Efficient learning of ground and thermal states within phases of matter

• Published in: Nature communications Vol. 15; no. 1; pp. 7755 - 8
• Main Authors: Rouzé, Cambyse, Stilck França, Daniel, Onorati, Emilio, Watson, James D.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 05.09.2024; Nature Publishing Group; Nature Portfolio
• Subjects: 639/705/1041; 639/766/483/481; Algorithms; Commuting; Humanities and Social Sciences; Learning; multidisciplinary; Observational learning; Parameterization; Science; Science (multidisciplinary); Thermodynamic properties; Tomography
• ISSN: 2041-1723; 2041-1723
• DOI: 10.1038/s41467-024-51439-x

We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we present sample-efficient ways to learn these properties to high precision. For the first task, we develop techniques to learn parameterisations of classes of systems, including quantum Gibbs states for classes of non-commuting Hamiltonians. We then give methods to sample-efficiently infer expectation values of extensive properties of the state, including quasi-local observables and entropies. For the second task, we exploit the locality of Hamiltonians to show that M local observables can be learned with probability 1 −  δ and precision ε using N = O log M δ e polylog ( ε − 1 ) samples — exponentially improving previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponentially decaying correlations. Scalable characterisation of quantum states can be achieved by leveraging on simplifications valid for specific classes of states. Here, the authors show how to combine the strengths of shadows and many-body tomography for all states exhibiting exponential decay of correlations and the approximate Markov property.