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arXiv:2305.01592

Learning hard distributions with quantum-enhanced Variational Autoencoders

Anantha Rao, Dhiraj Madan, Anupama Ray, Dhinakaran Vinayagamurthy, M. S. Santhanam

Published May 2, 2023• 0 citations
arXivHTMLPDF

Abstract

An important task in quantum generative machine learning is to model the probability distribution of measurements of many-body quantum systems. Classical generative models, such as generative adversarial networks (GANs) and variational autoencoders (VAEs), can model the distributions of product states with high fidelity, but fail or require an exponential number of parameters to model entangled states. In this paper, we introduce a quantum-enhanced VAE (QeVAE), a generative quantum-classical hybrid model that uses quantum correlations to improve the fidelity over classical VAEs, while requiring only a linear number of parameters. We provide a closed-form expression for the output distributions of the QeVAE. We also empirically show that the QeVAE outperforms classical models on several classes of quantum states, such as 4-qubit and 8-qubit quantum circuit states, haar random states, and quantum kicked rotor states, with a more than 2x increase in fidelity for some states. Finally, we find that the trained model outperforms the classical model when executed on the IBMq Manila quantum computer. Our work paves the way for new applications of quantum generative learning algorithms and characterizing measurement distributions of high-dimensional quantum states.

DOI
10.48550/arxiv.2305.01592

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An important task in quantum generative machine learning is to model the probability distribution of measurements of many-body quantum systems. Classical generative models, such as generative adversarial networks (GANs) and variational autoencoders (VAEs), can model the distributions of product states with high fidelity, but fail or require an exponential number of parameters to model entangled states. In this paper, we introduce a quantum-enhanced VAE (QeVAE), a generative quantum-classical hybrid model that uses quantum correlations to improve the fidelity over classical VAEs, while requiring only a linear number of parameters. We provide a closed-form expression for the output distributions of the QeVAE. We also empirically show that the QeVAE outperforms classical models on several classes of quantum states, such as 4-qubit and 8-qubit quantum circuit states, haar random states, and quantum kicked rotor states, with a more than 2x increase in fidelity for some states. Finally, we find that the trained model outperforms the classical model when executed on the IBMq Manila quantum computer. Our work paves the way for new applications of quantum generative learning algorithms and characterizing measurement distributions of high-dimensional quantum states.
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Variational Autoencoders (VAEs) are fundamental for generative modeling and image reconstruction, yet their performance often struggles to maintain high fidelity in reconstructions. This study introduces a hybrid model, Quantum Variational Autoencoder (Q-VAE), which integrates quantum encoding within the encoder while utilizing fully connected layers to extract meaningful representations. The decoder uses transposed convolution layers for up-sampling. The Q-VAE is evaluated against the classical VAE and the classical direct-passing VAE, which utilizes windowed pooling filters. Results on the MNIST and USPS datasets demonstrate that Q-VAE consistently outperforms classical approaches, achieving lower Fréchet Inception Distance scores, thereby indicating superior image fidelity and enhanced reconstruction quality. These findings highlight the potential of Q-VAE for high-quality synthetic data generation and improved image reconstruction in generative models.
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The exact description of many-body quantum systems represents one of the major challenges in modern physics, because it requires an amount of computational resources that scales exponentially with the size of the system. Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a practically usable deep architecture for representing and sampling from probability distributions of quantum states. Our representation is based on variational auto-encoders, a type of generative model in the form of a neural network. We show that this model is able to learn efficient representations of states that are easy to simulate classically and can compress states that are not classically tractable. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterizing states of the size expected in first generation quantum hardware.
Neural topic models enable scalable semantic discovery, but their integration with quantum hardware remains largely unexplored. We present a proof-of-concept hybrid classical-quantum variational autoencoder (VAE) for topic modeling, embedding parameterized quantum circuits within the VAE inference network while retaining a classical topic-word decoder. To address the resource constraints of quantum hardware, we propose a modified Gaussian Softmax posterior that decouples latent space dimensionality from the number of topics to be extracted, enabling the model to operate with a low-resource 10-qubit quantum device. On the AgNews dataset, the hybrid VAE outperforms state-of-the-art neural topic models (NTMs), reaching a CvC_vCv​ coherence score of 0.71 and an NPMI score of 0.20 while preserving high topic diversity. For comparison, we also construct a fully classical variant, which also outperforms state-of-the-art models on AgNews and exhibits clear class separation in the latent space. These results demonstrate that hybrid VAEs are computationally viable even on NISQ-era devices and represent a promising direction for quantum-enhanced topic modeling.
Modeling joint probability distributions is an important task in a wide variety of fields. One popular technique for this employs a family of multivariate distributions with uniform marginals called copulas. While the theory of modeling joint distributions via copulas is well understood, it gets practically challenging to accurately model real data with many variables. In this paper, we show that any copula can be naturally mapped to a multipartite maximally entangled state. Thus, the task of learning joint probability distributions becomes the task of learning maximally entangled states. We prove that a variational ansatz we christen as a ``qopula'' based on this insight leads to an exponential advantage over classical methods of learning some joint distributions. As an application, we train a quantum generative adversarial network (QGAN) and a quantum circuit Born machine (QCBM) using this variational ansatz to generate samples from joint distributions of two variables in historical data from the stock market. We demonstrate our generative learning algorithms on trapped ion quantum computers from IonQ for up to eight qubits. Our experimental results show interesting findings such as the resilience against noise, outperformance against equivalent classical models and 20--1000 times less iterations required to converge as compared to equivalent classical models.