Comprehensive Study on Heisenberg-limited Quantum Algorithms for Multiple Observables Estimation

TL;DR

This paper presents advanced Heisenberg-limited quantum algorithms for efficiently estimating multiple observables, including fermionic reduced density matrices, with theoretical guarantees and numerical evaluations demonstrating improved query complexity.

Contribution

It introduces generalized adaptive quantum gradient estimation algorithms with quantum enhancements and minimal costs for fermionic systems, along with performance analysis and numerical comparisons.

Findings

Achieved doubly quantum-enhanced query complexity for multiple observables.

Provided theoretical guarantees for estimation precision in root mean squared error.

Demonstrated improved performance through numerical evaluations and comparisons.

Abstract

In the accompanying paper of arXiv:2505.00697, we have presented a generalized scheme of adaptive quantum gradient estimation (QGE) 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 $k$-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.