Quantum Algorithm for Identifying Hidden Graphs: Spectral Theory and Numerical Evidence

TL;DR

This paper introduces a quantum algorithm for identifying hidden regular graphs from obfuscated versions using spectral theory and quantum walks, demonstrating potential exponential speedup over classical methods.

Contribution

The paper presents a novel quantum algorithm leveraging spectral analysis and continuous-time quantum walks to efficiently identify hidden graphs from obfuscated structures.

Findings

Numerical evidence suggests polynomial measurements suffice for graph distinction.

The algorithm achieves exponential speedup over classical query complexity.

Spectral theory underpins the algorithm's design and analysis.

Abstract

We give a quantum algorithm for a novel type of black-box problem: identifying a hidden $d$-regular base graph $G$ on $n$ vertices from oracle access to an obfuscated version of it, rather than traversing it. From $G$ we build the spired graph $G_{\rm spire}$ in three steps: each vertex is lifted into an exponentially large cluster, with adjacent clusters joined by a random bipartite graph; each cluster is then crowned with a balanced spire; finally, all vertices are randomly relabelled. Specializing to $G=K_2$ recovers the welded-trees graph. Our algorithm is conceptually simple: a continuous-time quantum walk on $G_{\rm spire}$, followed by a single Hadamard test at a classically precomputed time $t^*$; the algorithm returns the candidate whose predicted amplitude is closest to the measurement. The design rests on a rigorous spectral theory: from the apex of any spire, the walk is confined to a polynomial-dimensional invariant subspace evolving under the adjacency matrix of a simpler towered graph $G_{\rm tower}$; that matrix block-diagonalizes into $n$ independent tridiagonal systems of size $n$, each solved in closed form by a Chebyshev secular equation. Efficient numerics enabled by this decomposition supply $t^*$ and the predicted amplitudes. On the prism graphs $Y_m$ versus the M\"obius ladders $M_m$ (each on $n=2m$ vertices), the numerical study supports a precise conjecture that $\widetilde O(n^2/\log n)$ measurements at evolution time of order $m^2$ suffice to distinguish the two families; we have tested $4 \le m \le 5121$ ($n$ up to $10242$). By analogy with the welded-trees lower bounds, we further conjecture that any classical algorithm requires queries exponential in $n$. Together these conjectures point to an exponential quantum speedup for the identification of an obfuscated base graph.