Core Concepts

A quantum algorithm is proposed that can efficiently learn the parameters of the Fermi-Hubbard Hamiltonian, achieving the Heisenberg-limited scaling in the total evolution time.

Abstract

The paper presents a quantum algorithm for learning the parameters of the Fermi-Hubbard Hamiltonian, which is one of the most extensively studied models for Fermionic systems. The key highlights are:
The algorithm achieves the Heisenberg-limited scaling in the total evolution time required to learn all the Hamiltonian parameters with a given accuracy and failure probability. This matches the fundamental complexity lower bound in quantum metrology.
The algorithm only requires simple one or two-site Fermionic manipulations, which is desirable for experimental implementation.
The error bound in the parameter estimation is independent of the system size, in contrast to the Bosonic Hamiltonian learning case where an additional system size factor is involved.
The algorithm first divides the Fermionic sites into different colors based on a graph coloring approach. It then reshapes the Hamiltonian to decouple the interactions between different colors, allowing the learning of parameters for each color independently using single-site and two-site methods.
The reshaping error is rigorously analyzed and shown to be independent of the system size, enabling the Heisenberg-limited scaling.
The algorithm can be implemented using Fermionic Gaussian state preparation, number operator measurements, and random unitary insertions, which are feasible on experimental platforms.

Stats

The Fermi-Hubbard Hamiltonian is defined as:
H = -Σ_{i~j, σ} h_{ij} c^†{iσ} c{jσ} + Σ_i ξ_i n_{i↑} n_{i↓}

Quotes

"To the best of the authors' knowledge, a Hamiltonian learning algorithm for many-body Fermionic systems with Heisenberg-limited scaling is still missing."
"The error bound obtained in this work does not depend on the size of the system since the Fermion setting does not involve unbounded operators."

Key Insights Distilled From

by Hongkang Ni,... at **arxiv.org** 05-03-2024

Deeper Inquiries

To extend the proposed algorithm to learn Fermionic Hamiltonians with long-range interactions while maintaining the Heisenberg-limited scaling, we can adapt the divide-and-conquer approach used in the algorithm. By categorizing the interactions based on their range, we can reshape the Hamiltonian to isolate the long-range interactions into separate subsystems. This reshaping process would involve inserting random unitaries that target the specific interactions we want to learn. By treating each long-range interaction as a separate subsystem, we can apply the same methodology as in the original algorithm to estimate the parameters associated with these interactions. The key is to ensure that the reshaping error introduced by the random unitaries remains controlled to maintain the Heisenberg-limited scaling. Additionally, by carefully selecting the time steps and the distribution of the random unitaries, we can optimize the algorithm to handle long-range interactions efficiently while achieving the desired scaling.

The quadratic dependence on the inverse accuracy in the number of required single-site random unitary insertions can potentially be improved by exploring higher-order Trotter-like formulas. By incorporating higher-order terms in the Trotter expansion, we can reduce the error introduced by the finite time steps in the simulation of the Hamiltonian evolution. These higher-order Trotter-like formulas can provide a more accurate approximation of the unitary evolution, allowing for fewer random unitary insertions to achieve the same level of precision. By optimizing the choice of Trotter-like formulas and the corresponding time steps, we can potentially reduce the quadratic dependence on the inverse accuracy, leading to a more efficient algorithm for learning Fermionic Hamiltonians.

To handle experimental imperfections such as noisy state preparation and measurements while still achieving the Heisenberg-limited scaling, several modifications to the algorithm would be necessary. Firstly, incorporating error mitigation techniques such as error correction codes or error-robust algorithms like robust phase estimation can help mitigate the impact of noise on the estimation of parameters. Additionally, introducing calibration steps to account for experimental imperfections in state preparation and measurements can improve the accuracy of the results. Furthermore, optimizing the experimental setup to minimize noise sources and calibrating the system parameters regularly can enhance the overall performance of the algorithm. By robustly addressing experimental imperfections and noise, the algorithm can maintain its scalability and accuracy in real-world quantum systems.

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