Learning Hamiltonians in the Heisenberg limit with static single-qubit fields

Abstract

Learning the Hamiltonian governing a quantum system is a central task in quantum metrology, sensing, and device characterization. Existing Heisenberg-limited Hamiltonian learning protocols either require multi-qubit operations that are prone to noise, or single-qubit operations whose frequency or strength increases with the desired precision. These two requirements limit the applicability of Hamiltonian learning on near-term quantum platforms. We present a protocol that learns a quantum Hamiltonian with the optimal Heisenberg-limited scaling using only single-qubit control in the form of static fields with strengths that are independent of the target precision. Our protocol is robust against the state preparation and measurement (SPAM) error. By overcoming these limitations, our protocol provides new tools for device characterization and quantum sensing. We demonstrate that our method achieves the Heisenberg-limited scaling through rigorous mathematical proof and numerical experiments. We also prove an information-theoretic lower bound showing that a non-vanishing static field strength is necessary for achieving the Heisenberg limit unless one employs an extensive number of discrete control operations.

Figures (4)

• Figure 1: Efficient Hamiltonian learning with static local fields. For a system of qubits with unknown Hamiltonian $H(\boldsymbol{\lambda})$, by adding sufficiently strong static single-qubit fields along the $\pm\hat{x}$, $\pm\hat{y}$, and $\pm\hat{z}$ directions, and choosing a different field strength on a single qubit (highlighted in a different color in the figure, namely a field strength $\nu/2$ instead of $\nu$), we obtain a Hamiltonian whose ground and first excited states are nondegenerate and sufficiently close to unentangled states, denoted by $|\Phi_0\rangle$ and $|\Phi_1\rangle$, respectively. By preparing the system in $\frac{1}{\sqrt{2}}(\left|\Phi_0\right\rangle+\left|\Phi_1\right\rangle)$, which is also an unentangled state, and evolving it under the total Hamiltonian, we can efficiently extract information about the Hamiltonian terms in $H(\boldsymbol{\lambda})$. Repeating this procedure for different qubits and field directions allows us to reconstruct the original Hamiltonian $H(\boldsymbol{\lambda})$ with arbitrary precision $\epsilon$, using a total evolution time that saturates the Heisenberg limit $1/\epsilon$.
• Figure 2: Learning single-qubit and two-qubit Hamiltonians. Whisker plots showing the $\ell^2$-error $\varepsilon_{\ell^2}$ with different total evolution times $T$ and values of static field strengths $\nu$ for learning a single-qubit Hamiltonian (left) and a two-qubit Hamiltonian (right) on a log scale. The inner whiskers represents the 35th–65th percentiles, with the dot indicating the median. The outer whiskers extend from the 25th to the 75th percentiles. Dotted lines with slope 1 are included as references for the Heisenberg-limited scaling. Independent bit-flip channels with error rates 0.05 (for single-qubit case) and 0.03 (for two-qubit case) are applied to each qubit before measurement to model SPAM noise.
• Figure S3: Two-qubit Hamiltonian learning with different initial guesses. For fixed $\nu=5$ and total evolution time $1.4\times 10^8$ (corresponding to a target precision of $10^{-4}$ for energy gaps), we vary the initial guess and estimate the final accuracy. The $x$-axis is $\|\Delta\|$, which is the $\ell^2$ distance between the initial guess $\boldsymbol{\lambda}_{\text{guess}}$ and the true coefficient vector $\boldsymbol{\lambda}$ (here $\Delta=\boldsymbol{\lambda}_{\text{guess}}-\boldsymbol{\lambda}$).
• Figure S4: One-qubit experiment with increasing SPAM error rate $\eta$ for fixed $\nu=3$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2: Informal version of Theorem S6 in SM
• Lemma S1
• Theorem S1: Theorem 4.1 in MobusBluhmGefenEtAl2025heisenberg
• Lemma S2
• Lemma S3
• proof
• Proposition S1: Strong convexity
• proof
• Theorem S2