Characterizing Trainability of Instantaneous Quantum Polynomial Circuit Born Machine

Instantaneous Quantum Polynomial Quantum Circuit Born Machines (IQP-QCBMs) have been proposed as quantum generative models that combine a classically tractable training objective—based on the maximum mean discrepancy (MMD)—with a potential quantum advantage motivated by sampling-complexity arguments. While recent works have explored this model across various application domains, fundamental questions remain: does the model suffer from exponentially vanishing loss gradients, known as the barren plateau problem—a pervasive obstacle in quantum machine learning—and how do regimes of trainability relate to regimes of possible quantum advantage? Here, we address both questions analytically. To study trainability, we derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and establish general upper and lower bounds. We explicitly characterize how trainability depends on the generator set and the spectrum of the chosen kernel, identifying regimes in which low-weight kernels avoid exponential gradient suppression under structured topologies. Regarding potential quantum advantage, we reformulate the anti-concentration property in terms of the same generator-set quantities that govern trainability, enabling a unified analysis. We show that sparse IQP architectures can produce classically intractable output distributions while simultaneously remaining trainable, at least at lower-weight frequencies. Our analytical results corroborate the numerical observations of prior work and provides principled guidelines for designing scalable IQP-QCBM architectures.