Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1807.04564 · Physical Review Letters

Learning a Local Hamiltonian from Local Measurements

Eyal Bairey, Itai Arad, Netanel H. Lindner

Published Jan 18, 2019• 167 citations
arXivHTMLPDF

Abstract

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.

DOI
10.1103/physrevlett.122.020504

Related papers

Learning the dynamics of open quantum systems from their steady states

Eyal Bairey, Chu Guo, Dario Poletti +2 more

Abstract Recent works have shown that generic local Hamiltonians can be efficiently inferred from local measurements performed on their eigenstates or thermal states. Realistic quantum systems are often affected by dissipation and decoherence due to coupling to an external environment. This raises the question whether the steady states of such open quantum systems contain sufficient information allowing for full and efficient reconstruction of the system’s dynamics. We find that such a reconstruction is possible for generic local Markovian dynamics. We propose a recovery method that uses only local measurements; for systems with finite-range interactions, the method recovers the Lindbladian acting on each spatial domain using only observables within that domain. We numerically study the accuracy of the reconstruction as a function of the number of measurements, type of open-system dynamics and system size. Interestingly, we show that couplings to external environments can in fact facilitate the reconstruction of Hamiltonians composed of commuting terms.

arXiv:1907.11154•New Journal of Physics•Feb 7, 2020•54 citations

Determining a local Hamiltonian from a single eigenstate

Xiao-Liang Qi, Daniel Ranard

We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the `` k -correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.

arXiv:1712.01850•Quantum•Jul 8, 2019•157 citations

Optimized Hamiltonian learning from short-time measurements

Assaf Zubida, Elad Yitzhaki, Netanel H. Lindner +1 more

Characterizing noisy quantum devices requires methods for learning the underlying quantum Hamiltonian which governs their dynamics. Often, such methods compare measurements to simulations of candidate Hamiltonians, a task which requires exponential computational complexity. Here, we analyze and optimize a method which circumvents this difficulty using measurements of short time dynamics. We provide estimates for the optimal measurement schedule and reconstruction error. We demonstrate that the reconstruction requires a system-size independent number of experimental shots, and characterize an informationally-complete set of state preparations and measurements for learning local Hamiltonians. Finally, we show how grouping of commuting observables and use of Hamiltonian symmetries can improve the Hamiltonian reconstruction.

•Aug 19, 2021•0 citations

Learning Hamiltonians at Long Times

Constantin Cedillo Vayson de Pradenne, Jordan Cotler, Hsin-Yuan Huang

We study the problem of learning an unknown $n$-qubit Hamiltonian $H$ from $U = e^{-iHt}$ for a single time $t$, where $t$ may be arbitrarily large. For broad families of local Hamiltonians, we prove that, with high probability over $H$ and $t$, any sum of local observables $A$ that is normalized and orthogonal to $H$ satisfies $\tfrac{1}{2^n}\|[U(t),A]\|_F^2 \geq 1/\text{poly}(n)$. The Hamiltonian is therefore the unique approximately conserved local observable, and we can efficiently recover $H$, up to scale, as the approximate null vector of a data matrix built from random product-state inputs and classical shadows. As a corollary, we obtain a weak equilibration statement: the infinite-temperature autocorrelation of every sum of local observables orthogonal to $H$ decays by at least an inverse-polynomial amount.

quant-pharXiv:2606.05690•Jun 5, 2026•0 citations

Determining system Hamiltonian from eigenstate measurements without correlation functions

Shi‐Yao Hou, Ningping Cao, Sirui Lu +3 more

Abstract Local Hamiltonians arise naturally in physical systems. Despite their seemingly ‘simple’ local structures, exotic features such as non-local correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate ψ require information of nonlocal correlation functions. In this work, we argue that local measurements on ψ is enough to recover the Hamiltonian in most of the cases. Specially, we develop an algorithm to demonstrate the observation. Our algorithm is tested numerically for randomly generated local Hamiltonians of different system sizes and returns promising reconstructions with desired accuracy. Additionally, for random generated Hamiltonians (not necessarily local), our algorithm also provides precise estimations.

arXiv:1903.06569•New Journal of Physics•Jul 30, 2020•17 citations

Optimal short-time measurements for Hamiltonian learning

Assaf Zubida, Elad Yitzhaki, Netanel H. Lindner +1 more

Characterizing noisy quantum devices requires methods for learning the underlying quantum Hamiltonian which governs their dynamics. Often, such methods compare measurements to simulations of candidate Hamiltonians, a task which requires exponential computational complexity. Here, we propose efficient measurement schemes based on short-time dynamics which circumvent this exponential difficulty. We provide estimates for the optimal measurement schedule and reconstruction error, and verify these estimates numerically. We demonstrate that the reconstruction requires a system-size independent number of experimental shots, and identify a minimal set of state preparations and measurements which yields optimal accuracy for learning short-ranged Hamiltonians. Finally, we show how grouping of commuting observables and use of Hamiltonian symmetries improve the accuracy of the Hamiltonian reconstruction.

