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arXiv:2010.15968 · PRX Quantum

Entanglement-Induced Barren Plateaus

Carlos Ortiz Marrero, Mária Kieferová, Nathan Wiebe

Published Oct 25, 2021• 39 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.2.040316

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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.
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.
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.
Quantum neural networks (QNNs) have generated excitement around the possibility of efficiently analyzing quantum data.But this excitement has been tempered by the existence of exponentially vanishing gradients, known as barren plateau landscapes, for many QNN architectures.Recently, quantum convolutional neural networks (QCNNs) have been proposed, involving a sequence of convolutional and pooling layers that reduce the number of qubits while preserving information about relevant data features.In this work, we rigorously analyze the gradient scaling for the parameters in the QCNN architecture.We find that the variance of the gradient vanishes no faster than polynomially, implying that QCNNs do not exhibit barren plateaus.This result provides an analytical guarantee for the trainability of randomly initialized QCNNs, which highlights QCNNs as being trainable under random initialization unlike many other QNN architectures.To derive our results, we introduce a novel graph-based method to analyze expectation values over Haar-distributed unitaries, which will likely be useful in other contexts.Finally, we perform numerical simulations to verify our analytical results.
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.
Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.
Barren plateaus, which means the training gradients become extremely small, pose a major challenge in optimizing parameterized quantum circuits, often making the learning process impractically slow or stall. This work shows why using neural networks to generate quantum circuit parameters helps overcome this difficulty. We introduce a geometric viewpoint that describes how the parameters produced by neural networks evolve during training. Our analysis shows that these parameters follow smooth and efficient paths that avoid the flat regions in the training that cause barren plateaus. This provides a computational explanation for the improved trainability observed in recent neural network-assisted quantum learning methods. Overall, our findings bridge ideas from quantum machine learning and computational optimization, offering new insight into the structure of quantum models and guiding future approaches for designing more trainable quantum circuits or parameter initialization.