Measuring the largest coefficients of a quantum state

arXiv:2605.00341v2 Announce Type: replace Abstract: We introduce a hierarchical algorithm for identifying the indices of the largest Pauli coefficients of an unknown $n$-qubit quantum state, either those exceeding a threshold or the $t$ largest ones. The algorithm traverses a prefix-based tree whose nodes represent partial sums of squared Pauli coefficients, expanding either the nodes of largest estimated weight or all nodes above a threshold, and discarding the rest. Node weights are estimated using Bell sampling on two copies of the state, or alternatively via SWAP tests on subsystems. We prove finite-sample guarantees for the exact recovery of the target set in terms of state-dependent separation margins for the relevant prefix weights, and derive bounds on the number of nodes explored by the corresponding exact traversal. While the same Bell data can be processed exhaustively, the hierarchical strategy can reduce the classical search at the cost of potentially requiring more samples to distinguish the relevant prefix weights. The efficiency of our method depends on the number of nodes of the prefix tree with non-negligible weight as well as on the statistical conditioning of these values. We validate the method numerically on Pauli-singleton states, random stabilizer states, ground states of the transverse-field Ising model, and states with hierarchical Pauli support, illustrating different structural and statistical regimes. We further benchmark the method against alternative approaches for learning Pauli coefficients. Our work addresses the targeted discovery of dominant Pauli components without performing full state tomography.