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:2606.05690

Learning Hamiltonians at Long Times

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

quant-phSubmitted Jun 4, 2026• Announced Jun 5, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Comments
11+54 pages, 5 figures

Related papers

Learning Quantum Hamiltonians at Any Temperature in Polynomial Time

Ainesh Bakshi, Allen Liu, Ankur Moitra +1 more

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.

ACM Symposium on Theory of Computing (STOC)•Jun 10, 2024•15 citations

Near-Optimal Learning of Local Lindbladians

Itai Arad, Zhili Chen, Naixu Guo +2 more

We study the problem of learning local Lindbladians from black-box access to the physical evolution, where the goal is to estimate all Hamiltonian and dissipative coefficients. For local Lindbladians with low-intersection dissipators and local dynamical strength at most $Λ$, we design an algorithm that learns the coefficients to target accuracy $\varepsilon$, whose number of channel uses and total evolution time scale as $\widetilde{O}(Λ^2/\varepsilon^2)$ and $\widetilde{O}(Λ/\varepsilon^2)$, respectively, with only logarithmic dependence on the number of qubits. The algorithm is built directly from finite-time channel probes. It runs the unknown evolution for short times, estimates the corresponding Pauli transfer matrices from classical shadows, and converts these estimates into Lindbladian coefficients by stable local Fourier inversions. The algorithm is non-adaptive, uses no ancillas, and requires only random product states as inputs followed by random Pauli measurements. The method does not require knowing the structure of the Lindbladian in advance. We prove matching lower bounds that establish the near-optimality of the algorithm in both resources. By constructing a family of single-qubit dephasing Lindbladians, we show that any algorithm, even an adaptive one with arbitrary ancillas and measurements, requires $Ω(Λ^2/\varepsilon^2)$ channel uses and $Ω(Λ/\varepsilon^2)$ total evolution time. In particular, the lower bounds imply that the Heisenberg-limited scaling achievable for Hamiltonian learning is information-theoretically impossible once dissipative coefficients must be estimated.

quant-pharXiv:2606.20535•Jun 19, 2026•0 citations

Learning a Local Hamiltonian from Local Measurements

Eyal Bairey, Itai Arad, Netanel H. Lindner

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.

arXiv:1807.04564•Physical Review Letters•Jan 18, 2019•167 citations

Learning SYK Hamiltonians

Anurag Anshu, Srinivasan Arunachalam, Sitan Chen +1 more

We study the problem of learning the dense Sachdev--Ye--Kitaev (SYK) Hamiltonian from copies of its Gibbs state. Existing algorithms for Hamiltonian learning typically rely on geometric locality or bounded interaction degree and therefore do not apply to SYK, where each quartic interaction overlaps with $Θ(n^3)$ others. We show that this obstruction can be overcome by exploiting the random mean-field structure of the model. At any constant temperature, we prove that with high probability over the SYK couplings, the entire Hamiltonian can be learned to inverse-polynomial accuracy using polynomially many samples. Furthermore, when the inverse temperature is restricted to be a sufficiently small constant, we construct a quasipolynomial-time learning algorithm which is qualitatively different from the sample-efficient algorithm.

quant-pharXiv:2610.02178•Oct 2, 2026•0 citations

Learning Quantum Hamiltonians at Any Temperature in Polynomial Time

Ainesh Bakshi, Allen Liu, Ankur Moitra +1 more

Abstract. We study the problem of learning a local quantum Hamiltonian [Formula: see text] given copies of its Gibbs state [Formula: see text] at a known inverse temperature [Formula: see text]. Anshu et al. [2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pp. 685–691] gave an algorithm to learn a Hamiltonian on [Formula: see text] qubits to precision [Formula: see text] 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 [ Alhambra, PRX Quantum, 4 (2023), 040201 ; Anshu and Arunachalam, Nature Rev. Phys., 6 (2023), pp. 59–69 ], with prior work only resolving this in the limited cases of high temperature [ Haah, Kothari, and Tang, Markov field on finite graphs and lattices, 1971 ] or commuting terms [ Anshu et al., Efficient learning of commuting Hamiltonians on lattices, 2021 ]. We fully resolve this problem, giving a polynomial time algorithm for learning [Formula: see text] to precision [Formula: see text] from polynomially many copies of the Gibbs state at any constant [Formula: see text]. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multivariate 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.

arXiv:2310.02243•SIAM Journal on Computing•Oct 6, 2026•0 citations

Provable learning separation for predicting time-evolution of quantum many-body systems

Rahul Bandyopadhyay, Riccardo Molteni, Jens Eisert +2 more

Given that quantum computers are naturally suited to simulate the behavior of quantum many-body systems, an immediate question arises: can one formulate physically motivated quantum machine learning (QML) tasks that exhibit learning separations? We address this problem by studying the learnability of quantum many-body dynamics from the perspective of probably approximately correct (PAC)-learning. Concretely, we devise a supervised learning problem where the training set consists of specifications of randomized stabilizer probe states, evolution times sampled uniformly from a polynomially large time interval $[0,T]$, coupled with expectation values of certain observables evaluated on the resulting time-evolved state under an unknown Hamiltonian. For this learning task, we provide an efficient quantum procedure whose training phase learns the underlying Hamiltonian from short-time training samples, and whose deployment phase combines Hamiltonian simulation with the classical shadows protocol to perform inference on a newly given data point. By contrast, the existence of $O(\mathsf{poly}(n))$-time instances ensures classical hardness: by embedding a $\mathsf{BQP}$-complete computation into the polynomially long time-dynamics of a low-intersection variant of the Feynman-Kitaev clock Hamiltonian construction, we show that, for a certain family of input distributions, no randomized classical polynomial-time algorithm can fulfill our learning condition, unless $\mathsf{BQP}\subseteq\mathsf{P/poly}$. Furthermore, we show that the classically hard instance maintains quantum learnability. We also give an interpretation of our results in learning-assisted certified quantum simulation. Taken together, our results demonstrate a rigorous learning separation for a natural ML task based on Hamiltonian evolution, while building connections between quantum learning theory, quantum simulation, and QML.

