Provably Efficient Learning of Fermionic Correlations under Particle-Number Symmetry

Abstract

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 all $k$-body fermionic correlations of an $N$-mode $\eta$-particle state with a given variance ${\epsilon }^2$ using only $O_k({\eta }^k/{\epsilon }^2)$ samples, which are independent of the system size $N$. We further establish a matching information-theoretic lower bound ${\Omega }_k({\eta }^k/{\epsilon }^2)$ for any adaptive protocol based on single-copy measurements, showing that the $({\eta }^k,\epsilon )$-dependence is optimal up to constants depending only on $k$. Furthermore, numerical studies show a 20-fold query reduction for one-body correlation estimation at $N=200$, $\eta =20$, and $\epsilon =10^{-2}$, compared with the best alternative including Heisenberg-limited estimation methods. Relative to fermionic Gaussian-unitary shadows, sample costs for the square-lattice Fermi-Hubbard model at $N=288$ are reduced by factors of $148$ for energy density and $161$ for spin structure factor estimation. For molecular energy derivatives with respect to atomic coordinates, our method reduces sample requirements by factors of $53$ for ${\mathrm{Cr}}_2$ and $18.5$ for $[{\mathrm{Fe}}_2{\mathrm{S}}_2{]}^{2-}$ at $N=128$. These results demonstrate the practical advantage of particle-number symmetry for fermionic observable estimation.

arXiv comment102 pages, 8 figures. V2: Added tight bounds, exact 1-/2-RDM covariance formulas, and numerical benchmarks