Are Quantum Circuits Better than Neural Networks at Learning Multi-dimensional Discrete Data? An Investigation into Practical Quantum Circuit Generative Models

TL;DR

This paper investigates the expressive power of multi-layer parameterized quantum circuits (MPQCs) compared to classical neural networks, demonstrating their superior ability to generate complex discrete data distributions through theoretical proofs and numerical experiments.

Contribution

It provides a systematic analysis of MPQCs' expressive advantages over classical NNs, introduces a novel loss function (MCR loss), and proposes efficient training methods for quantum generative models.

Findings

MPQCs can generate distributions not efficiently simulated classically

Quantum models outperform classical GANs in discrete data generation

The proposed MCR loss improves training stability and reduces modal collapse

Abstract

Are multi-layer parameterized quantum circuits (MPQCs) more expressive than classical neural networks (NNs)? How, why, and in what aspects? In this work, we survey and develop intuitive insights into the expressive power of MPQCs in relation to classical NNs. We organize available sources into a systematic proof on why MPQCs are able to generate probability distributions that cannot be efficiently simulated classically. We first show that instantaneous quantum polynomial circuits (IQPCs), are unlikely to be simulated classically to within a multiplicative error, and then show that MPQCs efficiently generalize IQPCs. We support the surveyed claims with numerical simulations: with the MPQC as the core architecture, we build different versions of quantum generative models to learn a given multi-dimensional, multi-modal discrete data distribution, and show their superior performances over a classical Generative Adversarial Network (GAN) equipped with the Gumbel Softmax for generating discrete data. In addition, we address practical issues such as how to efficiently train a quantum circuit with only limited samples, how to efficiently calculate the (quantum) gradient, and how to alleviate modal collapse. We propose and experimentally verify an efficient training-and-fine-tuning scheme for lowering the output noise and decreasing modal collapse. As an original contribution, we develop a novel loss function (MCR loss) inspired by an information-theoretical measure -- the coding rate reduction metric, which has a more expressive and geometrically meaningful latent space representations -- beneficial for both model selection and alleviating modal collapse. We derive the gradient of our MCR loss with respect to the circuit parameters under two settings: with the radial basis function (RBF) kernel and with a NN discriminator and conduct experiments to showcase its effectiveness.