lin, Yong Wang, Chengwei Zheng +3 more

Jan 1, 20260 citations

Demonstration of Discrete-Time Quantum Walks and Observation of Topological Edge States in a Superconducting Qutrit Chain

Kun Zhou, Jian-Wen Xu, Qi-Ping Su +12 more

Quantum walk serves as a versatile tool for universal quantum computing and algorithmic research. However, the implementation of discrete-time quantum walks (DTQWs) with superconducting circuits is still constrained by some limitations such as operation precision, circuit depth and connectivity. With improved hardware efficiency by using superconducting qutrits (three-level systems), we experimentally demonstrate a scalable DTQW in a superconducting circuit, observing the ballistic spreading of quantum walk in a qutrit chain. The usage of qutrits in our implementation allows hardware efficiently encoding of the walker position and the coin degree of freedom. By exploiting the flexibility and intrinsic symmetries of qutrit-based DTQWs, we successfully prepare two topological phases in the chain. For the first time, particle-hole-symmetry-protected edge states, bounded at the interface between these two topological phases, are observed in the superconducting platform. Measured parameter dependencies further validate the properties of edge states. The scalability and gate-control compatibility of the demonstrated DTQWs enable a versatile tool for superconducting quantum computing and quantum simulation.Quantum walk serves as a versatile tool for universal quantum computing and algorithmic research. However, the implementation of discrete-time quantum walks (DTQWs) with superconducting circuits is still constrained by some limitations such as operation precision, circuit depth and connectivity. With improved hardware efficiency by using superconducting qutrits (three-level systems), we experimentally demonstrate a scalable DTQW in a superconducting circuit, observing the ballistic spreading of quantum walk in a qutrit chain. The usage of qutrits in our implementation allows hardware efficiently encoding of the walker position and the coin degree of freedom. By exploiting the flexibility and intrinsic symmetries of qutrit-based DTQWs, we successfully prepare two topological phases in the chain. For the first time, particle-hole-symmetry-protected edge states, bounded at the interface between these two topological phases, are observed in the superconducting platform. Measured parameter dependencies further validate the properties of edge states. The scalability and gate-control compatibility of the demonstrated DTQWs enable a versatile tool for superconducting quantum computing and quantum simulation.

quant-pharXiv:2601.01759Physical Review LettersJan 6, 20265 citations

δ

δ'

, crossed and asymmetric) realize qualitatively distinct transmission profiles and span the full range of behavior, including enhanced or strongly suppressed spreading and oscillatory entanglement.