Qisheng Wang, from Shanghai Jiao Tong University, and colleagues have achieved a key advancement in estimating the fidelity of an unknown quantum state to a known reference state. The new method achieves a sample complexity of O(r2/varepsilon2) with optimal dependence on the error $\varepsilon$ for a reference state of rank $r$. This improvement over previous bounds, alongside a corresponding lower bound of Ω(r/varepsilon2), has implications for quantum query complexity and enables more efficient tolerant quantum state certification, building upon existing exact certification methods. Reduced measurement requirements unlock practical quantum state fidelity estimation A dramatic improvement in the efficiency of estimating fidelity between quantum states has been achieved, reducing the required sample complexity from O(r²log²(1/ε)/ε⁴) to O(r²/ε²), where ‘r’ denotes the rank of the reference state and ‘ε’ represents the desired accuracy. This breakthrough crosses a key threshold, enabling practical fidelity estimation for systems where previously the computational cost was prohibitive. Accurate assessment of quantum states was limited by the sheer number of measurements needed. The rank of a quantum state, in this context, signifies the number of linearly independent states within its superposition, directly impacting the complexity of its characterisation. A higher rank necessitates more measurements to fully define the state. The parameter ε represents the acceptable deviation from the true fidelity value; a smaller ε demands greater precision and, consequently, more measurements. Alongside this improvement, a new lower bound of Ω(r/ε²) has been established, refining the fundamental limits of this quantum state analysis technique and providing a benchmark for future advancements. Fidelity estimation, a measure of how close two quantum states are, now requires fewer resources than previously thought, as demonstrated by researchers at Shanghai Jiao Tong University. The new method achieves a sample complexity of O(r²/ε²), meaning the number of measurements needed grows proportionally to the square of