Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）；Center for Nonlinear Studies, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究中心）
M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, Patrick J. Coles
备注：
14 + 25 pages, 8 + 2 figures. Updated to published version. Changed title
Variational quantum algorithms (VQAs) optimize the parameters $\vecθ$ of a parametrized quantum circuit $V(\vecθ)$ to minimize a cost function $C$. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming $V(\vecθ)$ is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining $C$ in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when $V(\vecθ)$ is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining $C$ with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of $V(\vecθ)$ is $\mathcal{O}(\log n)$. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.
Barren plateaus are a notorious problem in the optimization of variational quantum algorithms and pose a critical obstacle in the quest for more efficient quantum machine learning algorithms. Many potential reasons for barren plateaus have been identified but few solutions have been proposed to avoid them in practice. Existing solutions are mainly focused on the initialization of unitary gate parameters without taking into account the changes induced by input data. In this paper, we propose an alternative strategy which initializes the parameters of a unitary gate by drawing from a beta distri…
Variational quantum algorithms (VQAs) are expected to establish valuable applications on near-term quantum computers. However, recent works have pointed out that the performance of VQAs greatly relies on the expressibility of the ansatzes and is seriously limited by optimization issues such as barren plateaus (i.e., vanishing gradients). This work proposes the state efficient ansatz (SEA) for accurate ground state preparation with improved trainability. We show that the SEA can generate an arbitrary pure state with much fewer parameters than a universal ansatz, making it efficient for tasks li…
Optimizing parameterized quantum circuits (PQCs) is the leading approach to make use of near-term quantum computers. However, very little is known about the cost function landscape for PQCs, which hinders progress towards quantum-aware optimizers. In this work, we investigate the connection between three different landscape features that have been observed for PQCs: (1) exponentially vanishing gradients (called barren plateaus), (2) exponential cost concentration about the mean, and (3) the exponential narrowness of minina (called narrow gorges). We analytically prove that these three phenomen…
The existence of barren plateaus has recently revealed new training challenges in quantum machine learning (QML). Uncovering the mechanisms behind barren plateaus is essential in understanding the scope of problems that QML can efficiently tackle. Barren plateaus have recently been shown to exist when learning global properties of random unitaries, which is relevant when learning black hole dynamics. Establishing whether local cost functions can circumvent these barren plateaus is pertinent if we hope to apply QML to quantum many-body systems. We prove a no-go theorem showing that local cost f…
Pacific Northwest National Laboratory（西北太平洋国家实验室）；University of Technology Sydney（悉尼科技大学）；University of Toronto（多伦多大学）
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 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 t…
Noise-Induced Barren Plateaus in Variational Quantum Algorithms
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Imperial College London（伦敦帝国学院）；University of Strathclyde（思克莱德大学）；National Physical Laboratory（国家物理实验室）；Center for Nonlinear Studies, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究中心）
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Center for Nonlinear Studies（非线性研究中心）
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 s…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院）
Variational quantum algorithms rely on gradient based optimization to iteratively minimize a cost function evaluated by measuring output(s) of a quantum processor. A barren plateau is the phenomenon of exponentially vanishing gradients in sufficiently expressive parametrized quantum circuits. It has been established that the onset of a barren plateau regime depends on the cost function, although the particular behavior has been demonstrated only for certain classes of cost functions. Here we derive a lower bound on the variance of the gradient, which depends mainly on the width of the circuit …
Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren…
Exponentially many initializations to avoid barren plateaus
Fujitsu Research of America, Santa Clara, CA 95054, USA（美国富士通美洲研究部）；University of Delaware, Newark, DE 19716, USA（德雷克塞尔大学）；Department for Quantum Information and Computation at Kepler (QUICK), Johannes Kepler University, Linz, Austria（约翰·凯撒大学量子信息与计算部门）；Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA（洛斯阿拉莫斯国家实验室信息科学部）
Defeating Barren Plateaus with Task-Aligned Symmetry
