Treasure Hunt in Anonymous Graphs with Quantum Pebbles by Oblivious Agents

Abstract: We investigate the problem of finding a static treasure in anonymous graphs using oblivious agents and introduce a novel approach that leverages quantum information. In anonymous graphs, vertices are unlabelled, indistinguishable, and edges are locally labelled with port numbers. Agents typically rely on stationary classical pebbles placed by an oracle to guide their search. However, this classical approach is constrained by limited information transmission and high traversal complexity. Classical pebbles are not sufficient for search if the agents are oblivious. We propose the first use of quantum pebbles for search in anonymous graphs. Quantum pebbles periodically emit qubits in a fixed quantum state. Each pebble encodes the port number to the next node using a unique quantum state. The agent determines the correct path by performing measurements in multiple bases, exploiting the probabilistic nature of quantum measurement to distinguish states. We show that this strategy enables an oblivious agent to locate the treasure in $D$ steps using $D$ quantum pebbles, where $D$ is the length of the shortest path between the starting point and the treasure. Moreover, only $O((\log D + \log \Delta)/(\log 1/\delta))$ measurements per node are required to ensure high success probability in a graph with maximum degree $\Delta$ where $\delta = \cos<sup>2(\frac{\pi}{2\Delta})$. We propose the use of quantum information as a guidance mechanism in anonymous graph search. We demonstrate that quantum pebbles can not only emulate the functionality of classical pebbles but can do so with improved efficiency, offering a promising direction for future quantum-enhanced distributed algorithms.