🤖 AI Summary
This work investigates the trainability of Instantaneous Quantum Polynomial (IQP) quantum circuit Born machines (QCBMs) under Gaussian initialization, with a focus on vanishing gradients and barren plateaus. By integrating Stein’s lemma with Lipschitz concentration inequalities for Gaussian random variables, the study provides the first rigorous analysis of how arbitrary Gaussian initializations affect gradient variance and concentration. The authors derive an analytical lower bound on gradient variance and a probabilistic concentration bound quantifying the deviation of gradients from their mean. These results precisely characterize the conditions under which barren plateaus emerge, offering theoretical foundations and practical guidance for designing trainable IQP QCBMs through informed initialization strategies.
📝 Abstract
Quantum Circuit Born Machines (QCBMs) offer a natural approach to generative machine learning by leveraging the Born rule. Recent work has provided a method to classically train QCBMs with Instantaneous Quantum Polynomial (IQP) circuits via the Maximum Mean Discrepancy (MMD) loss. Despite the assumed intractability of sampling from IQP circuits classically, their expectation values can be computed classically, enabling training of these IQP QCBMs. However, quantum machine learning (QML) models have various other challenges, including trainability issues caused by exponential concentration or barren plateaus. While these issues have been explored for parameters sampled from a uniform distribution, little work has been done to rigorously treat the use of arbitrary Gaussian initialization schemes. This work leverages Stein's lemma and Lipschitz concentration bounds for Gaussian random variables to provide an analytical lower bound of the variance of the gradient and a probabilistic concentration bound of the deviation of the gradient from its mean. It discusses strategies to either avoid or encourage exponential concentration, as well as the conditions under which barren plateaus are more likely to occur.