hamiltonian learning

This work addresses the problem of learning Hamiltonian parameters of a many-body quantum system from thermal-state time evolution, using only local probes—reflecting realistic experimental constraints where global control or measurement is infeasible. We propose “quantum probe tomography,” a novel framework that integrates algebraic geometry with smooth analysis to rigorously establish identifiability: generic many-body Hamiltonians are uniquely reconstructible from local probe data. Methodologically, we design the first end-to-end efficient learning algorithm, achieving polynomial query complexity—$ ext{poly}(1/varepsilon)$—and polylogarithmic classical post-processing time—$ ext{polylog}(1/varepsilon)$. We validate the approach on translation- and rotation-invariant nearest-neighbor lattice models in one, two, and three dimensions, demonstrating $varepsilon$-accurate Hamiltonian reconstruction. Our key contributions are: (i) the first rigorous proof of local identifiability for generic many-body Hamiltonians under thermal dynamics; and (ii) the first locally constrained Hamiltonian learning scheme with provable polynomial-time guarantees.

This work addresses the challenges of Hamiltonian learning on near-term quantum devices, which typically rely on deep circuits, high time resolution, or ancillary qubits. The authors propose an in-situ learning algorithm that requires neither quantum control nor ancilla qubits. By leveraging Pauli product state preparation and measurement, combined with random sampling, band-limited kernel time sampling, and a shift-and-filter technique, the method efficiently reconstructs norm-bounded Hamiltonians. Theoretically, it achieves an optimal total evolution time scaling of Θ(Λ/ε² log(Λ/ε)) and, for the first time under no-control conditions, matches the information-theoretic lower bound of Ω(Λ/ε² log(Λ/ε)). The required probe time resolution depends only on the Hamiltonian norm, and the algorithm exhibits robustness against SPAM noise while maintaining optimal asymptotic performance for local Hamiltonians.

This work addresses the problem of learning the structure of an unknown quantum many-body Hamiltonian $ H $ from black-box time evolution, without prior assumptions on locality, interaction form, or specific term types—only requiring that the number of nonzero terms is polynomially bounded. We propose a novel learning framework based on pseudo-Choi states, enabling the first efficient reconstruction of arbitrary $ n $-qubit Hamiltonians. Our method integrates quantum phase estimation, Fourier spectral analysis, and controlled time evolution—including both time-reversal and purely forward-evolution models. Under the time-reversal model, the total evolution time scales as $ ilde{O}(m/varepsilon) $, yielding an $ varepsilon $-accurate Hamiltonian description; under the purely forward model, it achieves $ ilde{O}(|H|^3/varepsilon^4) $, breaking previous exponential complexity barriers and attaining near-Heisenberg-limited scaling.

This work studies the problem of testing *k-locality* of an unknown *n*-qubit Hamiltonian *H*: given black-box access to the time evolution under *H*, determine whether *H* is *k*-local or ε-far (in normalized Frobenius norm) from all *k*-local Hamiltonians. It is the first to formulate Hamiltonian property testing as a quantum property testing problem, revealing an exponential dependence of query complexity on the choice of distance metric. We propose the first average-case efficient algorithm, leveraging randomized measurements and incoherent quantum queries to achieve sample- and time-efficient *k*-locality testing with polynomial sample, query, and computational complexity. Our approach extends naturally to generalized Hamiltonian property testing. Crucially, it establishes the first exponential separation between quantum testing and quantum learning tasks—demonstrating that testing certain Hamiltonian properties is exponentially easier than learning them.

This work addresses the classical efficient estimation of expectation values of arbitrary observables under noiseless random quantum circuits. Prior methods were restricted to noisy circuits and struggled with fully connected or deep architectures. We propose the first universal classical algorithm, based on Pauli operator path expansion in the Heisenberg picture, integrated with single-qubit rotation-invariant measures and classical shadow techniques. Our algorithm is provably efficient for arbitrary circuit geometries—including all-to-all connectivity—and arbitrary depth, achieving polynomial-time high-precision estimation for constant accuracy (ε, δ), and quasi-polynomial time for inverse-polynomial precision. It succeeds on the overwhelming majority of circuit instances with controllable failure probability. Crucially, we provide the first rigorous proof that noiseless quantum circuits dominated by chaos and local shuffling remain classically estimable—thereby breaking the prior reliance on noise for classical simulability.

This work addresses the challenge of learning both the structure and parameters of Lindbladians governing open quantum systems. It proposes an efficient iterative algorithm that recovers the coefficients of an $n$-qubit, constant-locality Lindbladian from time-evolution data using only non-adaptive, ancilla-free random Pauli measurements, without requiring prior knowledge of the underlying interaction graph. The method achieves, for the first time, efficient structure learning for Lindbladians with quasi-local or power-law interactions and extends naturally to Hamiltonian structure learning from high-temperature Gibbs states. Based on Fourier coefficient optimization, the algorithm excels under limited interference conditions, attaining $\varepsilon$ accuracy with total evolution time $O(g d^2 \log n / \varepsilon^2)$ and temporal resolution $\Theta(1/g)$.

This work addresses the challenge of characterizing quantum device dynamics, particularly non-diagonal dissipative processes, which existing methods struggle to capture effectively due to their reliance on prior noise models, ancillary qubits, or complex control sequences. The authors propose an efficient reconstruction scheme requiring only product Pauli initial state preparation, a single uninterrupted evolution, and product Pauli measurements. Notably, the method identifies the support of a sparse Lindbladian without assuming locality. Leveraging compressed sensing principles, it robustly reconstructs all coefficients of the Hamiltonian and jump operators with $\tilde{O}(\Gamma^2 M_0^2 / \varepsilon^4)$ experimental repetitions and $\tilde{O}(\Gamma M_0^2 / \varepsilon^2)$ total evolution time to achieve accuracy $\varepsilon$, while providing theoretical robustness guarantees against calibrated initialization and measurement errors.

This work addresses the efficient learning of unknown Markovian open quantum system generators (Lindbladians) from physical time-evolution data, where the generator involves an exponential number of Hamiltonian and dissipative coefficients. The authors propose a two-stage non-adaptive algorithm that requires neither ancillary qubits nor controlled operations. In the first stage, product Pauli eigenstates are prepared and single-qubit Pauli measurements are performed to identify the support of significant terms. The second stage employs random stabilizer state preparation and measurements in random Clifford bases to accurately estimate the coefficients. This approach achieves, for the first time, near-optimal learning of arbitrary Lindbladians, with experimental complexity exceeding the theoretical lower bound only by logarithmic factors: it estimates all coefficients within error ε using Õ(Λ²/ε²) experiments and total evolution time Õ(Λ/ε²), while classical post-processing remains polynomial in cost.

This work investigates the efficient learning of unknown Hamiltonians governing quantum many-body systems from short-time evolution data, and rigorously characterizes the performance gap between quantum and classical machine learning in this setting. Framed within the PAC learning paradigm, the study formulates a supervised learning task that integrates Hamiltonian learning, Hamiltonian simulation, and classical shadow protocols for training and inference. Its central contribution is the first provable quantum–classical learning separation for a natural quantum machine learning problem rooted in physical dynamics: there exists a class of instances learnable by a quantum algorithm in polynomial time, yet provably intractable for any classical randomized algorithm unless BQP ⊆ P/poly. This result establishes a rigorous quantum advantage while preserving the physical interpretability and learnability of the underlying quantum system.