quant-pharXiv:2607.06472•Jul 8, 2026•0 citations
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.
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.
We study the problem of learning local Lindbladians from black-box access to the physical evolution, where the goal is to estimate all Hamiltonian and dissipative coefficients. For local Lindbladians with low-intersection dissipators and local dynamical strength at most ΛΛΛ, we design an algorithm that learns the coefficients to target accuracy ε\varepsilonε, whose number of channel uses and total evolution time scale as O~(Λ2/ε2)\widetilde{O}(Λ^2/\varepsilon^2)O(Λ2/ε2) and O~(Λ/ε2)\widetilde{O}(Λ/\varepsilon^2)O(Λ/ε2), respectively, with only logarithmic dependence on the number of qubits. The algorithm is built directly from finite-time channel probes. It runs the unknown evolution for short times, estimates the corresponding Pauli transfer matrices from classical shadows, and converts these estimates into Lindbladian coefficients by stable local Fourier inversions. The algorithm is non-adaptive, uses no ancillas, and requires only random product states as inputs followed by random Pauli measurements. The method does not require knowing the structure of the Lindbladian in advance. We prove matching lower bounds that establish the near-optimality of the algorithm in both resources. By constructing a family of single-qubit dephasing Lindbladians, we show that any algorithm, even an adaptive one with arbitrary ancillas and measurements, requires Ω(Λ2/ε2)Ω(Λ^2/\varepsilon^2)Ω(Λ2/ε2) channel uses and Ω(Λ/ε2)Ω(Λ/\varepsilon^2)Ω(Λ/ε2) total evolution time. In particular, the lower bounds imply that the Heisenberg-limited scaling achievable for Hamiltonian learning is information-theoretically impossible once dissipative coefficients must be estimated.
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.
We study the problem of learning the dense Sachdev--Ye--Kitaev (SYK) Hamiltonian from copies of its Gibbs state. Existing algorithms for Hamiltonian learning typically rely on geometric locality or bounded interaction degree and therefore do not apply to SYK, where each quartic interaction overlaps with Θ(n3)Θ(n^3)Θ(n3) others. We show that this obstruction can be overcome by exploiting the random mean-field structure of the model. At any constant temperature, we prove that with high probability over the SYK couplings, the entire Hamiltonian can be learned to inverse-polynomial accuracy using polynomially many samples. Furthermore, when the inverse temperature is restricted to be a sufficiently small constant, we construct a quasipolynomial-time learning algorithm which is qualitatively different from the sample-efficient algorithm.
Abstract. We study the problem of learning a local quantum Hamiltonian [Formula: see text] given copies of its Gibbs state [Formula: see text] at a known inverse temperature [Formula: see text]. Anshu et al. [2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pp. 685–691] gave an algorithm to learn a Hamiltonian on [Formula: see text] qubits to precision [Formula: see text] 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 [ Alhambra, PRX Quantum, 4 (2023), 040201 ; Anshu and Arunachalam, Nature Rev. Phys., 6 (2023), pp. 59–69 ], with prior work only resolving this in the limited cases of high temperature [ Haah, Kothari, and Tang, Markov field on finite graphs and lattices, 1971 ] or commuting terms [ Anshu et al., Efficient learning of commuting Hamiltonians on lattices, 2021 ]. We fully resolve this problem, giving a polynomial time algorithm for learning [Formula: see text] to precision [Formula: see text] from polynomially many copies of the Gibbs state at any constant [Formula: see text]. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multivariate 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.
Given that quantum computers are naturally suited to simulate the behavior of quantum many-body systems, an immediate question arises: can one formulate physically motivated quantum machine learning (QML) tasks that exhibit learning separations? We address this problem by studying the learnability of quantum many-body dynamics from the perspective of probably approximately correct (PAC)-learning. Concretely, we devise a supervised learning problem where the training set consists of specifications of randomized stabilizer probe states, evolution times sampled uniformly from a polynomially large time interval [0,T][0,T][0,T], coupled with expectation values of certain observables evaluated on the resulting time-evolved state under an unknown Hamiltonian. For this learning task, we provide an efficient quantum procedure whose training phase learns the underlying Hamiltonian from short-time training samples, and whose deployment phase combines Hamiltonian simulation with the classical shadows protocol to perform inference on a newly given data point. By contrast, the existence of O(poly(n))O(\mathsf{poly}(n))O(poly(n))-time instances ensures classical hardness: by embedding a BQP\mathsf{BQP}BQP-complete computation into the polynomially long time-dynamics of a low-intersection variant of the Feynman-Kitaev clock Hamiltonian construction, we show that, for a certain family of input distributions, no randomized classical polynomial-time algorithm can fulfill our learning condition, unless BQP⊆P/poly\mathsf{BQP}\subseteq\mathsf{P/poly}BQP⊆P/poly. Furthermore, we show that the classically hard instance maintains quantum learnability. We also give an interpretation of our results in learning-assisted certified quantum simulation. Taken together, our results demonstrate a rigorous learning separation for a natural ML task based on Hamiltonian evolution, while building connections between quantum learning theory, quantum simulation, and QML.