Faculty of Information Science and Engineering, Ocean University of China（中国海洋大学信息科学与工程学部）；China Telecom Quantum Information Technology Group Co., LTD.（中国电信量子信息技术集团有限公司）
Variational quantum algorithms are promising algorithms for achieving quantum advantage on near-term devices. The quantum hardware is used to implement a variational wave function and measure observables, whereas the classical computer is used to store and update the variational parameters. The optimization landscape of expressive variational ansätze is however dominated by large regions in parameter space, known as barren plateaus, with vanishing gradients which prevents efficient optimization. In this work we propose a general algorithm to avoid barren plateaus in the initialization and thro…
Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. …
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as the barren plateau phenomenon, in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown t…
One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum model…
Parametrized quantum circuits initialized with random initial parameter values are characterized by barren plateaus where the gradient becomes exponentially small in the number of qubits. In this technical note we theoretically motivate and empirically validate an initialization strategy which can resolve the barren plateau problem for practical applications. The technique involves randomly selecting some of the initial parameter values, then choosing the remaining values so that the circuit is a sequence of shallow blocks that each evaluates to the identity. This initialization limits the eff…
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Louisiana State University（路易斯安那州立大学）
Parameterized quantum circuits serve as ansätze for solving variational problems and provide a flexible paradigm for programming near-term quantum computers. Ideally, such ansätze should be highly expressive so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well established barren plateau phenomenon, which holds for ansätze that form …
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；University College London（伦敦大学学院）；Imperial College London（帝国理工学院）
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…
Variational quantum algorithms (VQAs) are among the most promising algorithms in the era of Noisy Intermediate Scale Quantum Devices. Such algorithms are constructed using a parameterization U($\pmbθ$) with a classical optimizer that updates the parameters $\pmbθ$ in order to minimize a cost function $C$. For this task, in general the gradient descent method, or one of its variants, is used. This is a method where the circuit parameters are updated iteratively using the cost function gradient. However, several works in the literature have shown that this method suffers from a phenomenon known …
Quantum compilation provides a method to translate quantum algorithms at a high level of abstraction into their implementations as quantum circuits on real hardware. One approach to quantum compiling is to design a parameterised circuit and to use techniques from optimisation to find the parameters that minimise the distance between the parameterised circuit and the target circuit of interest. While promising, such an approach typically runs into the obstacle of barren plateaus - i.e. large regions of parameter space in which the gradient vanishes. A number of recent works focusing on so-calle…
Mitigating Barren Plateaus in Quantum Denoising Diffusion Probabilistic Model
School of Computer Science and Technology, University of Science and Technology of China（中国科学技术大学计算机科学与技术学院）；College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算机科学与数据科学学院）；Research Institute of Quantum Technology, The Hong Kong Polytechnic University（香港理工大学量子技术研究所）
Variational quantum eigensolvers (VQEs) represent a powerful class of hybrid quantum-classical algorithms for computing molecular energies. Various numerical issues exist for these methods, however, including barren plateaus and large numbers of local minima. In this work, we consider Adaptive, Problem-Tailored (ADAPT)-VQE ansätze, and examine how they are impacted by these local minima. We find that while ADAPT-VQE does not remove local minima, the gradient-informed, one-operator-at-a-time circuit construction seems to accomplish two things: First, it provides an initialization strategy that …
A large ongoing research effort focuses on Variational Quantum Algorithms (VQAs), representing leading candidates to achieve computational speed-ups on current quantum devices. The scalability of VQAs to a large number of qubits, beyond the simulation capabilities of classical computers, is still debated. Two major hurdles are the proliferation of low-quality variational local minima, and the exponential vanishing of gradients in the cost function landscape, a phenomenon referred to as barren plateaus. Here we show that by employing iterative search schemes one can effectively prepare the grou…
Can Chemically Inspired Parameter Initialization Mitigate Barren Plateaus in Variational Quantum Eigensolvers?
