Provably Efficient Learning of Fermionic Correlations under Particle-Number Symmetry

Predicting local fermionic correlations is a central task in quantum many-body physics, as these correlations encode many physically relevant local observables. The ubiquitous particle-number symmetry imposes strong structural constraints on quantum states, suggesting that local correlations should be learned with fewer samples than by symmetry-agnostic approaches. However, it has remained unclear whether such a provable advantage exists in collective learning of local correlations. Here, we develop a framework of number-conserving fermionic-shadow tomography based on random orbital rotations. We prove that, for every given order $k$, we can simultaneously estimate {\it all} $k$-body fermionic correlations of an $N$-mode $\eta$-particle state with a given variance $\varepsilon^2$ using only $O_k(\eta^k/\varepsilon^2)$ samples, which are independent of the system size $N$. We further establish a matching information-theoretic lower bound $\Omega_k(\eta^k/\varepsilon^2)$ for any adaptive protocol based on single-copy measurements, showing that the $(\eta^k,\varepsilon)$-dependence is optimal up to constants depending only on $k$. Furthermore, our numerical calculation shows that the proposal reduces the query count by roughly an order of magnitude compared with state-of-the-art methods for one-body correlation estimation in a system of $N=100$, $\eta=20$ at $\varepsilon=10^{-2}$. This work establishes a provably efficient advantage of particle-number symmetry for fermionic observables estimation.