Parameterized quantum evolution is the main ingredient in variational quantum algorithms for near-term quantum devices. In digital quantum computing, it has been shown that random parameterized quantum circuits are able to express complex distributions intractable by a classical computer, leading to the demonstration of quantum supremacy. However, their chaotic nature makes parameter optimization challenging in variational approaches. Evidence of similar classically-intractable expressibility has been recently demonstrated in analog quantum computing with driven many-body systems. A thorough investigation of trainability of such analog systems is yet to be performed. In this work, we investigate how the interplay between external driving and disorder in the system dictates the trainability and expressibility of interacting quantum systems. We show that if the system thermalizes, the training fails at the expense of the a large expressibility, while the opposite happens when the system enters the many-body localized (MBL) phase. From this observation, we devise a protocol using quenched MBL dynamics which allows accurate trainability while keeping the overall dynamics in the quantum supremacy regime. Our work shows the fundamental connection between quantum many-body physics and its application in machine learning. We conclude our work with an example application in generative modeling employing a well studied analog many-body model of a driven Ising spin chain. Our approach can be implemented with a variety of available quantum platforms including cold ions, atoms and superconducting circuits
The hybrid quantum-classical algorithm is actively examined as a technique applicable even to intermediate-scale quantum computers. To execute this algorithm, the hardware efficient ansatz is often used, thanks to its implementability and expressibility; however, this ansatz has a critical issue in its trainability in the sense that it generically suffers from the so-called gradient vanishing problem. This issue can be resolved by limiting the circuit to the class of shallow alternating layered ansatz. However, even though the high trainability of this ansatz is proved, it is still unclear whe…
Breaking the One-Dimensional Expressibility-Trainability Tradeoff
Institute for Convergence Research and Education in Advanced Technology（融合研究中心和先进科技教育研究所）；Yonsei University（延世大学）；Department of Statistics and Data Science（统计与数据科学系）；Hanyang University（翰阳大学）；Department of Physics（物理系）；Hanyang Institute for Quantum Science and Quantum Technology（翰阳量子科学与技术研究所）；Department of Quantum Information（量子信息系）
Parameterized quantum circuits play an essential role in the performance of many variational hybrid quantum-classical (HQC) algorithms. One challenge in implementing such algorithms is to choose an effective circuit that well represents the solution space while maintaining a low circuit depth and number of parameters. To characterize and identify expressible, yet compact, parameterized circuits, we propose several descriptors, including measures of expressibility and entangling capability, that can be statistically estimated from classical simulations of parameterized quantum circuits. We comp…
An active area of investigation in the search for quantum advantage is Quantum Machine Learning. Quantum Machine Learning, and Parameterized Quantum Circuits in a hybrid quantum-classical setup in particular, could bring advancements in accuracy by utilizing the high dimensionality of the Hilbert space as feature space. But is the ability of a quantum circuit to uniformly address the Hilbert space a good indicator of classification accuracy? In our work, we use methods and quantifications from prior art to perform a numerical study in order to evaluate the level of correlation. We find a stron…
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 …
IBM Quantum, IBM Research（IBM量子，IBM研究院）；University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）；ETH Zurich（苏黎世联邦理工学院）
Fault-tolerant quantum computers offer the promise of dramatically improving machine learning through speed-ups in computation or improved model scalability. In the near-term, however, the benefits of quantum machine learning are not so clear. Understanding expressibility and trainability of quantum models-and quantum neural networks in particular-requires further investigation. In this work, we use tools from information geometry to define a notion of expressibility for quantum and classical models. The effective dimension, which depends on the Fisher information, is used to prove a novel gen…
With the advent of hybrid quantum classical algorithms using parameterized quantum circuits the question of how to optimize these algorithms and circuits emerges. In this paper we show that the number of single-qubit rotations in parameterized quantum circuits can be decreased without compromising the relative expressibility or entangling capability of the circuit. We also show that the performance of a variational quantum eigensolver is unaffected by a similar decrease in single-qubit rotations. We compare relative expressibility and entangling capability across different number of qubits in …
Observable-Guided Generator Selection for Improving Trainability in Quantum Machine Learning with a $ \mathfrak{g} $-Purity Interpretation under Restricted Settings
