Identifying an accurate model for the dynamics of a quantum system is a vexing problem that underlies a range of problems in experimental physics and quantum information theory. Recently, a method called quantum Hamiltonian learning has been proposed by the present authors that uses quantum simulation as a resource for modeling an unknown quantum system. This approach can, under certain circumstances, allow such models to be efficiently identified. A major caveat of that work is the assumption of that all elements of the protocol are noise free. Here we show that quantum Hamiltonian learning can tolerate substantial amounts of depolarizing noise and show numerical evidence that it can tolerate noise drawn from other realistic models. We further provide evidence that the learning algorithm will find a model that is maximally close to the true model in cases where the hypothetical model lacks terms present in the true model. Finally, we also provide numerical evidence that the algorithm works for noncommuting models. This work illustrates that quantum Hamiltonian learning can be performed using realistic resources and suggests that even imperfect quantum resources may be valuable for characterizing quantum systems.
Identifying an accurate model for the dynamics of a quantum system is a vexing problem that underlies a range of problems in experimental physics and quantum information theory. Recently, a method called quantum Hamiltonian learning has been proposed by the present authors that uses quantum simulation as a resource for modeling an unknown quantum system. This approach can, under certain circumstances, allow such models to be efficiently identified. A major caveat of that work is the assumption of that all elements of the protocol are noise free. Here we show that quantum Hamiltonian learning can tolerate substantial amounts of depolarizing noise and show numerical evidence that it can tolerate noise drawn from other realistic models. We further provide evidence that the learning algorithm will find a model that is maximally close to the true model in cases where the hypothetical model lacks terms present in the true model. Finally, we also provide numerical evidence that the algorithm works for noncommuting models. This work illustrates that quantum Hamiltonian learning can be performed using realistic resources and suggests that even imperfect quantum resources may be valuable for characterizing quantum systems.
In recent years quantum simulation has made great strides, culminating in experiments that existing supercomputers cannot easily simulate. Although this raises the possibility that special purpose analog quantum simulators may be able to perform computational tasks that existing computers cannot, it also introduces a major challenge: certifying that the quantum simulator is in fact simulating the correct quantum dynamics. We provide an algorithm that, under relatively weak assumptions, can be used to efficiently infer the Hamiltonian of a large but untrusted quantum simulator using a trusted quantum simulator. We illustrate the power of this approach by showing numerically that it can inexpensively learn the Hamiltonians for large frustrated Ising models, demonstrating that quantum resources can make certifying analog quantum simulators tractable.
Hamiltonian learning is a cornerstone for advancing accurate many-body simulations, improving quantum device performance, and enabling quantum-enhanced sensing. Existing readily deployable quantum metrology techniques primarily focus on achieving Heisenberg-limited precision in one- or two-qubit systems. In contrast, general Hamiltonian learning theories address broader classes of unknown Hamiltonian models but are highly inefficient due to the absence of prior knowledge about the Hamiltonian. There remains a lack of efficient and practically realizable Hamiltonian learning algorithms that directly exploit the known structure and prior information of the Hamiltonian, which are typically available for a given quantum computing platform. In this work, we present the first Hamiltonian learning algorithm that achieves both Cramer-Rao lower bound saturated optimal precision and robustness to realistic noise, while exploiting device structure for quadratic reduction in experimental cost for fully connected Hamiltonians. Moreover, this approach enables simultaneous in-situ estimation of all Hamiltonian parameters without requiring the decoupling of non-learnable interactions during the same experiment, thereby allowing comprehensive characterization of the system's intrinsic contextual errors. Notably, our algorithm does not require deep circuits and remains robust against both depolarizing noise and time-dependent coherent errors. We demonstrate its effectiveness with a detailed experimental proposal along with supporting numerical simulations on Rydberg atom quantum simulators, showcasing its potential for high-precision Hamiltonian learning in the NISQ era.
