Beyond Worst-Case Branching: Quantum Tree Search via Amplitude Amplification

Abstract: In this work, we investigate quantum tree search via amplitude amplification. Amplitude amplification generalizes Grover's algorithm by replacing the Hadamard initialization with an arbitrary unitary $A$, with Grover's algorithm recovered as the special case of uniform initialization. We demonstrate the construction of a dynamic search tree of depth $m$ with query complexity $\sqrt{\left(b_{avg}\right)^m}$ where $b_{avg}$ denotes the average branching factor, improving upon the commonly assumed $\sqrt{\left(b_{max}\right)^m}$, where $b_{max}$ is the maximum branching factor. We further challenge the widespread assumption that amplitude amplification is inferior to quantum backtracking. In fact, quantum backtracking is unsuitable for problems that do not naturally admit a backtracking structure; in such cases, amplitude amplification yields improved query complexity. We observe that amplitude amplification constructs the search tree dynamically, rendering its internal structure inaccessible, a constraint that applies equally to quantum backtracking. To address this, we propose sampling-based methods to estimate the tree structure, under the assumption that it approximates a normal distribution with increasing depth. Finally, we introduce a quantum greedy search based on a lookahead heuristic inspired by the classical cognitive architecture Soar, which models human-like problem-solving strategies.