Optimal and Robust In-situ Quantum Hamiltonian Learning through Parallelization

TL;DR

This work introduces the first in-situ Hamiltonian-learning algorithm that leverages a parallel-invariant-subspace structure to learn all coefficients of a fully connected $n$-qubit Hamiltonian with $O(n)$ experiments. By embedding the full Hamiltonian into $2\times2$ blocks and applying Quantum Signal Processing, the method achieves Heisenberg-limited precision while remaining robust to SPAM, decoherence, and time-dependent coherent errors; it also offers two implementation paths: an analog-digital-hybrid and a fully analog approach. The authors demonstrate a quadratic reduction in experimental cost for learning $O(n^2)$ parameters and provide an experimental proposal and supporting simulations on Rydberg-atom platforms to estimate interatomic distances with high precision. The results enable efficient, scalable, and resilient device-characterization crucial for high-fidelity quantum simulation and computation on NISQ-era hardware.

Abstract

Hamiltonian learning is a cornerstone for advancing accurate many-body simulations, improving quantum device performance, and enabling quantum-enhanced sensing. Existing readily deployable quantum metrology techniques primarily focus on achieving Heisenberg-limited precision in one- or two-qubit systems. In contrast, general Hamiltonian learning theories address broader classes of unknown Hamiltonian models but are highly inefficient due to the absence of prior knowledge about the Hamiltonian. There remains a lack of efficient and practically realizable Hamiltonian learning algorithms that directly exploit the known structure and prior information of the Hamiltonian, which are typically available for a given quantum computing platform. In this work, we present the first Hamiltonian learning algorithm that achieves both Cramer-Rao lower bound saturated optimal precision and robustness to realistic noise, while exploiting device structure for quadratic reduction in experimental cost for fully connected Hamiltonians. Moreover, this approach enables simultaneous in-situ estimation of all Hamiltonian parameters without requiring the decoupling of non-learnable interactions during the same experiment, thereby allowing comprehensive characterization of the system's intrinsic contextual errors. Notably, our algorithm does not require deep circuits and remains robust against both depolarizing noise and time-dependent coherent errors. We demonstrate its effectiveness with a detailed experimental proposal along with supporting numerical simulations on Rydberg atom quantum simulators, showcasing its potential for high-precision Hamiltonian learning in the NISQ era.

Figures (5)

• Figure 1: General framework for multi-qubit parallel learning. The $i$th full Hamiltonian $H_i$ is used to learn the $(n-1)$ parameters associated with the $i$th qubit. First, $H_i$ is mapped to a block-diagonal form with block size $2$, of which $(n-1)$ blocks are employed for estimation. Each block corresponds to an invariant subspace $\mathcal{B}_k$ with Definition \ref{['def:encoding']}. On each block for invariant subspace $\mathcal{B}_k$, QSPE is applied to estimate a parameter pair $(\theta_k,\zeta_k)$ encoding the Hamiltonian information via Eq. \ref{['eqn: map_j']}. The QSPEs across all invariant subspaces can be executed in parallel by evolving under the multi-qubit Hamiltonian, followed by the logical $Z_L(\omega)$ for phase accumulation. When projected onto each invariant subspace, the dynamics reduce to QSPE on a $2\times 2$ matrix.
• Figure 2: Performance of the algorithm for multi-atom case.
• Figure 3: Robustness test against coherent state preparation error(orange), readout error(green), depolarizing error(red) and time-dependent coherent error(purple) with the number of samples $N_{\text{shot}} = 10^5$ and the number of boostraps $N_{\text{bootstrap}} = 10^3$. The noise model is chosen based on realistic noise model for Rydberg system wurtz2023aquila, where the readout error is $(p_{\text{loss}},p_{\text{anti-trapping}}) =(0.01,0.08)$, the depolarizing noise has fidelity $\alpha = 0.8$, coherent state preparation error $\alpha = 0.01$ and the coherent error on $X_i$ is $\gamma = 10\%$. The normalized variance is defined as the relative variance $\text{Var}(\hat{c})/c^2$. The simulation results exhibit $\propto d^{-4}$ scaling in both noisy and noiseless scenarios, suggesting the robustness of the learning algorithm against experimental noise.
• Figure 4: Physical implementation on the Rydberg quantum simulator. The first dotted block is for state preparation, where $\ket{+}_l$ and $\ket{i}_l$ should be prepared according to Supplementary Material Sec. \ref{['sec:implement detials for two qubit']}. The second dotted block evolves $d$-repetition of target quantum dynamics and $Z_l$ phase accumulation. The whole evolution should run for $\omega = \frac{j}{2d-1}\pi, j = 0,\cdots,2d-2$. $d=3$ is picked here for reference.
• Figure 5: Application of the learning algorithm on Rydberg quantum simulators for atom distance. (a) Atom distance learning with three different distance ($R_{12} = 7.16\mu m, R_{13} = 7.52 \mu m, R_{23} = 8.04\mu m$) (b) The estimated distance of three estimators ($\hat{R}_{12},\hat{R}_{13},\hat{R}_{23}$). The bold x-ticks are the true value ($R_{12},R_{13},R_{23}$). After $d = 10$ cycles, the estimations of all distances are consistently correct. (c) The upper block of curves (light-green, light-blue, light-orange) represents the variance of the estimators $(\hat{c}_{12},\hat{c}_{13},\hat{c}_{23})$ with respect to the number of cycles $d$, where $c_{ij}$ is the parameter of term $Z_iZ_j$ in Eq. \ref{['eqn:full H']}. While the lower block of curves(green, blue, orange) is the variance of atom distance estimators ($\hat{R}_{12},\hat{R}_{13},\hat{R}_{23}$) with respect to $d$ after conversion given in Eq. \ref{['eqn:convertion']}. The simulation results exhibit $\propto d^{-4}$ scaling, in agreement with Theorem \ref{['thm:heisenberg limit']} for all estimators.

Theorems & Definitions (22)

• Definition 2.1
• Definition 2.2: Problem 1
• Definition 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6: Optimal precision performance
• Definition 2.7
• Lemma 2.8
• Theorem 2.9
• proof