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Quantum Hamiltonian Learning for the Fermi-Hubbard Model

Hongkang Ni, Haoya Li, Lexing Ying

TL;DR

This work tackles learning the Hamiltonian of a Fermi-Hubbard system from noisy quantum evolutions by introducing Hamiltonian reshaping with random unitaries and robust phase estimation to reach Heisenberg-limited scaling. The algorithm partitions the edge set of a bounded-degree graph into a constant number of colors, reshapes the Hamiltonian to decouple subsystems, and then learns the parameters $h_{ij}$ and $\xi_i$ using efficient single-site and two-site procedures. The paper proves that estimations of all parameters to accuracy $\epsilon$ can be achieved with total evolution time $\tilde{\mathcal{O}}(\epsilon^{-1})$, experiments $\tilde{\mathcal{O}}(\log(\epsilon^{-1})\log(\eta^{-1}))$, and single-site unitary insertions scaling as $\tilde{\mathcal{O}}(N\epsilon^{-2}\log(\eta^{-1}))$, with the reshaping error independent of system size. The approach relies only on native fermionic operations on one or two sites and robust phase estimation, making it practically relevant for calibrating and validating fermionic quantum simulators, especially on bounded-degree lattices.

Abstract

This work proposes a protocol for Fermionic Hamiltonian learning. For the Hubbard model defined on a bounded-degree graph, the Heisenberg-limited scaling is achieved while allowing for state preparation and measurement errors. To achieve $ε$-accurate estimation for all parameters, only $\tilde{\mathcal{O}}(ε^{-1})$ total evolution time is needed, and the constant factor is independent of the system size. Moreover, our method only involves simple one or two-site Fermionic manipulations, which is desirable for experiment implementation.

Quantum Hamiltonian Learning for the Fermi-Hubbard Model

TL;DR

This work tackles learning the Hamiltonian of a Fermi-Hubbard system from noisy quantum evolutions by introducing Hamiltonian reshaping with random unitaries and robust phase estimation to reach Heisenberg-limited scaling. The algorithm partitions the edge set of a bounded-degree graph into a constant number of colors, reshapes the Hamiltonian to decouple subsystems, and then learns the parameters and using efficient single-site and two-site procedures. The paper proves that estimations of all parameters to accuracy can be achieved with total evolution time , experiments , and single-site unitary insertions scaling as , with the reshaping error independent of system size. The approach relies only on native fermionic operations on one or two sites and robust phase estimation, making it practically relevant for calibrating and validating fermionic quantum simulators, especially on bounded-degree lattices.

Abstract

This work proposes a protocol for Fermionic Hamiltonian learning. For the Hubbard model defined on a bounded-degree graph, the Heisenberg-limited scaling is achieved while allowing for state preparation and measurement errors. To achieve -accurate estimation for all parameters, only total evolution time is needed, and the constant factor is independent of the system size. Moreover, our method only involves simple one or two-site Fermionic manipulations, which is desirable for experiment implementation.
Paper Structure (13 sections, 4 theorems, 53 equations, 3 algorithms)

This paper contains 13 sections, 4 theorems, 53 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries and Notations
  4. Main algorithm and theoretical results
  5. Method description
  6. Single-site case
  7. Hamiltonian reshaping
  8. Two-site case
  9. Multiple-site case
  10. Conclusion and discussion
  11. Competing Interests:
  12. Funding:
  13. Acknowledgements:

Key Result

Theorem 1

(Informal) Assume $H$ is a Hamiltonian in the form eq:hamiltonian. For a given failure probability $\eta$, we can generate estimations $\hat{h}_{ij}$ and $\hat{\xi_i}$ with precision $\epsilon$ for all $i,\ j$ at the cost of Here we use the notation $\tilde{\mathcal{O}}$ to hide the higher order $\log$ terms for conciseness.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3
  • Theorem 4
  • proof