Achieving quantum advantage in efficiently estimating collective properties of quantum many-body systems remains a fundamental goal in quantum computing. While the quantum gradient estimation (QGE) algorithm has been shown to achieve doubly quantum enhancement in the precision and the number of observables, it remains unclear whether one benefits in practical applications. In this Letter, we present a generalized framework of the adaptive QGE algorithm and further propose two variants which enable us to estimate the collective properties of fermionic systems using the smallest cost among existing quantum algorithms. The first method utilizes the symmetry inherent in the target state, and the second method enables estimation in a single-shot manner using the parallel scheme. We show that our proposal offers a quadratic speedup compared with prior QGE algorithms in the task of fermionic partial tomography for systems with limited particle numbers. Furthermore, we provide numerical demonstrations showing that, for a problem of estimating 2-body fermionic reduced density matrices, our proposals improve the number of queries to the target state preparation oracle by a factor of 4.4 for the nitrogenase FeMo cofactor and by a factor of 7.8 for Fermi-Hubbard model of 200 sites in chemical accuracy.

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• Quantum algorithms & computation
• Quantum circuits
• Quantum computation
• Quantum simulation