Quantum search algorithm for similar subgraph identification under fixed edge removal

Abstract

We introduce a novel quantum algorithm for similar subgraph identification in form of an NP-hard cardinality-constrained binary quadratic optimization problem. Given a weighted reference graph with Laplacian $\boldsymbol{B}$, our algorithm determines the subgraph featuring Laplacian $\boldsymbol{B'}$ on the same vertex set, but $x$ out of $N$ inactive edges, minimizing the Frobenius distance $||\boldsymbol{B} - \boldsymbol{B'}||_\mathrm{F}^2$. We represent the $\binom{N}{x}$ graph topologies by an equal-weight superposition in form of a Dicke state, enabling controlled transformations applied to the quantum state associated with the vectorized Laplacian of the reference graph. Combined with amplitude estimation and a minimum finding approach, our algorithm provides a polynomial speed up $\mathcal{O}(\sqrt{N^{x}/x!}N\log\log N)$ compared to $\mathcal{O}(N^{x+1}/x!)$ of classical brute-force search algorithms. We demonstrate the application of our method on standard test cases, which represent electric power grids, by reconstructing $||\boldsymbol{B} -\boldsymbol{B'}||_\mathrm{F}^2$ from measurements and show how our approach can be additionally used to calculate energy functional like quadratic forms of the Laplacians with respect to a given vector.