Robust Treasure Hunt in Anonymous Graphs with Quantum Pebbles by Oblivious Agents

Abstract

We study how to find a hidden treasure in anonymous graphs using an agent that has no persistent memory. The nodes are indistinguishable, and only edges have local port numbers. Classical pebbles placed by an oracle cannot guide an oblivious agent to the treasure. We introduce \emph{quantum pebbles}, which are sources that emit qubits in a fixed (unknown) state, encoding at every node the outgoing port on the shortest path to the treasure. By measuring in several non-orthogonal bases, an oblivious agent recovers the port and can reach the treasure in $D$ steps using $D$ quantum pebbles. This requires $O(\Delta^{3}(\log D + \log \Delta))$ measurements per node, where $\Delta$ is the maximum degree. We further establish \emph{error robustness}, distinguishing two models of state preparation error. Under \emph{per-node persistent} error, where a device returns the same faulty encoding on every read, a single mislabelled coloured pebble can trap an oblivious agent in an infinite loop and every randomized strategy decays exponentially in $D$. Quantum pebbles inherit the same exponential decay. Under \emph{per-emission} error, the intended encoding is correct, but each emitted qubit independently changes state as $\rho = (1-e)\,\lvert\psi\rangle\langle\psi\rvert + e\,\sigma $ for an arbitrary noise matrix $\sigma$. Here the quantum protocol is provably robust. A threshold decoding rule with $O((\log D + \log \Delta)/\gamma^{2})$ measurements per basis, where $\gamma = (1-e) - \delta_e$ and $\delta_e = (1-e)\delta + e$, has success probability close to $1$ as $D \to \infty$, provided $e < e^{*} = \sin^2(\pi/2\Delta)/(1+\sin^2(\pi/2\Delta))$. The separation that we establish is thus among quantum pebbles with per-emission error and a persistent marker.