  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.
