In the accompanying Letter of Koizumi et al., Phys. Rev. Lett. 136, 080605 (2026), we presented a generalized scheme of the adaptive quantum gradient estimation algorithm and further proposed two practical variants which not only achieve doubly quantum enhancement in query complexity regarding estimation precision and number of observables but also enable minimal cost to estimate $k$-body  reduced density matrix ($k$-RDMs) in fermionic systems among existing quantum algorithms. Here we provide full descriptions on the algorithm and provide theoretical guarantee for the estimation precision in terms of the root-mean-squared error. Furthermore, we analyze the performance of the quantum amplitude estimation algorithm, another variant of the Heisenberg-limited scaling algorithm, and show how the estimation error is minimized under the circuit structure that resembles the phase estimation algorithm. We finally describe the details for the numerical evaluation of the query complexity of the Heisenberg-limited algorithms and sampling-based methods to make a thorough comparison in the task of estimating fermionic $k$-RDMs.

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