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arXiv:2212.06380

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

Pengyuan Zhai

Published Dec 13, 2022• 0 citations
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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.

DOI
10.48550/arxiv.2212.06380

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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.
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Quantum machine learning has recently attracted much attention from the community of quantum computing. In this paper, we explore the ability of generative adversarial networks (GANs) based on quantum computing. More specifically, we propose a quantum GAN for generating classical discrete distribution, which has a classical-quantum hybrid architecture and is composed of a parameterized quantum circuit as the generator and a classical neural network as the discriminator. The parameterized quantum circuit only consists of simple one-qubit rotation gates and two-qubit controlled-phase gates that are available in current quantum devices. Our scheme has the following characteristics and potential advantages: (i) It is intrinsically capable of generating discrete data (e.g., text data), while classical GANs are clumsy for this task due to the vanishing gradient problem. (ii) Our scheme avoids the input/output bottlenecks embarrassing most of the existing quantum learning algorithms that either require to encode the classical input data into quantum states, or output a quantum state corresponding to the solution instead of giving the solution itself, which inevitably compromises the speedup of the quantum algorithm. (iii) The probability distribution implicitly given by data samples can be loaded into a quantum state, which may be useful for some further applications.
Quantum machine learning has recently attracted much attention from the community of quantum computing. In this paper, we explore the ability of generative adversarial networks (GANs) based on quantum computing. More specifically, we propose a quantum GAN for generating classical discrete distribution, which has a classical-quantum hybrid architecture and is composed of a parameterized quantum circuit as the generator and a classical neural network as the discriminator. The parameterized quantum circuit only consists of simple one-qubit rotation gates and two-qubit controlled-phase gates that are available in current quantum devices. Our scheme has the following characteristics and potential advantages: (i) It is intrinsically capable of generating discrete data (e.g., text data), while classical GANs are clumsy for this task due to the vanishing gradient problem. (ii) Our scheme avoids the input/output bottlenecks embarrassing most of the existing quantum learning algorithms that either require to encode the classical input data into quantum states, or output a quantum state corresponding to the solution instead of giving the solution itself, which inevitably compromises the speedup of the quantum algorithm. (iii) The probability distribution implicitly given by data samples can be loaded into a quantum state, which may be useful for some further applications.
Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.
Quantum mechanics is inherently probabilistic in light of Born's rule. Using quantum circuits as probabilistic generative models for classical data exploits their superior expressibility and efficient direct sampling ability. However, training of quantum circuits can be more challenging compared to classical neural networks due to the lack of an efficient differentiable learning algorithm. We devise an adversarial quantum-classical hybrid training scheme via coupling a quantum circuit generator and a classical neural network discriminator together. After training, the quantum circuit generative model can infer missing data with quadratic speed-up via amplitude amplification. We numerically simulate the learning and inference of generative adversarial quantum circuits using the prototypical bars-and-stripes dataset. Generative adversarial quantum circuits are a fresh approach to machine learning which may enjoy the practically useful quantum advantage of near-term quantum devices.
Quantum generative modeling (QGM) relies on preparing quantum states and generating samples from these states as hidden---or known---probability distributions. As distributions from some classes of quantum states (circuits) are inherently hard to sample classically, QGM represents an excellent test bed for quantum supremacy experiments. Furthermore, generative tasks are increasingly relevant for industrial machine learning applications, and thus QGM is a strong candidate for demonstrating a practical quantum advantage. However, this requires that quantum circuits are trained to represent industrially relevant distributions, and the corresponding training stage has an extensive training cost for current quantum hardware in practice. In this work, we propose protocols for classical training of QGMs based on circuits of the specific type that admit an efficient gradient computation, while remaining hard to sample. In particular, we consider instantaneous quantum polynomial (IQP) circuits and their extensions. Showing their classical simulability in terms of the time complexity, sparsity, and anticoncentration properties, we develop a classically tractable way of simulating their output probability distributions, allowing classical training to a target probability distribution. The corresponding quantum sampling from IQPs can be performed efficiently, unlike when using classical sampling. We numerically demonstrate the end-to-end training of IQP circuits using probability distributions for up to 30 qubits on a regular desktop computer. When applied to industrially relevant distributions this combination of classical training with quantum sampling represents an avenue for reaching advantage in the noisy intermediate-scale quantum (NISQ) era.