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arXiv:2112.14553 · Physical Review Research

Active learning of quantum system Hamiltonians yields query advantage

Arkopal Dutt, Edwin Pednault, Chai Wah Wu, Sarah Sheldon, John A. Smolin, Lev S. Bishop, Isaac L. Chuang

Published Jul 28, 2023• 16 citations
arXivHTMLPDF

Abstract

Hamiltonian learning is an important procedure in quantum system identification, calibration, and successful operation of quantum computers. Through queries to the quantum system, this procedure seeks to obtain the parameters of a given Hamiltonian model and description of noise sources. Standard techniques for Hamiltonian learning require careful design of queries and $O({\ensuremath{\epsilon}}^{\ensuremath{-}2})$ queries in achieving learning error $\ensuremath{\epsilon}$ due to the standard quantum limit. With the goal of efficiently and accurately estimating the Hamiltonian parameters within learning error $\ensuremath{\epsilon}$ through minimal queries, we introduce an active learner that is given an initial set of training examples and the ability to interactively query the quantum system to generate new training data. To ensure applicability on near-term quantum hardware, the active learner operates in batch mode as opposed to sequentially, proposing batches of queries to be made during learning. We formally specify and experimentally assess the performance of this Hamiltonian active learning (HAL) algorithm for learning the six parameters of a two-qubit cross-resonance Hamiltonian on four different superconducting IBM quantum devices. Compared with standard techniques for the same problem and a specified learning error, HAL achieves more than a $95%$ reduction in queries required, and upwards of $33%$ reduction over a sequential active learner. Moreover, with access to prior information on a subset of Hamiltonian parameters and given the ability to select queries with linearly (or exponentially) longer system interaction times during learning, HAL can exceed the standard quantum limit and achieve Heisenberg (or super-Heisenberg) limited convergence rates during learning.

