We study the problem of learning the parameters for the Hamiltonian of a quantum many-body system, given limited access to the system. In this work, we build upon recent approaches to Hamiltonian learning via derivative estimation. We propose a protocol that improves the scaling dependence of prior works, particularly with respect to parameters relating to the structure of the Hamiltonian (e.g., its locality k). Furthermore, by deriving exact bounds on the performance of our protocol, we are able to provide a precise numerical prescription for theoretically optimal settings of hyperparameters in our learning protocol, such as the maximum evolution time (when learning with unitary dynamics) or minimum temperature (when learning with Gibbs states). Thanks to these improvements, our protocol has practical scaling for large problems: we demonstrate this with a numerical simulation of our protocol on an 80-qubit system.
We study the problem of learning the parameters for the Hamiltonian of a quantum many-body system, given limited access to the system. In this work, we build upon recent approaches to Hamiltonian learning via derivative estimation. We propose a protocol that improves the scaling dependence of prior works, particularly with respect to parameters relating to the structure of the Hamiltonian (e.g., its locality k). Furthermore, by deriving exact bounds on the performance of our protocol, we are able to provide a precise numerical prescription for theoretically optimal settings of hyperparameters in our learning protocol, such as the maximum evolution time (when learning with unitary dynamics) or minimum temperature (when learning with Gibbs states). Thanks to these improvements, our protocol has practical scaling for large problems: we demonstrate this with a numerical simulation of our protocol on an 80-qubit system.
We study the problem of learning a Hamiltonian H to precision ε, supposing we are given copies of its Gibbs state ρ=exp(−βH)/Tr(exp(−βH)) at a known inverse temperature β. Anshu, Arunachalam, Kuwahara, and Soleimanifar [AAKS21] recently studied the sample complexity (number of copies of ρ needed) of this problem for geometrically local N-qubit Hamiltonians. In the high-temperature (low β) regime, their algorithm has sample complexity poly (N, 1/β, 1/ε) and can be implemented with polynomial, but suboptimal, time complexity. In this paper, we study the same question for a more general class of Hamiltonians. We show how to learn the coefficients of a Hamiltonian to error ε with sample complexity S=O(logN/(βε)2) and time complexity linear in the sample size, O(SN). Furthermore, we prove a matching lower bound showing that our algorithm’s sample complexity is optimal, and hence our time complexity is also optimal. In the appendix, we show that virtually the same algorithm can be used to learn H from a real-time evolution unitary e−itH in a small t regime with similar sample and time complexity.
Learning the unknown interactions that govern a quantum system is crucial for quantum information processing, device benchmarking, and quantum sensing. The problem, known as Hamiltonian learning, is well understood under the assumption that interactions are local, but this assumption may not hold for arbitrary Hamiltonians. Previous methods all require high-order inverse polynomial dependency with precision, unable to surpass the standard quantum limit and reach the gold-standard Heisenberg-limited scaling. Whether Heisenberg-limited Hamiltonian learning is possible without prior assumptions about the interaction structures, a challenge we term , remains an open question. In this work, we present a quantum algorithm to learn arbitrary sparse Hamiltonians without any structure constraints using only black-box queries of the system’s real-time evolution and minimal digital controls to attain Heisenberg-limited scaling in estimation error. Our method is also resilient to state-preparation-and-measurement errors, enhancing its practical feasibility. We numerically demonstrate our ansatz-free protocol for learning physical Hamiltonians and validating analog quantum simulations, benchmarking our performance against the state-of-the-art Heisenberg-limited learning approach. Moreover, we establish a fundamental trade-off between total evolution time and quantum control on learning arbitrary interactions, revealing the intrinsic interplay between controllability and total evolution-time complexity for any learning algorithm. These results pave the way for further exploration into Heisenberg-limited Hamiltonian learning in complex quantum systems under minimal assumptions, potentially enabling new benchmarking and verification protocols.
Learning the Hamiltonian underlying a quantum many-body system in thermal equilibrium is a fundamental task in quantum learning theory and experimental sciences. To learn the Gibbs state of local Hamiltonians at any constant inverse temperature, the state-of-the-art provable algorithms fall short of the optimal sample and computational complexity, in sharp contrast with the locality and simplicity in the classical cases. In this work, we present a learning algorithm that learns each local term of a n-qubit Hamiltonian on any bounded-degree graph to a constant additive error with the optimal sample complexity O(logn). The protocol uses parallelizable local quantum measurements that act within bounded neighborhoods of the graph and near-linear-time classical post-processing. We also give a learning algorithm for lattice Hamiltonians with near-optimal scaling on the learning precision and the inverse temperature. At the heart of our algorithm is the interplay between locality, the Kubo-MartinSchwinger condition, and the operator Fourier transform at arbitrary temperatures.
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) independent parameters of a bounded-degree two-body Hamiltonian on N 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) for an N-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.
We study the problem of learning a local quantum Hamiltonian H given copies of its Gibbs state ρ = e−β H/(e−β H) at a known inverse temperature β>0. Anshu, Arunachalam, Kuwahara, and Soleimanifar gave an algorithm to learn a Hamiltonian on n qubits to precision with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem, with prior work only resolving this in the limited cases of high temperature or commuting terms. We fully resolve this problem, giving a polynomial time algorithm for learning H to precision from polynomially many copies of the Gibbs state at any constant β > 0. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian.
In this work, we study the problems of certifying and learning quantum k-local Hamiltonians, for a constant k. Our main contributions are as follows:
- Certification of Hamiltonians. We show that certifying a local Hamiltonian in normalized Frobenius norm via access to its time-evolution operator can be achieved with only O(1/ε) evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of Ω(1/ε). To our knowledge, this is the first optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Hypercontractivity Lemma from Fourier analysis.
- Learning Gibbs states. We design an algorithm for learning Gibbs states of local Hamiltonians in trace norm that is sample-efficient in all relevant parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state), and thus inevitably suffered from exponential sample complexity scaling in the inverse temperature.
- Certification of Gibbs states. We give an algorithm for certifying Gibbs states of local Hamiltonians in trace norm that is both sample and time-efficient in all relevant parameters, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022).