Characterizing Trainability of Instantaneous Quantum Polynomial Circuit Born Machines

TL;DR

This paper investigates the trainability of IQP-based quantum generative models, deriving analytical expressions for gradient variances, identifying regimes avoiding barren plateaus, and discussing conditions for potential quantum advantage.

Contribution

It provides analytical insights into the trainability of IQP-QCBMs, including bounds on gradient variances and conditions for avoiding barren plateaus, advancing understanding of their practical use.

Findings

Barren plateaus depend on generator set and kernel spectrum.

Low-weight-biased kernels can avoid exponential gradient suppression.

Small-variance Gaussian initialization ensures polynomial gradient scaling.

Abstract

Instantaneous quantum polynomial quantum circuit Born machines (IQP-QCBMs) have been proposed as quantum generative models with a classically tractable training objective based on the maximum mean discrepancy (MMD) and a potential quantum advantage motivated by sampling-complexity arguments, making them an exciting model worth deeper investigation. While recent works have further proven the universality of a (slightly generalized) model, the next immediate question pertains to its trainability, i.e., whether it suffers from the exponentially vanishing loss gradients, known as the barren plateau issue, preventing effective use, and how regimes of trainability overlap with regimes of possible quantum advantage. Here, we provide significant strides in these directions. To study the trainability at initialization, we analytically derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and provide general upper and lower bounds. With uniform initialization, we show that barren plateaus depend on the generator set and the spectrum of the chosen kernel. We identify regimes in which low-weight-biased kernels avoid exponential gradient suppression in structured topologies. Also, we prove that a small-variance Gaussian initialization ensures polynomial scaling for the gradient under mild conditions. As for the potential quantum advantage, we further argue, based on previous complexity-theoretic arguments, that sparse IQP families can output a probability distribution family that is classically intractable, and that this distribution remains trainable at initialization at least at lower-weight frequencies.