  The training of a parameterized model largely depends on the landscape of the
underlying loss function. In particular, vanishing gradients (also known as
barren plateaus) are a central bottleneck in the scalability of variational
quantum algorithms (VQAs), and are known to arise in various ways, from circuit
depth and hardware noise to global observables. However, a caveat of most
existing gradient bound results is the requirement of t-design circuit
assumptions that are typically not satisfied in practice. In this work, we
loosen these assumptions altogether and derive tight upper and lower bounds on
loss and gradient concentration for a large class of parameterized quantum
circuits and arbitrary observables, which are significantly stronger than prior
work. Moreover, we show that these bounds, as well as the variance of the loss
itself, can be estimated efficiently and classically--providing practical tools
to study the loss landscapes of VQA models, including verifying whether or not
a circuit/observable induces barren plateaus. This insight has direct
implications for hybrid Quantum Generative Adversarial Networks (qGANs), a
generative model that can be reformulated as a VQA with an observable composed
of local and global terms. We prove that designing the discriminator
appropriately leads to 1-local weights that stay constant in the number of
qubits, regardless of discriminator depth. Combined with our first
contribution, this implies that qGANs with shallow generators can be trained at
scale without suffering from barren plateaus, making them a promising candidate
for applications in generative quantum machine learning. We demonstrate this
result by training a qGAN to learn a 2D mixture of Gaussian distributions with
up to 16 qubits, and provide numerical evidence that global contributions to
the gradient may kick in substantially over the course of training.
