  Quantum machine learning through variational quantum algorithms (VQAs) has
gained substantial attention in recent years. VQAs employ parameterized quantum
circuits, which are typically optimized using gradient-based methods. However,
these methods often exhibit sub-optimal convergence performance due to their
dependence on Euclidean geometry. The quantum natural gradient descent (QNGD)
optimization method, which considers the geometry of the quantum state space
via a quantum information (Riemannian) metric tensor, provides a more effective
optimization strategy. Despite its advantages, QNGD encounters notable
challenges for learning from quantum data, including the no-cloning principle,
which prohibits the replication of quantum data, state collapse, and the
measurement postulate, which leads to the stochastic loss function. This paper
introduces the quantum natural stochastic pairwise coordinate descent (2-QNSCD)
optimization method. This method leverages the curved geometry of the quantum
state space through a novel ensemble-based quantum information metric tensor,
offering a more physically realizable optimization strategy for learning from
quantum data. To improve computational efficiency and reduce sample complexity,
we develop a highly sparse unbiased estimator of the novel metric tensor using
a quantum circuit with gate complexity $\Theta(1)$ times that of the
parameterized quantum circuit and single-shot quantum measurements. Our
approach avoids the need for multiple copies of quantum data, thus adhering to
the no-cloning principle. We provide a detailed theoretical foundation for our
optimization method, along with an exponential convergence analysis.
Additionally, we validate the utility of our method through a series of
numerical experiments.
