Certifying and learning quantum Ising Hamiltonians

In this work, we study the problems of certifying and learning quantum Ising Hamiltonians. Our main contributions are as follows: Certification of Ising Hamiltonians. We show that certifying an Ising Hamiltonian in normalized Frobenius norm via access to its time-evolution operator requires only $\widetilde O(1/\varepsilon)$ time evolution. This matches the Heisenberg-scaling lower bound of $\Omega(1/\varepsilon)$ up to logarithmic factors. To our knowledge, this is the first nearly-optimal algorithm for testing a Hamiltonian property. A key ingredient in our analysis is the Bonami Lemma from Fourier analysis. Learning Ising Gibbs states. We design an algorithm for learning Ising Gibbs states in trace norm that is sample-efficient in all parameters. In contrast, previous approaches learned the underlying Hamiltonian (which implies learning the Gibbs state) but suffered from exponential sample complexity in the inverse temperature. Certification of Ising Gibbs states. We give an algorithm for certifying Ising Gibbs states in trace norm that is both sample and time-efficient, thereby solving a question posed by Anshu (Harvard Data Science Review, 2022). Finally, we extend our results on learning and certification of Gibbs states to general $k$-local Hamiltonians for any constant $k$.