Quantum process characterization is a fundamental task in quantum information processing, yet conventional methods, such as quantum process tomography, require prohibitive resources and lack scalability. Here, we introduce an efficient quantum process learning method specifically designed for short-time Hamiltonian dynamics. Our approach reconstructs an equivalent quantum circuit representation from measurement data of unknown Hamiltonian evolution without requiring additional assumptions and achieves polynomial sample and computational efficiency. Our results have broad applications in various directions. We demonstrate applications in quantum machine learning, where our protocol enables efficient training of variational quantum neural networks by directly learning unitary transformations. Additionally, it facilitates the prediction of quantum expectation values with provable efficiency and provides a robust framework for verifying quantum computations and benchmarking realistic noisy quantum hardware. This work establishes a new theoretical foundation for practical quantum dynamics learning, paving the way for scalable quantum process characterization in both near-term and fault-tolerant quantum computing.
A major problem facing the development of quantum computers or large scale quantum simulators is that general methods for characterizing and controlling are intractable. We provide a new approach to this problem that uses small quantum simulators to efficiently characterize and learn control models for larger devices. Our protocol achieves this by using Bayesian inference in concert with Lieb–Robinson bounds and interactive quantum learning methods to achieve compressed simulations for characterization. We also show that the Lieb–Robinson velocity is epistemic for our protocol, meaning that information propagates at a rate that depends on the uncertainty in the system Hamiltonian. We illustrate the efficiency of our bootstrapping protocol by showing numerically that an 8 qubit Ising model simulator can be used to calibrate and control a 50 qubit Ising simulator while using only about 750 kilobits of experimental data. Finally, we provide upper bounds for the Fisher information that show that the number of experiments needed to characterize a system rapidly diverges as the duration of the experiments used in the characterization shrinks, which motivates the use of methods such as ours that do not require short evolution times.
We develop a framework for learning from noisy quantum experiments in which fault-tolerant devices access uncharacterized systems through noisy couplings. Introducing the complexity class NBQP ("noisy BQP''), we model noisy fault-tolerant quantum computers that cannot generally error-correct the oracle systems they query. Using this class, we prove that while noise can eliminate the exponential quantum learning advantages of unphysical, noiseless learners, a superpolynomial gap remains between NISQ and fault-tolerant devices. Turning to canonical learning tasks in noisy settings, we find that the exponential two-copy advantage for purity testing collapses under local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient physical structure restores this quantum learning advantage. We then analyze noisy Pauli shadow tomography, deriving lower bounds characterizing how instance size, quantum memory and noise jointly control sample complexity, and design algorithms with parametrically matching scalings. We study similar tradeoffs in quantum metrology, and show that the Heisenberg-limited sensitivity of existing error-correction-based protocols persists only up to a timescale inverse-polynomial in the error rate per probe qubit. Together, our results demonstrate that the primitives underlying quantum-enhanced experiments are fundamentally fragile to noise, and that realizing meaningful quantum advantages in future experiments will require interfacing noise-robust physical properties with available algorithmic techniques.
With the fast development of quantum technology, the sizes of both digital and analog quantum systems increase drastically. In order to have better control and understanding of the quantum hardware, an important task is to characterize the interaction, i.e., to learn the Hamiltonian, which determines both static and dynamic properties of the system. Conventional Hamiltonian learning methods either require costly process tomography or adopt impractical assumptions, such as prior information on the Hamiltonian structure and the ground or thermal states of the system. In this work, we present a robust and efficient Hamiltonian learning method that circumvents these limitations based only on mild assumptions. The proposed method can efficiently learn any Hamiltonian that is sparse on the Pauli basis using only short-time dynamics and local operations without any information on the Hamiltonian or preparing any eigenstates or thermal states. The method has a scalable complexity and a vanishing failure probability regarding the qubit number. Meanwhile, it performs robustly given the presence of state preparation and measurement errors and resiliently against a certain amount of circuit and shot noise. We numerically test the scaling and the estimation accuracy of the method for transverse field Ising Hamiltonian with random interaction strengths and molecular Hamiltonians, both with varying sizes and manually added noise. All these results verify the robustness and efficacy of the method, paving the way for a systematic understanding of the dynamics of large quantum systems.