Heisenberg-Limited Quantum Hamiltonian Learning via Randomly Spread Product-States

TL;DR

This work tackles Hamiltonian learning under a fixed resource budget by showing that Heisenberg-like scaling can be achieved without entanglement, coherent joint measurements, or dynamical control. The key idea is to ensemble-average over Haar-random local product states and random Pauli measurements, which cancels interference terms and yields an FI that scales as $\mathcal{I}(t) = \Theta(t^2)$ for short times, enabling HL performance with local operations. Furthermore, averaging over spread-state ensembles diagonalizes the Fisher information matrix, allowing all Hamiltonian parameters to be learned simultaneously from a single dataset. The authors validate the theory numerically on disordered multi-qubit Hamiltonians and demonstrate that, with non-uniform time stamping, the accumulated FI can approach HL scaling with total experiment time $T_{\rm tot}$; these results suggest a practical, near-term route to resource-efficient quantum characterization and metrology.

Abstract

We show how one can asymptotically reach the Heisenberg limit in quantum Hamiltonian learning without entanglement, globally coherent measurements, or dynamical control, using only local quantum operations. Our protocol uses ensemble-averaging over the outcomes of experiments initialized in Haar-random local product states, accompanied by random Pauli measurements, leading to the effective cancellation of interference terms so that a Heisenberg-limited regime emerges for short-time experiments. Furthermore, we show that the act of ensemble averaging makes unbiased estimation data, meaning all Hamiltonian parameters can be simultaneously estimated from the same data-set, removing the need for parameter isolation. We supplement the theoretical results by showing empirically that, even away from the asymptotic limit, one can surpass the SQL using randomly spread product-state ensembles. We do so numerically by learning a selection of different disordered multi-qubit Hamiltonians in a black-box learning scenario.

Figures (3)

• Figure 1: Reconstruction error $\varepsilon$ versus total experiment time $T_{\rm tot}$ for four representative Hamiltonian families at $\alpha = 1.0$. Each data point is given with ten random Hamiltonian realizations. Across all Hamiltonians, the error decays as $\varepsilon \propto T_{\rm tot}^{-\beta}$ with $\beta \approx 0.66$, surpassing the standard quantum limit and consistent with Heisenberg-limited single-probe scaling.
• Figure 2: (a) Reconstruction error $\varepsilon$ versus total experiment time $T_{\rm tot}$ for goriwng numbers of spread states $R$ (XYZ model Eq. \ref{['eq:first_test_ham']}). (b) Corresponding scaling exponents $\beta$ obtained from $\varepsilon \propto T_{\rm tot}^{-\beta}$ across Hamiltonian families defined in Sec. \ref{['sec:model']}. As $R$ increases, $\beta$ converges toward the prediction $\beta \approx 0.75$ (see Eqs. \ref{['eq:err_1']} and \ref{['eq:err_2']} with $\alpha=1.0$, and assuming Heisenberg limited scaling with single evolution time $t$: $\gamma_0=2$,), confirming the increase of the learning rate with the number of spread states (see Theorem \ref{['thm:random_fisher_scaling']}) and also that all parameters can be learned at once from the same dataset at increasing numbers of spread state (see Theorem \ref{['thm:fisher_diagonalization']}), since the recovery error considers the whole multi-parameter Hamiltonian in each case. See Appendix \ref{['app:spread_scaling_data']} for the visualized data.
• Figure 3: Empirical scaling exponent $\beta_{T_{\rm tot}}$ as a function of the scheduling parameter $\alpha$. The solid curve shows the theoretical prediction $\beta_{T_{\rm tot}}(\alpha) = \frac{1}{2}\,\frac{\alpha \gamma_0 + 1}{\alpha + 1}$ given based on assuming Heisenberg limited scaling:$\gamma_0 = 2$, and the dashed curve includes a small vertical offset accounting for finite ensemble and sampling effects that vanish asymptotically (see Props. \ref{['prop:cumulative_scaling']} and Thm. \ref{['thm:random_fisher_scaling']}). See Appendix \ref{['app:alpha_scaling_data']} for visualized data.

Theorems & Definitions (14)

• Definition 1: Spread State
• Theorem 1: Time Evolution of Fisher Information under Local Randomness is Heisenberg-Limited
• proof
• Remark 1
• Theorem 2: Fisher Information Diagonalization
• proof
• Proposition 1: Cumulative Fisher-Information Scaling
• proof
• proof
• proof