Efficient learning of ground and thermal states within phases of matter

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mfenced><mml:mrow><mml:mi>log</mml:mi> <mml:mfenced> <mml:mrow> <mml:mfrac><mml:mrow><mml:mi>M</mml:mi></mml:mrow> <mml:mrow><mml:mi>δ</mml:mi></mml:mrow> </mml:mfrac> </mml:mrow> </mml:mfenced> <mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow> <mml:mrow><mml:mi>polylog</mml:mi> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:msup><mml:mrow><mml:mi>ε</mml:mi></mml:mrow> <mml:mrow><mml:mo>-</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:msup> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> 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.