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arXiv:2610.01074

The stationarity test: a framework for learning quantum many-body systems from their thermal states

Thiago Bergamaschi

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

The task of learning the Hamiltonian interactions governing a quantum system, given samples of its thermal (or 'Gibbs') states, is a foundational question at the intersection of quantum learning theory and many-body physics. In this paper, we draw connections to the quantum Gibbs sampling literature to introduce a natural learning algorithm we call the stationarity test: which simply "guesses" the Hamiltonian, and measures the rate-of-change of local observables, under the associated detailed-balanced quantum Markov chain [CKG23]. We leverage the stationarity test to address the following applications: 1. To give the first learning algorithm for the underlying interaction graph, i.e. structure learning, of lattice Hamiltonians at all temperatures, given copies of their Gibbs states. 2. To give the first learning algorithm for the coefficients of a lattice Hamiltonian, given only copies of its thermal metastable states, modeled as the "local minima" of the free energy. 3. In addition, we present a refinement to the recent area law for thermal metastable states [BCV25], which holds in the thermodynamic limit. Our learning algorithms are rigorous, time-efficient, and nearly sample-optimal in system size and accuracy. At a technical level, our arguments are based on new approximate locality and convexity properties for the quantum Fisher information of these Gibbs sampling algorithms.

Comments
62 pages, 4 figures

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The task of learning the Hamiltonian interactions governing a quantum system, given samples of its thermal (or 'Gibbs') states, is a foundational question at the intersection of quantum learning theory and many-body physics. In this paper, we draw connections to the quantum Gibbs sampling literature to introduce a natural learning algorithm we call the stationarity test: which simply "guesses" the Hamiltonian, and measures the rate-of-change of local observables, under the associated detailed-balanced quantum Markov chain [CKG23]. We leverage the stationarity test to address the following applications:
  1. To give the first learning algorithm for the underlying interaction graph, i.e. structure learning, of lattice Hamiltonians at all temperatures, given copies of their Gibbs states.
  2. To give the first learning algorithm for the coefficients of a lattice Hamiltonian, given only copies of its thermal metastable states, modeled as the "local minima" of the free energy.
  3. In addition, we present a refinement to the recent area law for thermal metastable states [BCV25], which holds in the thermodynamic limit.
Our learning algorithms are rigorous, time-efficient, and nearly sample-optimal in system size and accuracy. At a technical level, our arguments are based on new approximate locality and convexity properties for the quantum Fisher information of these Gibbs sampling algorithms.
Learning quantum interactions from finite-temperature many-body systems is a central task in emerging quantum platforms. Recently, the problem of learning from lattice quantum Gibbs states has found rigorous, efficient protocols. Nevertheless, exact Gibbs states, as the input premise, are in fact computationally intractable to prepare and may not faithfully represent generic finite-temperature quantum systems. In contrast, a system coupled to a heat bath can be stuck at an approximate stationary state (metastable state) long before it truly equilibrates. Here, we formulate a physically and algorithmically consistent alternative: learning from such metastable states of detailed-balanced master equations (Lindbladians) arising from system-bath interactions. We distill the algorithmic mechanism and structural condition underlying Gibbs-state learning and extend it in full to metastable states, attaining nearly optimal sample and computational complexity (in the system size and the precision). More broadly, we sharpen notions of metastability and develop a unified framework for finite-temperature learning.
We study the problem of learning the Hamiltonian of a quantum many-body system given samples from its Gibbs (thermal) state. The classical analog of this problem, known as learning graphical models or Boltzmann machines, is a well-studied question in machine learning and statistics. In this work, we give the first sample-efficient algorithm for the quantum Hamiltonian learning problem. In particular, we prove that polynomially many samples in the number of particles (qudits) are necessary and sufficient for learning the parameters of a spatially local Hamiltonian in l_2-norm. Our main contribution is in establishing the strong convexity of the log-partition function of quantum many-body systems, which along with the maximum entropy estimation yields our sample-efficient algorithm. Classically, the strong convexity for partition functions follows from the Markov property of Gibbs distributions. This is, however, known to be violated in its exact form in the quantum case. We introduce several new ideas to obtain an unconditional result that avoids relying on the Markov property of quantum systems, at the cost of a slightly weaker bound. In particular, we prove a lower bound on the variance of quasi-local operators with respect to the Gibbs state, which might be of independent interest. Our work paves the way toward a more rigorous application of machine learning techniques to quantum many-body problems.
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(log⁡n)\mathcal{O}(\log n)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.
In this work, we study the problems of certifying and learning quantum kkk-local Hamiltonians, for a constant kkk. 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)O(1/\varepsilon)O(1/ε) evolution time. This is optimal, as it matches the Heisenberg-scaling lower bound of Ω(1/ε)Ω(1/\varepsilon)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).
Statistical mechanics assumes that a quantum many-body system at low temperature can be effectively described by its Gibbs state. However, many complex quantum systems exist only as metastable states of dissipative open system dynamics, which appear stable and robust yet deviate substantially from true thermal equilibrium. In this work, we model metastable states as approximate stationary states of a quasi-local, (KMS)-detailed-balanced master equation representing Markovian system-bath interaction, and unveil a universal structural theory: all metastable states satisfy an area law of mutual information and a Markov property. The more metastable the states are, the larger the regions to which these structural results apply. Therefore, the hallmark correlation structure and noise resilience of Gibbs states are not exclusive to true equilibrium but emerge dynamically. Behind our structural results lies a systematic framework encompassing sharp equivalences between local minima of free energy, a non-commutative Fisher information, and approximate detailed balance conditions. Our results build towards a comprehensive theory of thermal metastability and, in turn, formulate a well-defined, feasible, and repeatable target for quantum thermal simulation.
Understanding and harnessing the power of quantum systems has the potential to transform many domains in science and technology. However, before we can achieve these aspirations, we must first build a better understanding of how quantum systems fundamentally behave. In this thesis, we approach this question through the lens of learning theory to develop new paradigms for learning about quantum systems and understanding their structural properties. We deliver several surprising results, upending previous beliefs about even fundamental laws and giving provably efficient algorithms for learning about quantum systems in settings previously conjectured to be intractable. Typically in quantum many-body systems, the particles in the system interact locally with respect to some geometry as described by a local Hamiltonian. Two key questions are first, understanding equilibrium properties of a system with a given Hamiltonian and second, recovering the Hamiltonian from measurements of the properties of the system. For the first, we prove a universal law that there is a sudden death of entanglement, at a critical temperature depending only on the geometry but not on the system size. For the second, we give the first efficient algorithm for recovering the Hamiltonian at any temperature, breaking a conjectured barrier at low temperatures. Beyond systems with local interactions, we also consider learning and testing properties of general quantum states, focusing on the interplay between statistical complexity and near-term quantum device constraints, only allowing for entangled measurements over a limited number of copies of the state. We characterize the optimal rates for learning and testing with single-copy measurements and for multi-copy measurements in many relevant near-term regimes.