DOI
10.1103/physrevresearch.5.033060

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Hamiltonian learning is an important procedure in quantum system identification, calibration, and successful operation of quantum computers. Through queries to the quantum system, this procedure seeks to obtain the parameters of a given Hamiltonian model and description of noise sources. Standard techniques for Hamiltonian learning require careful design of queries and O(\ensuremathϵ\ensuremath−2)O({\ensuremath{\epsilon}}^{\ensuremath{-}2})O(\ensuremathϵ\ensuremath−2) queries in achieving learning error \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ due to the standard quantum limit. With the goal of efficiently and accurately estimating the Hamiltonian parameters within learning error \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ through minimal queries, we introduce an active learner that is given an initial set of training examples and the ability to interactively query the quantum system to generate new training data. To ensure applicability on near-term quantum hardware, the active learner operates in batch mode as opposed to sequentially, proposing batches of queries to be made during learning. We formally specify and experimentally assess the performance of this Hamiltonian active learning (HAL) algorithm for learning the six parameters of a two-qubit cross-resonance Hamiltonian on four different superconducting IBM quantum devices. Compared with standard techniques for the same problem and a specified learning error, HAL achieves more than a 9595%95 reduction in queries required, and upwards of 3333%33 reduction over a sequential active learner. Moreover, with access to prior information on a subset of Hamiltonian parameters and given the ability to select queries with linearly (or exponentially) longer system interaction times during learning, HAL can exceed the standard quantum limit and achieve Heisenberg (or super-Heisenberg) limited convergence rates during learning.
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.
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.
Characterizing quantum systems by learning their underlying Hamiltonians is a central task in quantum information science. While recent algorithmic advances have achieved near-optimal efficiency in this task, they critically rely on accessing arbitrarily short-time dynamics. This reliance poses severe experimental challenges due to finite control bandwidth and transient pulse errors. In this work, we demonstrate that Heisenberg-limited Hamiltonian learning can be achieved without short-time control. We introduce a framework in which every query to the unknown dynamics has duration at least a prescribed minimum time TTT, and show that this restriction does not preclude Heisenberg-limited scaling. The key ingredient is a method for emulating the continuous quantum control required by iterative learning algorithms using only such lower-bounded evolution times. This reduces the learning task to sparse pure-state tomography. Notably, for logarithmically sparse Hamiltonians, our algorithm achieves the information-theoretically optimal 1/ε1/\varepsilon1/ε scaling in total evolution time for any arbitrary constant minimum evolution time TTT. For many-body (polynomially sparse) systems, we uncover a rigorous quantitative tradeoff, showing that the minimum required evolution time can be significantly relaxed from the standard limit at a polynomial cost in total evolution time. Our results affirmatively resolve a prominent open problem in the field and reveal that high-bandwidth, ultra-short pulses are not fundamentally necessary for optimal quantum learning.
Determining the Hamiltonian of a quantum system is essential for understanding its dynamics and validating its behavior. Hamiltonian learning provides a data-driven approach to reconstruct the generator of the dynamics from measurements on the evolved system. Among its applications, it is particularly important for benchmarking and characterizing quantum hardware, such as quantum computers and simulators. However, as these devices grow in size and complexity, this task becomes increasingly challenging. To address this, we propose a scalable and experimentally friendly Hamiltonian learning protocol for Hamiltonian operators made of local interactions. It leverages the quantum Zeno effect as a reshaping tool to localize the system's dynamics and then applies quantum process tomography to learn the coefficients of a local subset of the Hamiltonian acting on selected qubits. Unlike existing approaches, our method does not require complex state preparations and uses experimentally accessible, coherence-preserving operations. We derive theoretical performance guarantees and demonstrate the feasibility of our protocol both with numerical simulations and through an experimental implementation on IBM's superconducting quantum hardware, successfully learning the coefficients of a 109-qubit Hamiltonian.
Efficiently learning an unknown Hamiltonian given access to its dynamics is a problem of interest for quantum metrology, many-body physics, and machine learning. A fundamental question is whether learning can be performed at the Heisenberg limit, where the Hamiltonian evolution time scales inversely with the error $\ensuremath{\varepsilon}$ in the reconstructed parameters. The Heisenberg limit has previously been shown to be achievable for certain classes of qubit and bosonic Hamiltonians. Most recently, a Heisenberg-limited learning algorithm was proposed for a simplified class of fermionic Hubbard Hamiltonians restricted to real hopping amplitudes and zero chemical potential at all sites, along with on-site interactions. In this work, we provide an algorithm to learn a more general class of fermionic Hubbard Hamiltonians at the Heisenberg limit, allowing complex hopping amplitudes and nonzero chemical potentials in addition to the on-site interactions, thereby including several models of physical interest. The required evolution time across all experiments in our protocol is $O(1/\ensuremath{\varepsilon})$ and the number of experiments required to learn all the Hamiltonian parameters is $O[\text{polylog}(1/\ensuremath{\varepsilon})]$, independent of system size as long as each fermionic mode interacts with O(1)O(1)O(1) other modes. Unlike prior algorithms for bosonic and fermionic Hamiltonians, to obey fermionic parity superselection constraints in our more general setting, our protocol utilizes O(N)O(N)O(N) ancillary fermionic modes, where NNN is the system size. Each experiment involves preparing fermionic Gaussian states, interleaving time evolution with fermionic linear optics unitaries, and performing local occupation number measurements on the fermionic modes. The protocol is robust to a constant amount of state preparation and measurement error.
We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to applye−iHte^{-\mathrm{i}Ht}e−iHtfor an unknown local HamiltonianH=Σa=1mλaEaH=\Sigma_{a=1}^{m}\lambda_{a}E_{a}H=Σa=1m​λa​Ea​onnnnqubits, the goal is to recoverHHH. This problem is already well-understood under the assumption that the interaction terms,EaE_{a}Ea​, are given, and only the interaction strengths,λa\lambda_{a}λa​, are unknown. But how efficiently can we learn a local Hamiltonian without prior knowledge of its interaction structure? We present a new, general approach to Hamiltonian learning that not only solves the challenging structure learning variant, but also resolves other open questions in the area, all while achieving the gold standard of Heisenberg-limited scaling. In particular, our algorithm recovers the Hamiltonian toε\varepsilonεerror with total evolution timeO(log⁡(n)/ε)\mathcal{O}(\log(n)/\varepsilon)O(log(n)/ε), and has the following appealing properties: 1)It does not need to know the Hamiltonian terms; 2)It works beyond the short-range setting, extending to any HamiltonianHHHwhere the sum of terms interacting with a qubit has bounded norm; 3)It evolves according toHHHin constant timetttincrements, thus achieving constant time resolution. As an application, we can also learn Hamiltonians exhibiting power-law decay up to accuracyε\varepsilonεwith total evolution time beating the standard limit of1/ε21/\varepsilon^{2}1/ε2.