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arXiv:2607.16047

Rigorous Time-dependent Hamiltonian Learning via Continuous Weak Measurements

Jesús Jiménez-Rodríguez, Giacomo Franceschetto, Antonio Acín, Luciano Pereira

quant-phSubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
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Abstract

Characterizing the Hamiltonian that a quantum processor actually implements is central to calibrating and validating current quantum hardware. Many devices, however, operate with generators that are time dependent by design. Here we develop a rigorous and experimentally friendly protocol for learning time-dependent many-body Hamiltonians from continuous weak measurement records. The key observation is that interaction sparsity reduces the global reconstruction to a set of local inverse problems, whose number is controlled by the interaction connectivity rather than by the system size. Pure separable probe states suffice to drive these inversions, and a graph-coloring construction embeds them into a small number of global product-state preparations. We derive explicit reconstruction-error bounds and a sample-complexity theorem that cleanly separates the finite-sampling statistical noise from the deterministic bias of the iterative state update, and we validate the protocol on time-dependent spin chains with up to $n=8$ qubits. Beyond these results, our analysis provides a rigorous foundation for time-dependent Hamiltonian learning from continuous monitoring in many-body systems, establishing a framework that extends naturally to many platforms and probe ensembles.

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Characterizing the Hamiltonian that a quantum processor actually implements is central to calibrating and validating current quantum hardware. Many devices, however, operate with generators that are time dependent by design. Here we develop a rigorous and experimentally friendly protocol for learning time-dependent many-body Hamiltonians from continuous weak measurement records. The key observation is that interaction sparsity reduces the global reconstruction to a set of local inverse problems, whose number is controlled by the interaction connectivity rather than by the system size. Pure separable probe states suffice to drive these inversions, and a graph-coloring construction embeds them into a small number of global product-state preparations. We derive explicit reconstruction-error bounds and a sample-complexity theorem that cleanly separates the finite-sampling statistical noise from the deterministic bias of the iterative state update, and we validate the protocol on time-dependent spin chains with up to n=8n=8n=8 qubits. Beyond these results, our analysis provides a rigorous foundation for time-dependent Hamiltonian learning from continuous monitoring in many-body systems, establishing a framework that extends naturally to many platforms and probe ensembles.
Hamiltonian learning provides a systematic framework for reconstructing unknown quantum dynamics. However, existing protocols typically assume direct measurement access to the entire system. With fast single-qubit control and a connected reference backbone, we show that a single measurable qubit suffices to learn all O(N)O(N)O(N) independent parameters of a bounded-degree two-body Hamiltonian on NNN qubits at the Heisenberg limit. Crucially, our protocol uses robust SWAP gates synthesized by quantum signal processing, enabling coherent transfer of states evolving under distant Hamiltonian parameters to the measurable qubit. This transfer requires no prior calibration of the Hamiltonian parameters of the intermediate links. A parallel learning architecture achieves total query time O~(N)\widetilde{O}(N)O(N) for an NNN-qubit chain, while retaining Heisenberg-limited precision scaling. On the chain, these scalings match the fundamental precision and information-propagation lower bounds up to logarithmic factors. The framework further extends to arbitrary bounded-degree interaction graphs. Our results establish a scalable route to learning an extensive number of many-body Hamiltonian parameters through only a local measurement interface.
Hamiltonian learning protocols are becoming valuable tools for benchmarking and verifying quantum computers and simulators, yet few rigorous protocols address explicitly time-dependent dynamics. We give a scalable protocol for learning the time-dependent coefficients of a locally interacting nnn-qubit system on a graph G\mathsf{G}G of effective dimension DDD. The unknown generator contains geometrically local Hamiltonian terms and weak onsite Pauli-dephasing terms with time-dependent, nonnegative rates. Its coefficient functions and their derivatives admit a certified simultaneous C1C^1C1 approximation in a known finite-dimensional function class that supports stable interpolation. The only experimental primitives are preparation of product Pauli eigenstates, evolution for chosen times, and product-Pauli measurements. Under an explicit weak-dissipation condition, the Lieb--Robinson estimates proved here, and a certified effective approximation/interpolation degree meffm_{\rm eff}meff​, the protocol reconstructs every coefficient uniformly on a constant interval with error εεε and success probability 1−δ1-δ1−δ using O~ ⁣(ε−2poly⁡(meff)log⁡(nδ−1))\widetilde{\mathcal{O}}\!\left(ε^{-2}\operatorname{poly}(m_{\rm eff})\log(nδ^{-1})\right)O(ε−2poly(meff​)log(nδ−1)) samples and polynomial classical processing. The method estimates derivatives of finite-time channel overlaps by stable interpolation, uses process shadows to learn the required local transfer matrices, and constructs sparse diagonally dominant systems through localized semidefinite programs.
Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works have proposed protocols achieving the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian HHH with ∣∣H∣∣≤Λ||H||\leqΛ∣∣H∣∣≤Λ in total evolution time of Θ(Λε2log⁡(Λε))Θ(\fracΛ{ε^2}\log(\fracΛε))Θ(ε2Λ​log(εΛ​)). The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of Ω(Λε2log⁡(Λε))Ω(\fracΛ{ε^2}\log(\fracΛε))Ω(ε2Λ​log(εΛ​)). Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on ΛΛΛ instead of ε\varepsilonε, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.
Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli X/ZX/ZX/Z interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.
Characterizing the dynamics of open quantum systems at the level of microscopic interactions and error mechanisms is essential for calibrating quantum hardware, designing robust simulation protocols, and developing tailored error-correction methods. Under Markovian noise/dissipation, a natural characterization approach is to identify the full Lindbladian generator that gives rise to both coherent (Hamiltonian) and dissipative dynamics. Prior protocols for learning Lindbladians from dynamical data assumed pre-specified interaction structure, which can be restrictive when the relevant noise channels or control imperfections are not known in advance. In this paper, we present a sample-efficient protocol for learning sparse Lindbladians without assuming any a priori structure. Our protocol is ancilla-free, uses only product-state preparations and Pauli-basis measurements, admits provably stable coefficient reconstruction, and achieves near-optimal time resolution, making it compatible with near-term experimental capabilities. Together, this provides a systematic route to scalable characterization of open-system quantum dynamics, especially in settings where the error mechanisms of interest are unknown.
Quantum devices are open systems whose dynamics interleave coherent evolution with dissipation, and benchmarking, error mitigation, and error correction all rest on a faithful model of both. Existing characterization protocols either assume prior knowledge of the interaction and noise structure, or demand ancillas, entangled probes, or mid-circuit control, or capture only the Pauli-diagonal part of the noise. Here, we present a protocol that reconstructs an arbitrary sparse Markovian generator, including every Hamiltonian together with the jump operator coefficients, using only product Pauli state preparation, single uninterrupted forward evolutions, and product Pauli measurements. Given a sparsity budget M0M_0M0​ and a strength bound ΓΓΓ of the Lindbladian, every coefficient is learned to precision εεε from O~(Γ2M02/ε4)\widetilde{O}(Γ^2M_0^2/ε^4)O(Γ2M02​/ε4) experiments and O~(ΓM02/ε2)\widetilde{O}(ΓM_0^2/ε^2)O(ΓM02​/ε2) total evolution time, with both supports identified from data without locality assumptions. The protocol runs at a logarithmic number of positive evolution times on a hardware clock lattice and is provably robust to calibrated state-preparation and measurement errors.