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 B, our algorithm determines the subgraph featuring Laplacian B' on the same vertex set, but x out of N inactive edges, minimizing the Frobenius distance ||B - B'||F². We represent the binomNx 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 O(√(Nˣ/x!)Nloglog N) compared to O(Nˣ⁺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 ||B -B'||F² 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.