Determining the Hamiltonian that governs a quantum system represents a fundamental challenge in quantum sensing and device characterisation. The authors have developed a new protocol to address this critical task. Their research overcomes limitations of existing methods, which typically rely on complex multi‑qubit operations or single‑qubit controls that become increasingly demanding with greater precision. This new approach achieves optimal Heisenberg‑limited scaling using only static single‑qubit fields, crucially maintaining consistent field strengths regardless of the desired accuracy. The protocol’s robustness against state‑preparation and measurement (SPAM) errors, alongside a rigorous mathematical foundation, offers significant advancements for near‑term quantum technologies and provides a pathway towards more effective device characterisation and sensing capabilities.

Scientists have now demonstrated a novel protocol for determining this Hamiltonian with optimal, Heisenberg‑limited precision, overcoming limitations inherent in existing methods. The technique utilises only static, single‑qubit control fields, with strengths independent of the desired precision, to achieve this breakthrough in Hamiltonian learning. This approach circumvents the need for complex multi‑qubit operations prone to noise, or single‑qubit operations requiring increasingly high frequencies as precision demands increase.

The team achieved this by developing a protocol that learns a quantum Hamiltonian with a scaling that meets the Heisenberg limit, allowing the estimation of unknown Hamiltonians with an error ε using a total evolution time of O(1/ε). Unlike previous techniques, this method relies solely on the application of static control fields, modifying the system Hamiltonian without requiring entangled gates, even when the original Hamiltonian itself exhibits entanglement. The protocol involves preparing qubits in product states, allowing them to evolve under the modified Hamiltonian, and then performing single‑qubit Pauli measurements to extract information about the system’s properties. Rigorous mathematical proof and extensive numerical experiments confirm that this method achieves the desired scaling, demonstrating its efficacy in both single‑ and two‑qubit systems.

Furthermore, the study establishes an information‑theoretic lower bound, proving that a non‑vanishing static field strength is essential for reaching the Heisenberg limit unless an impractical number of discrete control operations are employed. This finding highlights the fundamental role of static fields in efficient Hamiltonian learning and provides a theoretical basis for the protocol’s performance. The research also demonstrates robustness against SPAM errors, a significant advantage for practical implementation on near‑term quantum platforms. Simulations reveal that the protocol maintains Heisenberg‑limited scaling with sufficiently strong static fields (ν ≥ 1.9 for single qubits and ν ≥ 4.5 for two qubits), even in the presence of SPAM noise.

These results pave the way for improved device characterisation and enhanced quantum sensing capabilities, with potential applications spanning diverse fields from precision magnetometry to advanced quantum computing. This work establishes a new benchmark for Hamiltonian learning, offering a practical and efficient pathway to unlock the full potential of quantum systems. By eliminating the need for complex control operations and demonstrating resilience to common experimental errors, the protocol provides a valuable tool for researchers and engineers developing quantum technologies. The findings open exciting possibilities for more accurate and reliable quantum measurements, ultimately driving progress in a wide range of scientific and technological domains.

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Static Fields Determine Quantum Hamiltonian Accurately

Scientists developed a novel protocol for learning a quantum Hamiltonian, achieving optimal Heisenberg‑limited scaling with a significantly simplified control scheme. The research addresses limitations inherent in existing methods, which typically rely on complex multi‑qubit operations or single‑qubit operations demanding increasingly precise control as desired precision increases. This work pioneers a technique employing only static, single‑qubit fields, applied with strengths independent of the target precision, to accurately determine the Hamiltonian governing a quantum system. The study demonstrates robustness against SPAM errors, a crucial advancement for practical implementation on near‑term quantum platforms.

The core of this innovation lies in the application of static control fields along the x, y, and z directions, modifying the system Hamiltonian without requiring dynamic manipulation. Researchers engineered a system where a carefully chosen static field strength, denoted ν, is applied to each qubit, effectively reshaping the Hamiltonian to facilitate efficient information extraction. Initial qubit preparation involves establishing a tensor product of Pauli eigenstates, followed by evolution under the modified Hamiltonian and subsequent single‑qubit Pauli measurements. This approach enables the reconstruction of the original Hamiltonian with arbitrary precision, scaling with a total evolution time of O(ε⁻¹), thus satisfying the Heisenberg limit.

