Introducing nonlinear activations into quantum generative models

Abstract

Due to the linearity of quantum mechanics, it remains a challenge to design quantum generative machine learning models that embed nonlinear activations into the evolution of the state vector. However, some of the most successful classical generative models, such as those based on neural networks, involve highly nonlinear dynamics for quality training. In this paper we explore the effect of these dynamics in quantum generative modeling by introducing a model that adds nonlinear activations via a neural network structure onto the standard Born machine framework---the quantum neuron Born machine (QNBM). To achieve this, we utilize a previously introduced quantum neuron subroutine, which is a repeat-until-success circuit with midcircuit measurements and classical control. After introducing the QNBM, we investigate how its performance depends on network size, by training a three-layer QNBM with four output neurons and various input and hidden layer sizes. We then compare our nonlinear QNBM to the linear quantum circuit Born machine (QCBM). We allocate similar time and memory resources to each model such that the only major difference is the qubit overhead required by the QNBM. With gradient-based training, we show that while both models can easily learn a trivial uniform probability distribution, on a more challenging class of distributions, the QNBM achieves an almost three-times-smaller error rate than a QCBM with a similar number of tunable parameters. We therefore provide evidence that suggests that nonlinearity is a useful resource in quantum generative models, and we put forth the QNBM as a model with good generative performance and potential for quantum advantage.