  The dual tasks of quantum Hamiltonian learning and quantum Gibbs sampling are
relevant to many important problems in physics and chemistry. In the low
temperature regime, algorithms for these tasks often suffer from
intractabilities, for example from poor sample- or time-complexity. With the
aim of addressing such intractabilities, we introduce a generalization of
quantum natural gradient descent to parameterized mixed states, as well as
provide a robust first-order approximating algorithm, Quantum-Probabilistic
Mirror Descent. We prove data sample efficiency for the dual tasks using tools
from information geometry and quantum metrology, thus generalizing the seminal
result of classical Fisher efficiency to a variational quantum algorithm for
the first time. Our approaches extend previously sample-efficient techniques to
allow for flexibility in model choice, including to spectrally-decomposed
models like Quantum Hamiltonian-Based Models, which may circumvent intractable
time complexities. Our first-order algorithm is derived using a novel quantum
generalization of the classical mirror descent duality. Both results require a
special choice of metric, namely, the Bogoliubov-Kubo-Mori metric. To test our
proposed algorithms numerically, we compare their performance to existing
baselines on the task of quantum Gibbs sampling for the transverse field Ising
model. Finally, we propose an initialization strategy leveraging geometric
locality for the modelling of sequences of states such as those arising from
quantum-stochastic processes. We demonstrate its effectiveness empirically for
both real and imaginary time evolution while defining a broader class of
potential applications.
