Variational quantum algorithms (VQAs) are a broad class of algorithms with many applications in science and industry. Applying a VQA to a problem involves optimizing a parameterized quantum circuit by maximizing or minimizing a cost function. A particular challenge associated with VQAs is understanding the properties of associated cost functions. Having the landscapes of VQA cost functions can greatly assist in developing and testing new variational quantum algorithms, but they are extremely expensive to compute. Reconstructing the landscape of a VQA using existing techniques requires a large number of cost function evaluations, especially when the dimension or the resolution of the landscape is high. To address this challenge, we propose a low-rank tensor-completion-based approach for local landscape reconstruction. By leveraging compact low-rank representations of tensors, our technique can overcome the curse of dimensionality and handle high-resolution landscapes. We demonstrate the power of landscapes in VQA development by showcasing practical applications of analyzing penalty terms for constrained optimization problems and examining the probability landscapes of certain basis states.
We employ spectral analysis and compressed sensing to identify settings where a variational algorithm's cost function can be recovered purely classically or with minimal quantum computer access. We present theoretical and numerical evidence supporting the viability of sparse recovery techniques. To demonstrate this approach, we use basis pursuit denoising to efficiently recover simulated Quantum Approximate Optimization Algorithm (QAOA) instances of large system size from very few samples. Our results indicate that sparse recovery can enable a more efficient use and distribution of quantum res…
Energy Landscape Structure of Small Graph Isomorphism Under Variational Optimization
Calcutta Institute of Engineering and Management, Maulana Abul Kalam Azad University of Technology（加尔各答工程与管理学院，穆拉纳·阿布尔·卡拉姆·阿扎德技术大学）；University of Dhaka（达卡大学）；Institute of Theoretical and Applied Informatics, Polish Academy of Sciences（波兰科学院理论与应用信息研究所）；Silesian University of Technology（西里西亚理工大学）
Variational Quantum Algorithms (VQAs) are promising candidates for finding practical applications of near to mid-term quantum computers. There has been an increasing effort to study the intricacies of VQAs, such as the presence or absence of barren plateaus and the design of good quantum circuit ansätze. Many of these studies can be linked to the loss landscape that is optimized as part of the algorithm, and there is high demand for quality software tools for flexibly studying these loss landscapes. In our work, we collect a variety of techniques that have been used to visualize the training o…
Quantifying the flatness of the objective-function landscape associated with unstructured parameterized quantum circuits is important for understanding the performance of variational algorithms utilizing a "hardware-efficient ansatz", particularly for ensuring that a prohibitively flat landscape -- a so-called "barren plateau" -- is avoided. For a model of such ansätze, we relate the typical landscape flatness to a certain family of random walks, enabling us to derive a Monte Carlo algorithm for efficiently, classically estimating the landscape flatness for any architecture. The statistical pi…
The Dual Role of Low-Weight Pauli Propagation: A Flawed Simulator but a Powerful Initializer for Variational Quantum Algorithms
Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Beijing National Laboratory for Condensed Matter Physics（北京凝聚态物理国家实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
Machine learning techniques enhanced by noisy intermediate-scale quantum (NISQ) devices and especially variational quantum circuits (VQC) have recently attracted much interest and have already been benchmarked for certain problems. Inspired by classical deep learning, VQCs are trained by gradient descent methods which allow for efficient training over big parameter spaces. For NISQ sized circuits, such methods show good convergence. There are however still many open questions related to the convergence of the loss function and to the trainability of these circuits in situations of vanishing gr…
Energy Landscape Plummeting in Variational Quantum Eigensolver: Subspace Optimization, Non-iterative Corrections and Generator-informed Initialization for Improved Quantum Efficiency
Optimization Landscape Geometry in VQE for Frustrated Quantum Spin Models
VSB–Technical University of Ostrava（俄斯特拉发理工大学）；IT4Innovations National Supercomputing Center, VSB–Technical University of Ostrava（俄斯特拉发理工大学 IT4创新国家超级计算中心）；Marine Research Institute, Klaipeda University（克莱佩达大学海洋研究所）；Indian Statistical Institute（印度统计研究所）；Università dell’Aquila（阿奎拉大学）
Variational Quantum Algorithms have emerged as a leading paradigm for near-term quantum computation. In such algorithms, a parameterized quantum circuit is controlled via a classical optimization method that seeks to minimize a problem-dependent cost function. Although such algorithms are powerful in principle, the non-convexity of the associated cost landscapes and the prevalence of local minima means that local optimization methods such as gradient descent typically fail to reach good solutions. In this work we suggest a method to improve gradient-based approaches to variational quantum circ…
