[论文解读] Learning Hamiltonians in the Heisenberg limit with static single-qubit fields

Shrigyan Brahmachari, Shuchen Zhu+2 more|arXiv (Cornell University)|Jan 15, 2026

Quantum Computing Algorithms and Architecture被引用 0

一句话总结

论文提出一种协议，利用仅静态单量子比特场，在Heisenberg级别的尺度下学习未知量子哈密顿量，对SPAM鲁棒，且适用于1-或2-量子比特系统。

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.

研究动机与目标

• Motivate learning unknown Hamiltonians for quantum metrology, sensing, and device characterization.
• Introduce a protocol that achieves Heisenberg-limited scaling using only static single-qubit fields.
• Ensure robustness to state preparation and measurement (SPAM) errors.
• Demonstrate performance via rigorous proofs and numerical simulations for small-qubit systems.

提出的方法

• Apply strong static single-qubit fields along x, y, z directions to modify the Hamiltonian as H_tot = H − ν H_ctrl.
• Prepare initial product states, evolve under H_tot, and perform single-qubit Pauli measurements.
• Use phase-estimation experiments to estimate the energy gap EΔ generated by H_tot.
• Perform a least-squares recovery from measured energy gaps to estimate the Hamiltonian coefficients λ via a strongly convex loss near the true parameters.
• Prove an information-theoretic lower bound showing tradeoffs between evolution time, field strength, and number of discrete control operations.
• Provide numerical simulations for 1- and 2-qubit Hamiltonians to illustrate Heisenberg-limited scaling under SPAM.

Figure 1: Efficient Hamiltonian learning with static local fields. For a system of qubits with unknown Hamiltonian $H(\bm{\lambda})$ , 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

实验结果

研究问题

• RQ1 Can a Hamiltonian be learned with Heisenberg-limited scaling using only static single-qubit fields?
• RQ2 What is the minimum total evolution time and experimental resources required to achieve a target precision ε?
• RQ3 How robust is the protocol to SPAM errors and to nonadaptive versus adaptive strategies?
• RQ4 How can coefficients of H be recovered reliably from energy-gap measurements via a convex optimization approach?
• RQ5 What fundamental tradeoffs exist between field strength, evolution time, and discrete control operations for Hamiltonian learning?

主要发现

• The protocol achieves Heisenberg-limited scaling with total evolution time T = O(ε^{-1}).
• Static fields of fixed strength ν, independent of ε, suffice for high-precision Hamiltonian learning without entangling gates.
• The method is robust to SPAM errors as long as SPAM is below a fixed, ε-independent threshold.
• Energy gaps EΔ(λ, ν, k, s, β) can be estimated to precision ε via O(polylog(ε^{-1} δ^{-1})) experiments in time O(1/ε).
• A least-squares estimator on the measured energy gaps recovers the Hamiltonian coefficients with ε accuracy, with a locally strongly convex loss guaranteeing unique recovery near the true parameters.
• They prove an information-theoretic lower bound showing that a non-vanishing static field is necessary for Heisenberg scaling unless many discrete control operations are used.

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