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Expressive power of parametrized quantum circuits

Yuxuan Du, Hsieh, M-H, Tongliang Liu, Dacheng Tao

Published Oct 29, 2018• 76 citations
OpenAlex

Abstract

Parameterized quantum circuits (PQCs) have been broadly used as a hybrid\nquantum-classical machine learning scheme to accomplish generative tasks.\nHowever, whether PQCs have better expressive power than classical generative\nneural networks, such as restricted or deep Boltzmann machines, remains an open\nissue. In this paper, we prove that PQCs with a simple structure already\noutperform any classical neural network for generative tasks, unless the\npolynomial hierarchy collapses. Our proof builds on known results from tensor\nnetworks and quantum circuits (in particular, instantaneous quantum polynomial\ncircuits). In addition, PQCs equipped with ancillary qubits for post-selection\nhave even stronger expressive power than those without post-selection. We\nemploy them as an application for Bayesian learning, since it is possible to\nlearn prior probabilities rather than assuming they are known. We expect that\nit will find many more applications in semi-supervised learning where prior\ndistributions are normally assumed to be unknown. Lastly, we conduct several\nnumerical experiments using the Rigetti Forest platform to demonstrate the\nperformance of the proposed Bayesian quantum circuit.

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Parameterized quantum circuits (PQCs) have been broadly used as a hybrid\nquantum-classical machine learning scheme to accomplish generative tasks.\nHowever, whether PQCs have better expressive power than classical generative\nneural networks, such as restricted or deep Boltzmann machines, remains an open\nissue. In this paper, we prove that PQCs with a simple structure already\noutperform any classical neural network for generative tasks, unless the\npolynomial hierarchy collapses. Our proof builds on known results from tensor\nnetworks and quantum circuits (in particular, instantaneous quantum polynomial\ncircuits). In addition, PQCs equipped with ancillary qubits for post-selection\nhave even stronger expressive power than those without post-selection. We\nemploy them as an application for Bayesian learning, since it is possible to\nlearn prior probabilities rather than assuming they are known. We expect that\nit will find many more applications in semi-supervised learning where prior\ndistributions are normally assumed to be unknown. Lastly, we conduct several\nnumerical experiments using the Rigetti Forest platform to demonstrate the\nperformance of the proposed Bayesian quantum circuit.
Parameterized quantum circuits (PQCs), as one of the most promising schemes to realize quantum machine learning algorithms on near-term quantum computers, have been designed to solve machine earning tasks with quantum advantages. In this paper, we explain why PQCs with different structures can achieve generative tasks in the discrete case from the perspective of probability estimation. Although different kinds of PQCs are proposed for generative tasks, the current methods often encounter the following three hurdles: (1) the mode contraction problem; (2) unexpected data are often generated with a high proportion; (3) target data cannot be sampled directly. For the purpose of tackling the above hurdles, we devise Bayesian quantum circuit (BQC) through introducing ancillary qubits to represent prior distributions. BQC advances both generative and semi-supervised quantum circuit learning tasks, where its effectiveness is validated by numerical simulations using the Rigetti Forest platform.
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.
Abstract Hybrid quantum–classical systems make it possible to utilize existing quantum computers to their fullest extent. Within this framework, parameterized quantum circuits can be regarded as machine learning models with remarkable expressive power. This Review presents the components of these models and discusses their application to a variety of data-driven tasks, such as supervised learning and generative modeling. With an increasing number of experimental demonstrations carried out on actual quantum hardware and with software being actively developed, this rapidly growing field is poised to have a broad spectrum of real-world applications.
Parameterized quantum circuits (PQC) have emerged as a quantum analogue of deep neural networks and can be trained for discriminative or generative tasks and can be trained with gradient-based optimization on near-term quantum devices [1], [2], [3]. In the current era of quantum computing, known as the noisy intermediate scale quantum (NISQ) era [4], these devices contain a moderate number of qubits (< 100), and algorithmic performance is strongly impacted by hardware noise. Additionally, the training of PQCs are hybrid algorithms, in which the computational workflow is split between quantum and classical computing platforms.
Many applications in quantum simulation, quantum chemistry, and quantum machine learning require not a single quantum state but an ensemble of states characterizing the heterogeneity of a target system. Preparing such ensembles state-by-state is prohibitive in both variational and fault-tolerant settings, thereby motivating a generative modeling approach. We introduce latent-conditioned parameterized quantum circuits (LPQCs), a hybrid quantum-classical framework in which classical neural networks map a latent variable sampled from a prior distribution to the parameters of a parameterized quantum circuit. We prove that LPQCs are universal approximators for probability measures over density operators in the 1-Wasserstein distance, extending classical universal approximation theorems to the quantum-distribution setting. We additionally introduce a multimodal latent prior and a mixture-of-experts circuit architecture, and show empirically that the latent-conditioned parameterization alleviates the barren plateau problem during optimization, a behavior for which we provide rigorous partial guarantees. Numerical experiments validate the framework on a synthetic multi-cluster ensemble of mixed quantum states and on a QM9-derived ensemble of 3-D molecular structures. In these tasks, LPQC outperforms recent quantum generative baselines and matches the generation quality of a classical neural-network baseline, while requiring an output dimension that grows only linearly with the number of qubits rather than exponentially. By leveraging classical expressivity in the latent space, LPQCs offer a tractable route to quantum generative modeling.