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Entanglement-Induced Barren Plateaus

Carlos Ortiz Marrero

Published Oct 27, 2021• 282 citations
Publisher

Abstract

In recent years the prospects of quantum machine learning, as exemplified by quantum deep neural networks, have gain notoriety in the scientific community. By combining ideas from quantum computing with machine learning methodology, these models promise new ways to interpret classical and quantum data sets. In our work, we connect quantum thermodynamics and quantum deep learning to demonstrate that, with high probability, entanglement between the visible and hidden units will lead to states that are exponentially close to a maximally mixed state. Moreover, the gradients of such models with respect to their parameters will vanish as the dimension of the system increases for reasonable random ensembles of quantum models. We call this effect entanglement induced barren plateaus and argue that their existence necessitates a re-evaluation of how we think about the role that entanglement and other quantum effects play within quantum machine learning algorithms.

DOI
10.1103/prxquantum.2.040316

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In recent years the prospects of quantum machine learning, as exemplified by quantum deep neural networks, have gain notoriety in the scientific community. By combining ideas from quantum computing with machine learning methodology, these models promise new ways to interpret classical and quantum data sets. In our work, we connect quantum thermodynamics and quantum deep learning to demonstrate that, with high probability, entanglement between the visible and hidden units will lead to states that are exponentially close to a maximally mixed state. Moreover, the gradients of such models with respect to their parameters will vanish as the dimension of the system increases for reasonable random ensembles of quantum models. We call this effect entanglement induced barren plateaus and argue that their existence necessitates a re-evaluation of how we think about the role that entanglement and other quantum effects play within quantum machine learning algorithms.
We argue that an excess in entanglement between the visible and hidden units in a quantum neural network can hinder learning.In particular, we show that quantum neural networks that satisfy a volume law in the entanglement entropy will give rise to models that are not suitable for learning with high probability.Using arguments from quantum thermodynamics, we then show that this volume law is typical and that there exists a barren plateau in the optimization landscape due to entanglement.More precisely, we show that for any bounded objective function on the visible layers, the Lipshitz constants of the expectation value of that objective function will scale inversely with the dimension of the hidden subsystem with high probability.We show how this can cause both gradient-descent and gradient-free methods to fail.We note that similar problems can happen with quantum Boltzmann machines, although stronger assumptions on the coupling between the hidden and/or visible subspaces are necessary.We highlight how pretraining such generative models may provide a way to navigate these barren plateaus.
Barren Plateaus are a formidable challenge for hybrid quantum-classical algorithms that lead to flat plateaus in the loss function landscape making it difficult to take advantage of the expressive power of parameterized quantum circuits with gradient-based methods. Like in classical neural network models, parameterized quantum circuits suffer the same vanishing gradient issue due to large parameter spaces with non-convex landscapes. In this review, we present an overview of the different genesis for barren plateaus, mathematical formalisms of common themes around barren plateaus, and dives into gradients. The central objective is to provide a conceptual perspective between classical and quantum interpretations of vanishing gradients as well as dive into techniques involving cost functions, entanglement, and initialization strategies to mitigate barren plateaus. Addressing barren plateaus paves the way towards feasibility of many classically intractable applications for quantum simulation, optimization, chemistry, and quantum machine learning.
Abstract A new paradigm for data science has emerged, with quantum data, quantum models, and quantum computational devices. This field, called quantum machine learning (QML), aims to achieve a speedup over traditional machine learning for data analysis. However, its success usually hinges on efficiently training the parameters in quantum neural networks, and the field of QML is still lacking theoretical scaling results for their trainability. Some trainability results have been proven for a closely related field called variational quantum algorithms (VQAs). While both fields involve training a parametrized quantum circuit, there are crucial differences that make the results for one setting not readily applicable to the other. In this work, we bridge the two frameworks and show that gradient scaling results for VQAs can also be applied to study the gradient scaling of QML models. Our results indicate that features deemed detrimental for VQA trainability can also lead to issues such as barren plateaus in QML. Consequently, our work has implications for several QML proposals in the literature. In addition, we provide theoretical and numerical evidence that QML models exhibit further trainability issues not present in VQAs, arising from the use of a training dataset. We refer to these as dataset-induced barren plateaus. These results are most relevant when dealing with classical data, as here the choice of embedding scheme (i.e., the map between classical data and quantum states) can greatly affect the gradient scaling.
Realizing successful quantum algorithms on current generation of quantum hardware depends on developing appropriate parametrized quantum circuits, which are the quantum computing counterpart to classical neural networks. A key consideration in this context is the algorithm’s trainability that can be hindered by the existence of so-called barren plateaus. In this case, the algorithm suffers from exponentially vanishing gradients as the number of qubits increases, resulting in a flat optimization landscape and making an efficient training intractable. Due to its serious impact on the success of such algorithms, this problem has gained substantial attention in recent years of quantum computing research. Theoretical and heuristic approaches have been proposed to investigate, avoid and mitigate this issue. In this work, we explore this phenomenon in detail and study the underlying mathematical concepts in depth, investigating the factors that influence the structure of the optimization landscape. Using corresponding mathematical tools, we diagnose whether a given variational circuit will exhibit barren plateaus when going beyond tens of qubits. Additionally, a review of mitigation strategies will be provided. The theory will then be applied to a use case that involves image classification to evaluate whether the model is affected by this phenomenon and which hyperparameters improve or degrade the overall performance.
This thesis addresses one of the central challenges in Quantum Machine Learning: the Barren Plateaus phenomenon, which causes the exponential vanishing of gradients and hin ders the effective training of variational quantum algorithms. We propose the Q-MOLLD method in which we design a hybrid protocol that leverages the generative capabilities of Large Language Models to generate quantum algorithms for optimization problems and to produce informed initializations for quantum circuits to improve their performance. Experimental results demonstrate that the proposed approach can mitigate the Barren Plateaus problem, preserving larger gradient magnitudes during the early stages of training and facilitating convergence compared to random or classical strategies. The findings suggest that Large Language Models can play a crucial role in the design and training of quantum algorithms, especially in the Noisy Intermediate-Scale Quantum era, where hardware limitations and noise pose significant challenges. Future work will focus on extending the analysis to different ansatz architectures, exploring active learning techniques, and evaluating performance under realistic noise conditions on quantum hardware.
Quantum neural networks (QNNs) are a framework for creating quantum algorithms that promises to combine the speedups of quantum computation with the widespread successes of machine learning. A major challenge in QNN development is a concentration of measure phenomenon known as a barren plateau that leads to exponentially small gradients for a range of QNNs models. In this work, we examine the assumptions that give rise to barren plateaus and show that an unbounded loss function can circumvent the existing no-go results. We propose a training algorithm that minimizes the maximal Rényi divergence of order two and present techniques for gradient computation. We compute the closed form of the gradients for Unitary QNNs and Quantum Boltzmann Machines and provide sufficient conditions for the absence of barren plateaus in these models. We demonstrate our approach in two use cases: thermal state learning and Hamiltonian learning. In our numerical experiments, we observed rapid convergence of our training loss function and frequently archived a 99%99\%99% average fidelity in fewer than 100100100 epochs.