Guangdong Basic Research Center of Excellence for Aggregate Science, School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen)（广东聚合科学卓越研究中心，科学与工程学部，香港中文大学（深圳））；School of Physical and Chemical Sciences, Queen Mary University of London（物理与化学科学学院，伦敦大学玛丽女王学院）
Taming Barren Plateaus in Arbitrary Parameterized Quantum Circuits without Sacrificing Expressibility
Department of Computer Science and Technology, Tsinghua University, Beijing, China（计算机科学与技术系，清华大学，北京，中国）；Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China（叶阳数学科学中心，清华大学，北京，中国）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 100407, China（燕琦湖北京数学科学与应用研究所，北京，中国）；Department of Mathematics, Tsinghua University, Beijing 100084, China（数学系，清华大学，北京，中国）
Does provable absence of barren plateaus imply classical simulability?
Information Sciences, Los Alamos National Laboratory（信息科学，洛斯阿拉莫斯国家实验室）；Quantum Science Center, Oak Ridge（量子科学中心，奥克兰岭）；Theoretical Division, Los Alamos National Laboratory（理论部，洛斯阿拉莫斯国家实验室）；Center for Nonlinear Studies, Los Alamos National Laboratory（非线性研究中心，洛斯阿拉莫斯国家实验室）；University of Strathclyde（斯特拉斯克莱德大学）；Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL)（物理研究所，洛桑联邦理工学院（EPFL））；Donostia International Physics Center（多斯通国际物理中心）；Department of Physics and Astronomy, University of Waterloo（物理与天文学系，滑铁卢大学）；Vector Institute, MaRS Centre（向量研究所，马尔伯里中心）；Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University（复杂系统，物理系，科学学院，朱拉隆梭大学）；Walter Burke Institute for Theoretical Physics, Caltech（沃尔特·伯克理论物理研究所，加州理工学院）
Tensor networks are efficient representations of high-dimensional tensors with widespread applications in quantum many-body physics. Recently, they have been adapted to the field of machine learning, giving rise to an emergent research frontier that has attracted considerable attention. Here, we study the trainability of tensor-network based machine learning models by exploring the landscapes of different loss functions, with a focus on the matrix product states (also called tensor trains) architecture. In particular, we rigorously prove that barren plateaus (i.e., exponentially vanishing grad…
Variational quantum algorithms (VQAs) are widely applied in the noisy intermediate-scale quantum era and are expected to demonstrate quantum advantage. However, training VQAs faces difficulties, one of which is the so-called barren plateaus (BP) phenomenon, where gradients of cost functions vanish exponentially with the number of qubits. In this paper, inspired by transfer learning, where knowledge of pre-solved tasks could be further used in a different but related work with training efficiency improved, we report a parameter initialization method to mitigate BP. In the method, a small-sized …
Hybrid quantum-classical variational algorithms are one of the most propitious implementations of quantum computing on near-term devices, offering classical machine learning support to quantum scale solution spaces. However, numerous studies have demonstrated that the rate at which this space grows in qubit number could preclude learning in deep quantum circuits, a phenomenon known as barren plateaus. In this work, we implicate random entanglement as the source of barren plateaus and characterize them in terms of many-body entanglement dynamics, detailing their formation as a function of syste…
A Representation-Theoretic Framework for Characterizing Barren Plateaus
International Institute of Physics, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）；Instituto de Física de São Carlos, Universidade de São Paulo（圣保罗大学圣卡洛斯物理研究所）
Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on…
Scrambling processes, which rapidly spread entanglement through many-body quantum systems, are difficult to investigate using standard techniques, but are relevant to quantum chaos and thermalization. In this Letter, we ask if quantum machine learning (QML) could be used to investigate such processes. We prove a no-go theorem for learning an unknown scrambling process with QML, showing that any variational ansatz is highly probable to have a barren plateau landscape, i.e., cost gradients that vanish exponentially in the system size. This implies that the required resources scale exponentially …