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…
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）
Variational quantum algorithms are suitable for use on noisy quantum systems. One of the most important use-cases is the quantum simulation of materials, using the variational quantum eigensolver (VQE). To optimize VQE performance, a suitable parameterized quantum circuit (ansatz) must be selected. We investigate a class of ansatze that incorporates knowledge of the quantum hardware, namely the hardware efficient ansatze. The performance of hardware efficient ansatze is affected differently by noise, and our goal is to study the effect of noise on evaluating which ansatz gives more accurate re…
Optimizing parameterized quantum circuits promises efficient use of near-term quantum computers to achieve the potential quantum advantage. However, there is a notorious tradeoff between the expressibility and trainability of the parameter ansatz. We find that in combinatorial optimization problems, since the solutions are described by bit strings, one can trade the expressiveness of the ansatz for high trainability. To be specific, by focusing on the max-cut problem we introduce a simple yet efficient algorithm named Quantum Qubit Rotation Algorithm (QQRA). The quantum circuits are comprised …
Scaling of variational quantum algorithms to large problem sizes requires efficient optimization of random parameterized quantum circuits. For such circuits with uncorrelated parameters, the presence of exponentially vanishing gradients in cost function landscapes is an obstacle to optimization by gradient descent methods. In this work, we prove that reducing the dimensionality of the parameter space by utilizing circuit modules containing spatially or temporally correlated gate layers can allow one to circumvent the vanishing gradient phenomenon. Examples are drawn from random separable circu…
This article explores search strategies for the design of parameterized quantum circuits. We propose several optimization approaches including random search plus survival of the fittest, reinforcement learning both with classical and hybrid quantum classical controllers and Bayesian optimization as decision makers to design a quantum circuit in an automated way for a specific task such as multi-labeled classification over a dataset. We introduce nontrivial circuit architectures that are arduous to be hand-designed and efficient in terms of trainability. In addition, we introduce reuploading of…
Variational Quantum Algorithms (VQAs) have emerged as a powerful class of algorithms that is highly suitable for noisy quantum devices. Therefore, investigating their design has become key in quantum computing research. Previous works have shown that choosing an effective parameterized quantum circuit (PQC) or ansatz for VQAs is crucial to their overall performance, especially on near-term devices. In this paper, we utilize pulse-level access to quantum machines and our understanding of their two-qubit interactions to optimize the design of two-qubit entanglers in a manner suitable for VQAs. O…
Quantum machine learning (QML) is the use of quantum computing for the computation of machine learning algorithms. With the prevalence and importance of classical data, a hybrid quantum-classical approach to QML is called for. Parameterized Quantum Circuits (PQCs), and particularly Quantum Kernel PQCs, are generally used in the hybrid approach to QML. In this paper we discuss some important aspects of PQCs with quantum kernels including PQCs, quantum kernels, quantum kernels with quantum advantage, and the trainability of quantum kernels. We conclude that quantum kernels with hybrid kernel met…
Twirlator: A Pipeline for Analyzing Subgroup Symmetry Effects in Quantum Machine Learning Ansatzes
Aalto University \& University of Helsinki Espoo Finland；University of Helsinki Helsinki Finland；Aalto University Espoo Finland；Aalto University \& University of Helsinki；University of Helsinki；Aalto University
The Variational Quantum Eigensolver (VQE) is a promising candidate for quantum applications on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. Despite a lot of empirical studies and recent progress in theoretical understanding of VQE's optimization landscape, the convergence for optimizing VQE is far less understood. We provide the first rigorous analysis of the convergence of VQEs in the over-parameterization regime. By connecting the training dynamics with the Riemannian Gradient Flow on the unit-sphere, we establish a threshold on the sufficient number of parameters for efficie…
Variational quantum algorithms, which utilize Parametrized Quantum Circuits (PQCs), are promising tools to achieve quantum advantage for optimization problems on near-term quantum devices. Their PQCs have been conventionally constructed from parametrized rotational angles of single-qubit gates around predetermined set of axes, and two-qubit entangling gates, such as CNOT gates. We propose a method to construct a PQC by continuous parametrization of both the angles and the axes of its single-qubit rotation gates. The method is based on the observation that when rotational angles are fixed, opti…
