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Hamiltonian Learning at Heisenberg Limit for Hybrid Quantum Systems

Lixing Zhang, Ze-Xun Lin, Prineha Narang, Di Luo

TL;DR

This work establishes a rigorous theoretical framework proving that, given access to an unknown spin-boson type Hamiltonian, the algorithm achieves Heisenberg-limited estimation for all coupling parameters up to error $\epsilon$ with a total evolution time of less than one second.

Abstract

Hybrid quantum systems with different particle species are fundamental in quantum materials and quantum information science. In this work, we establish a rigorous theoretical framework proving that, given access to an unknown spin-boson type Hamiltonian, our algorithm achieves Heisenberg-limited estimation for all coupling parameters up to error $ε$ with a total evolution time ${O}(ε^{-1})$ using only ${O}({\rm polylog}(ε^{-1}))$ measurements. It is also robust against small state preparation and measurement errors. In addition, we provide an alternative algorithm based on distributed quantum sensing, which significantly reduces the evolution time per measurement. To validate our method, we demonstrate its efficiency in hybrid Hamiltonian learning and spectrum learning, with broad applications in AMO, condensed matter and high energy physics. Our results provide a scalable and robust framework for precision Hamiltonian characterization in hybrid quantum platforms.

Hamiltonian Learning at Heisenberg Limit for Hybrid Quantum Systems

TL;DR

This work establishes a rigorous theoretical framework proving that, given access to an unknown spin-boson type Hamiltonian, the algorithm achieves Heisenberg-limited estimation for all coupling parameters up to error with a total evolution time of less than one second.

Abstract

Hybrid quantum systems with different particle species are fundamental in quantum materials and quantum information science. In this work, we establish a rigorous theoretical framework proving that, given access to an unknown spin-boson type Hamiltonian, our algorithm achieves Heisenberg-limited estimation for all coupling parameters up to error with a total evolution time using only measurements. It is also robust against small state preparation and measurement errors. In addition, we provide an alternative algorithm based on distributed quantum sensing, which significantly reduces the evolution time per measurement. To validate our method, we demonstrate its efficiency in hybrid Hamiltonian learning and spectrum learning, with broad applications in AMO, condensed matter and high energy physics. Our results provide a scalable and robust framework for precision Hamiltonian characterization in hybrid quantum platforms.

Paper Structure

This paper contains 10 sections, 1 theorem, 53 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Random Unitary Tranformations
  2. Estimation error propagation
  3. Extraction of $\Xi_{b,l}$ using robust phase estimation
  4. Proof of the applicability of robust frequency estimation
  5. Solving Hamiltonian coefficients with linear equations
  6. Deviation from effective dynamics
  7. Derivation for Eq. \ref{['displacement']}
  8. Discretization of spectral density function
  9. Derivations for models learnable by our Algorithms
  10. Pseudocode for Hamiltonian learning of hybrid quantum systems

Key Result

Theorem 1

Given a unitary dynamics access to arbitrary Hamiltonian in the form of Eq. (gen_H), there is an algorithm $\mathcal{A}$ that can estimate $\xi_a$, $\lambda_a^n$, $\eta_a^n$ and $\omega_n$ up to a RMSE $\epsilon$ such that:

Figures (5)

  • Figure 1: Hybrid Quantum systems Hamiltonian learning. Left panel: Schematic of the learning protocol presented in this work. Right panel: Properties and applications of the main algorithm.
  • Figure 2: Estimation error scaling at different $\tau$. The presented data is averaged over $100$ independent runs. Ideal error refers to $\epsilon = 1/2T$, which is derived from Eq. \ref{['displacement']}. For GDM, the target parameter is $\lambda_{XXI}$. For SBM, the target parameter is $\lambda_X^{n=1}$.
  • Figure 3: Estimation error scaling with SPAM error. For GDM, the target parameter is $\lambda_{XXI}$. For SBM, the target parameter is $\lambda_X^{n=1}$. $\mathfrak K = 6$ is used in the $2^{\mathfrak K}$ fold space to implement RFE.
  • Figure 4: Learning the spectral density function of SBM. (a) Relative error of the estimation generated from the DQS-based scheme. (b) The value of $\lambda_X^{n}$ across all bosonic modes, showing the shape of the discrete spectral density $J(\omega)$. For the parameter used: $W = 1000$ and $n_{pt} = 1$.
  • Figure 5: The variance and mean value of trace distance $||\Delta \rho||_1$ are plotted against $\tau$ and $R\tau = t$

Theorems & Definitions (1)

  • Theorem 1