Rigorous mathematical proof and numerical experiments validate the method’s performance, confirming its ability to achieve the optimal scaling. Furthermore, the team proved an information‑theoretic lower bound, establishing that a non‑vanishing static field strength is essential for attaining the Heisenberg limit unless an extensive number of discrete control operations are employed. Experiments utilized n qubits, demonstrating the protocol’s efficacy even with entangled Hamiltonians, and the study’s Theorem 1 characterizes performance, showing that the protocol achieves the stated precision ε with probability at least 1 − δ using O(ε⁻¹ log(1/δ)) total evolution time. This breakthrough provides new tools for device characterization and quantum sensing, circumventing the challenges associated with complex control schemes and paving the way for more robust and precise quantum technologies.

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Heisenberg Scaling with Single‑Qubit Control Demonstrated

Scientists have developed a novel protocol for learning the Hamiltonian governing a quantum system, achieving optimal Heisenberg‑limited scaling using only single‑qubit control. The research overcomes limitations of existing methods which require complex multi‑qubit operations or control strengths that increase with desired precision. This breakthrough utilizes static fields with strengths independent of the target precision, offering a robust approach against SPAM errors. The work demonstrates new tools for precise device characterization and sensing applications.

Experiments revealed the protocol achieves Heisenberg‑limited scaling, confirmed through both rigorous mathematical proof and extensive numerical simulations. For single‑qubit Hamiltonians, the scaling was evident for static field strengths of ν ≥ 1.9, while two‑qubit Hamiltonians required ν ≥ 4.5 to demonstrate the same precision. The unknown single‑qubit Hamiltonian was defined as

[

H = 0.1,\sigma_x + 0.5,\sigma_y + 0.3,\sigma_z,

]

and detailed simulations, including variations in the two‑qubit Hamiltonian and initial optimization guesses, are provided in the supplemental materials. Measurements confirm the protocol’s resilience to SPAM noise, with independent bit‑flip channels introducing error rates of 0.05 for single‑qubit and 0.03 for two‑qubit cases.

Data show a fundamental lower bound on the minimum total evolution time required to achieve a target error ε. The team derived a quantitative trade‑off between achievable precision and relevant resources, including field strength ν, total evolution time T, and the number of discrete control operations L. Theorem 2 establishes that, without an extensive number of discrete control operations, achieving Heisenberg‑limited scaling necessitates a static field strength of at least ν = Ω(1). This finding reveals a crucial relationship between evolution time, field strength, and discrete control operations needed for a given precision.

Further analysis, utilizing Le Cam’s two‑point method, demonstrates that the distance between unitary evolution operators generated by two Hamiltonians grows proportionally to (ν ε + ε²) T + L ε. The research considered discriminating between two fixed single‑qubit Hamiltonians, establishing a connection between learning protocols and the ability to distinguish these Hamiltonians. The study focused on a system of n qubits evolving under an unknown Hamiltonian, aiming to estimate coefficients λ such that the ℓ₂‑distance between estimated and actual values is bounded by ε with probability at least 1 − δ.

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Static Fields Enable Optimal Hamiltonian Learning

This work details a new protocol for learning the Hamiltonian governing a quantum system, achieving optimal Heisenberg‑limited scaling with significantly reduced demands on quantum control. Existing methods typically require complex multi‑qubit operations or single‑qubit manipulations that become increasingly difficult as precision increases; this research circumvents these limitations by utilising only static fields, with strengths independent of the desired accuracy. Through rigorous mathematical proof and numerical experiments, the authors demonstrate the efficacy of their approach for both device characterisation and precision sensing. The significance of this achievement lies in its potential to facilitate Hamiltonian learning on near‑term quantum platforms.

By removing the need for increasingly complex control operations, the protocol offers a pathway to higher‑precision measurements with hardware currently under development. The authors acknowledge that, to achieve the Heisenberg limit, a non‑vanishing static field strength is necessary unless extensive discrete control operations are employed. Future research directions include further exploration of the protocol’s robustness and potential applications in more complex quantum systems, building upon the established foundation of constant‑precision static field control.

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Reference

Learning Hamiltonians in the Heisenberg limit with static single‑qubit fields – arXiv: 2601.10380 (2026)

https://arxiv.org/abs/2601.10380