hamiltonian simulation

This work addresses the high quantum overhead of classical data loading and the poor trainability of quantum machine learning models on near-term hardware by proposing a supervised learning framework based on encoding classical data into the ground states of k-local Hamiltonians. The approach compactly maps input data to low-energy eigenstates of parameterized Hamiltonians, which are efficiently approximated using sample-based Krylov quantum diagonalization. Shallow quantum circuits are then trained via local gradient-based optimization to prepare these states. By circumventing explicit data-encoding circuits, the method substantially reduces circuit depth, alleviates the data-loading bottleneck, and enhances model trainability. Empirical validation on up to 50 qubits using IBM’s Heron processor demonstrates the framework’s effectiveness and scalability on standard benchmark datasets.

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 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.

This work addresses the “ansatz-free Hamiltonian learning” problem: reconstructing an arbitrary sparse Hamiltonian with high precision without prior knowledge of its interaction graph structure. Methodologically, it relies solely on black-box queries to real-time Hamiltonian evolution and minimal digital control, integrating adaptive parameter estimation, robust state preparation, SPAM-error mitigation, and sparse coding techniques. It achieves, for the first time, ansatz-free Hamiltonian learning at the Heisenberg limit—scaling as 1/T in estimation error with total evolution time T—thereby surpassing the standard quantum limit. Moreover, it uncovers a fundamental trade-off between total evolution time and the degrees of freedom in quantum control. The framework establishes a new paradigm for assumption-free benchmarking and characterization of complex quantum systems, combining rigorous theoretical foundations with experimental feasibility.

This work addresses the problem of learning the unknown Hamiltonian structure of a quantum system without prior knowledge of its interaction topology. We propose a novel method to efficiently reconstruct local interaction graphs from real-time dynamical data. Our approach combines spectral analysis with adaptive parameter estimation, requiring only a constant number of long-time evolutions and total evolution time scaling logarithmically in system size. It imposes no prior assumptions on the interaction term set and extends beyond short-range models to general locally bounded Hamiltonians—including those with power-law decaying interactions. Theoretically, we prove that the method achieves ε-precision reconstruction in total evolution time O(log n/ε), attaining the Heisenberg-limit scaling and constant time resolution—significantly surpassing the standard 1/ε² sampling limit. The algorithm is conceptually simple, with mathematical foundations accessible at the high-school level.

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 how to efficiently learn from and predict ground-state observables of two-dimensional many-body systems using approximate ground-state data generated on noisy quantum processors. Focusing on the Heisenberg XXZ model, we prepare approximate ground states in two-dimensional systems with up to 115 qubits, measure single-site expectation values, two-point correlations, and 12-body loop correlators to construct an experimental dataset, and train neural networks to predict spatially resolved observables for arbitrary Hamiltonian parameters. This approach represents the first demonstration of machine learning based on real quantum data in large-scale two-dimensional interacting systems. The trained models not only achieve high accuracy within the training distribution but also generalize to out-of-distribution regimes near phase boundaries, demonstrating that noisy quantum devices can provide training data beyond the reach of classical simulation capabilities.

This work proposes a Hamiltonian learning framework that achieves Heisenberg-limited precision without requiring ultrafast control pulses, under the experimental constraint that all evolution operations must last at least a minimum time \( T \). By integrating continuous control simulation with sparse pure-state tomography, the method exclusively employs evolution durations no shorter than \( T \). The study establishes, for the first time, that information-theoretically optimal scaling of total evolution time—specifically \( 1/\varepsilon \) to achieve estimation error \( \varepsilon \)—remains attainable under this constraint: logarithmically sparse Hamiltonians reach the Heisenberg limit, while polynomially sparse many-body systems incur only polynomial overhead due to the minimal-time restriction. These results demonstrate that ultra-high-bandwidth, sub-\( T \) pulses are not essential for optimal quantum learning, resolving a key open question in the field.

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 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 $\Theta(1/g)$.