Samuel Scalet

We introduce the notion of polynomial-depth duality transformations, which relates two sets of operator algebras through a conjugation by a poly-depth quantum circuit, and make use of this to construct efficient Gibbs samplers for a variety of interesting quantum Hamiltonians as they are poly-depth dual to classical Hamiltonians. This is for example the case for the 2D toric code, which is demonstrated to be poly-depth dual to two decoupled classical Ising spin chains for any system size, and we give evidence that such dualities hold for a wide class of stabilizer Hamiltonians. Additionally, we extend the above notion of duality to Lindbladians in order to show that mixing times and other quantities such as the spectral gap or the modified logarithmic Sobolev inequality are preserved under duality.

We show that spin chains in thermal equilibrium have a correlation structure in which individual regions are strongly correlated at most with their near vicinity. We quantify this with alternative notions of the conditional mutual information defined through the so-called Belavkin-Staszewski relative entropy. We prove that these measures decay super-exponentially, under the assumption that the spin chain Hamiltonian is translation-invariant. Using a recovery map associated with these measures, we sequentially construct tensor network approximations in terms of marginals of small (sub-logarithmic) size. As a main application, we show that classical representations of the states can be learned efficiently from local measurements with a polynomial sample complexity. We also prove an approximate factorization condition for the purity of the entire Gibbs state, which implies that it can be efficiently estimated to a small multiplicative error from a small number of local measurements. As a technical step of independent interest, we show an upper bound to the decay of the Belavkin-Staszewski relative entropy upon the application of a conditional expectation.