Weighted hypergraph states as a resource for quantum metrology

Abstract

Quantum correlations can be used to improve sensing precision beyond classical limits, yet the correlated resource states needed can be challenging to prepare and preserve. Here, we study the family of weighted hypergraph states and identify subclasses characterized by their underlying graph topology that exhibit enhanced metrological performance. Using the quantum Fisher information and a novel Dicke state basis representation, we show that these subclasses outperform both separable and standard weighted graph state benchmarks over broad ranges of edge weights corresponding to weak controlled operations between qubits representing the vertices of the graph. We provide analytical formulas for an arbitrary number $N$ of qubits, supplemented by a general study, while focusing a detailed analysis on the example case of $N=4$. By identifying the critical probabilities for local dephasing, we show that adding higher-order weighted hyperedges for $N$ qubits can enhance metrological robustness, allowing the resulting states to retain a quantum advantage at higher noise levels. These probe states exhibit favorable scaling, enhanced precision, and noise tolerance. Our findings establish weighted hypergraph states as viable, structurally tunable and robust resources for quantum-enhanced metrology.

The paper asks whether adding uniformly weighted higher-order hyperedges to graph-state probes improves phase sensitivity and tolerance to local dephasing, particularly at preparation weights below $\phi=\pi$. It studies star-backbone states with hyperedges included in complete leaf-permutation orbits, alongside a fully symmetric complete-hypergraph benchmark. For noiseless probes, quantum Fisher information (QFI) is derived from binomial sums of one- and two-qubit correlators. Symmetry gives a $2N$-dimensional Dicke-basis representation for star-based pure states. A detailed four-qubit comparison covers eight orbit selections. Adding one leaf-only triple or all higher-order hyperedges moves the first standard-quantum-limit crossing from $\phi\simeq1.01471$ for the weighted star to $0.76158$ or $0.34805$, respectively. These probes outperform the star over intermediate-weight intervals, but not at its optimum $\phi=\pi$, where the star reaches $\mathcal F_Q=N^2$. Independent phase-flip noise acts before encoding. Mixed-state QFI and critical noise probabilities are evaluated numerically, using both $\mathcal F_Q=N$ and the optimal equally dephased separable benchmark $N-1+(1-2p)^2$ for the star generator. Weight-optimized calculations extend to even $N\leq24$; at $N=24$, SQL thresholds are approximately $p=0.421410$ for the all-higher-order star family and $0.413111$ for the complete hypergraph. The complete and star families use different encoding generators. Noisy states need not remain in the reduced pure-state subspaces, and numerical optimization does not certify global maxima. Preparation and measurement costs, accessible near-optimal readout, nonuniform weights, and generator optimization remain outside the analysis.