Quantum-walk search in motion

TL;DR

This work extends quantum-walk search algorithms to handle multiple marked nodes with an inherent chronological ordering by attaching label states to marked vertices. By introducing static and dynamic labeling schemes, the authors show how the Hilbert space can be augmented to simultaneously amplify targets and encode their order, enabling applications such as tracking a moving particle on a 2D lattice. The approach preserves the square-root-type scaling of QWSA while distributing probability across layers, with per-layer and total success probabilities analyzed, and it is complemented by a proposed quantum circuit implementation. The results offer a pathway toward real-time object tracking and dynamic network tasks using quantum-walk-based search, with broader implications for temporal graphs and time-resolved data processing.

Abstract

In quantum computing, the quantum walk search algorithm is designed for locating fixed marked nodes within a graph. However, when multiple marked nodes exist, the conventional search algorithm lacks the capacity to simultaneously amplify the marked nodes as well as identify the correct chronological ordering between the marked nodes, if any. To address this limitation, we explore a potential extension of the algorithm by introducing additional quantum states to label the marked nodes. The labels resolve the ambiguity of simultaneous amplification of the marked nodes. Additionally, by associating the label states with a chronological ordering, we can extend the algorithm to track a moving particle on a two-dimensional surface. Our algorithm efficiently searches for the trajectory of the particle and is supported by a proposed quantum circuit. This concept holds promise for a range of applications, from real-time object tracking to network management and routing.

Figures (16)

• Figure 1: Replication of the $2d$ lattice ($xy$- grid) on the $z$-direction which denotes the labels. The original search problem, including five marked nodes on a $2d$ lattice, reduces to the search problem, including five $2d$ sheets with one marked node each. We can allow for hopping between the sheets to enhance the probability distribution for the marked nodes at the cost of diminishing the probability for the unmarked ones.
• Figure 2: Two-dimensional lattice with double periodic boundary condition is topologically equivalent to a torus.
• Figure 3: The structure of the shift operator in the case of open boundary conditions is different in the interior and exterior of the grid. The figure shows the self-loop at the boundary point of the lattice which ensures the unitarity of the shift operator.
• Figure 4: Schematic diagram of the structure of Hilbert space for quantum-walk search for ordered marked nodes.
• Figure 5: Amplification of marked nodes with steps for static labelling in the case of (a) open grid and (b) torus. In case of open grid, the probability of marked nodes $[6,8]$ and $[8,9]$ coincides, while in case of torus, the probability of all marked nodes coincide with each other.