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 η-particle state with a given variance ε2 using only Ok(ηk/ε2) samples, which are independent of the system size N. We further establish a matching information-theoretic lower bound Ωk(ηk/ε2) for any adaptive protocol based on single-copy measurements, showing that the (ηk,ε)-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, η=20, and ε=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 Cr2 and $18.5$ for [Fe2S2]2− at N=128. These results demonstrate the practical advantage of particle-number symmetry for fermionic observable estimation.
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BibTeX
@misc{koizumi2026provably,
title = {{Provably Efficient Learning of Fermionic Correlations under Particle-Number Symmetry}},
author = {Yuki Koizumi and Kaito Wada and Toshinori P. Takama and Nobuyuki Yoshioka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.30601},
eprint = {2606.30601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30601},
}
RIS
TY - UNPB
TI - Provably Efficient Learning of Fermionic Correlations under Particle-Number Symmetry
AU - Yuki Koizumi
AU - Kaito Wada
AU - Toshinori P. Takama
AU - Nobuyuki Yoshioka
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2606.30601
UR - https://arxiv.org/abs/2606.30601
AB - 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 η-particle state with a given variance ε^2 using only O_k(η^k/ε^2) samples, which are independent of the system size N. We further establish a matching information-theoretic lower bound Ω_k(η^k/ε^2) for any adaptive protocol based on single-copy measurements, showing that the (η^k,ε)-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, η=20, and ε=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 Cr_2 and $18.5$ for [Fe_2S_2]^2− at N=128. These results demonstrate the practical advantage of particle-number symmetry for fermionic observable estimation.
ER -