arXiv:2108.08824•Aug 19, 2021•10 citations
Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.
Abstract Recent works have shown that generic local Hamiltonians can be efficiently inferred from local measurements performed on their eigenstates or thermal states. Realistic quantum systems are often affected by dissipation and decoherence due to coupling to an external environment. This raises the question whether the steady states of such open quantum systems contain sufficient information allowing for full and efficient reconstruction of the system’s dynamics. We find that such a reconstruction is possible for generic local Markovian dynamics. We propose a recovery method that uses only local measurements; for systems with finite-range interactions, the method recovers the Lindbladian acting on each spatial domain using only observables within that domain. We numerically study the accuracy of the reconstruction as a function of the number of measurements, type of open-system dynamics and system size. Interestingly, we show that couplings to external environments can in fact facilitate the reconstruction of Hamiltonians composed of commuting terms.
We ask whether the knowledge of a single eigenstate of a local Hamiltonian is sufficient to uniquely determine the Hamiltonian. We present evidence that the answer is ``yes" for generic local Hamiltonians, given either the ground state or an excited eigenstate. In fact, knowing only the two-point equal-time correlation functions of local observables with respect to the eigenstate should generically be sufficient to exactly recover the Hamiltonian for finite-size systems, with numerical algorithms that run in a time that is polynomial in the system size. We also investigate the large-system limit, the sensitivity of the reconstruction to error, and the case when correlation functions are only known for observables on a fixed sub-region. Numerical demonstrations support the results for finite one-dimensional spin chains (though caution must be taken when extrapolating to infinite-size systems in higher dimensions). For the purpose of our analysis, we define the `` k -correlation spectrum" of a state, which reveals properties of local correlations in the state and may be of independent interest.
Characterizing noisy quantum devices requires methods for learning the underlying quantum Hamiltonian which governs their dynamics. Often, such methods compare measurements to simulations of candidate Hamiltonians, a task which requires exponential computational complexity. Here, we analyze and optimize a method which circumvents this difficulty using measurements of short time dynamics. We provide estimates for the optimal measurement schedule and reconstruction error. We demonstrate that the reconstruction requires a system-size independent number of experimental shots, and characterize an informationally-complete set of state preparations and measurements for learning local Hamiltonians. Finally, we show how grouping of commuting observables and use of Hamiltonian symmetries can improve the Hamiltonian reconstruction.
We study the problem of learning an unknown nnn-qubit Hamiltonian HHH from U=e−iHtU = e^{-iHt}U=e−iHt for a single time ttt, where ttt may be arbitrarily large. For broad families of local Hamiltonians, we prove that, with high probability over HHH and ttt, any sum of local observables AAA that is normalized and orthogonal to HHH satisfies 12n∥[U(t),A]∥F2≥1/poly(n)\tfrac{1}{2^n}\|[U(t),A]\|_F^2 \geq 1/\text{poly}(n)2n1​∥[U(t),A]∥F2​≥1/poly(n). The Hamiltonian is therefore the unique approximately conserved local observable, and we can efficiently recover HHH, up to scale, as the approximate null vector of a data matrix built from random product-state inputs and classical shadows. As a corollary, we obtain a weak equilibration statement: the infinite-temperature autocorrelation of every sum of local observables orthogonal to HHH decays by at least an inverse-polynomial amount.
Abstract Local Hamiltonians arise naturally in physical systems. Despite their seemingly ‘simple’ local structures, exotic features such as non-local correlations and topological orders exhibit in eigenstates of these systems. Previous studies for recovering local Hamiltonians from measurements on an eigenstate ψ require information of nonlocal correlation functions. In this work, we argue that local measurements on ψ is enough to recover the Hamiltonian in most of the cases. Specially, we develop an algorithm to demonstrate the observation. Our algorithm is tested numerically for randomly generated local Hamiltonians of different system sizes and returns promising reconstructions with desired accuracy. Additionally, for random generated Hamiltonians (not necessarily local), our algorithm also provides precise estimations.
Characterizing noisy quantum devices requires methods for learning the underlying quantum Hamiltonian which governs their dynamics. Often, such methods compare measurements to simulations of candidate Hamiltonians, a task which requires exponential computational complexity. Here, we propose efficient measurement schemes based on short-time dynamics which circumvent this exponential difficulty. We provide estimates for the optimal measurement schedule and reconstruction error, and verify these estimates numerically. We demonstrate that the reconstruction requires a system-size independent number of experimental shots, and identify a minimal set of state preparations and measurements which yields optimal accuracy for learning short-ranged Hamiltonians. Finally, we show how grouping of commuting observables and use of Hamiltonian symmetries improve the accuracy of the Hamiltonian reconstruction.