Symmetry Speeds Up Quantum Measurements

Estimating the properties of quantum states is a fundamental task in both theoretical and experimental quantum physics. While numerous Pauli-based measurement strategies have been proposed, none explicitly exploit prior knowledge of structural features such as symmetry. Here, we introduce a compact measurement scheme that exploits symmetry to speed up quantum measurements. By incorporating symmetry into the measurement design, our approach can rigorously reduce sample complexity for symmetric states and suitable observables. We provide a theoretical analysis showing that, for specific observables, the variance of our estimator admits polynomial or even exponential improvements in the required number of state copies. To demonstrate practicality, we implement the scheme on photonic quantum platforms, preparing high-fidelity Greenberger-Horne-Zeilinger (GHZ) and W states. The experimental results confirm substantial gains in estimation accuracy for both linear and nonlinear properties. Our work suggests that symmetry and other structural properties of quantum states can be systematically exploited as algorithmic resources, opening new avenues for scalable characterization and verification of complex quantum systems on near-term and fault-tolerant devices.