We present a variational approach which shows that the wave functions belonging to quantum systems in different potential landscapes, are pairwise linked to each other through a generalized continuity equation. This equation contains a source term proportional to the potential difference. In case the potential landscapes are related by a linear symmetry transformation in a finite domain of the embedding space, the derived continuity equation leads to generalized currents which are divergence free within this spatial domain. In a single spatial dimension these generalized currents are invariant…
The performance of variational quantum algorithms relies on the success of using quantum and classical computing resources in tandem. Here, we study how these quantum and classical components interrelate. In particular, we focus on algorithms for solving the combinatorial optimization problem MaxCut, and study how the structure of the classical optimization landscape relates to the quantum circuit used to evaluate the MaxCut objective function. In order to analytically characterize the impact of quantum features on the critical points of the landscape, we consider a family of quantum circuit a…
When is global evolutionary search useful for variational quantum algorithms? A landscape-first study
VSB - Technical University of Ostrava（俄斯特拉发理工大学）；IT4Innovations National Supercomputing Center, VSB - Technical University of Ostrava（俄斯特拉发理工大学 IT4创新国家超算中心）；Klaipeda University（克莱佩达大学）
Measurement-Induced Landscape Transitions and Coding Barren Plateaus in Hybrid Variational Quantum Circuits
Hewlett Packard Enterprise Labs（惠普企业实验室）；Laboratory of Atomic and Solid State Physics, Cornell University（康奈尔大学原子与固体物理实验室）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Physics, Applied Physics, and Astronomy, Binghamton University（宾汉姆顿大学物理、应用物理和天文学系）；School of Physics and Astronomy, University of Minnesota（明尼苏达大学物理与天文学院）
Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms
Chula Intelligent and Complex Systems Lab, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand, 10330（朱拉隆功大学科学学院物理系复杂系统实验室）；Siam Quantum Square, Faculty of Science, Chulalongkorn University, Bangkok, Thailand, 10330（朱拉隆功大学科学学院暹罗量子广场）；Centre for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland（瑞士洛桑联邦理工学院量子科学与工程中心）
Experimental demonstration of the absence of noise-induced barren plateaus using information content landscape analysis
Honda Research Institute Europe GmbH（本田欧洲研究院）；Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento（皮塔耶夫斯基玻色-爱因斯坦凝聚中心，意大利国家研究委员会光学研究所及特伦托大学物理系）；INFN-TIFPA, Trento Institute for Fundamental Physics and Applications（INFN-TIFPA，特伦托基础物理与应用研究院）
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…
Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks…
We study the effects of entanglement and control parameters on the energy landscape and optimization performance of the variational quantum circuit. Through a systematic analysis of the Hessian spectrum, we characterize the local geometry of the energy landscape at a random point and along an optimization trajectory. We argue that decreasing the entangling capability and increasing the number of circuit parameters have the same qualitative effect on the Hessian eigenspectrum. Both the low-entangling capability and the abundance of control parameters increase the curvature of non-flat direction…
National Institute of Technology, Agartala（阿加尔塔拉国家理工学院）；Chitkara University Institute of Engineering & Technology, Chitkara University（奇特卡拉大学工程与技术学院）；Mansoura University（曼苏拉大学）
In quantum computing, the variational quantum algorithms (VQAs) are well suited for finding optimal combinations of things in specific applications ranging from chemistry all the way to finance. The training of VQAs with gradient descent optimization algorithm has shown a good convergence. At an early stage, the simulation of variational quantum circuits on noisy intermediate-scale quantum (NISQ) devices suffers from noisy outputs. Just like classical deep learning, it also suffers from vanishing gradient problems. It is a realistic goal to study the topology of loss landscape, to visualize th…
Quantum circuit Born machines (QCBMs) and training via variational quantum algorithms (VQAs) are key applications for near-term quantum hardware. QCBM ansätze designs are unique in that they do not require prior knowledge of a physical Hamiltonian. Many ansätze are built from fixed designs. In this work, we train and compare the performance of QCBM models built using two commonly employed parameterizations and two commonly employed entangling layer designs. In addition to comparing the overall performance of these models, we look at features and characteristics of the loss landscape -- connect…
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）
Geometric Analysis of Variational Quantum Eigensolver
Michigan Institute for Computational Discovery and Engineering, Department of Electrical Engineering and Computer Science and Department of Statistics, University of Michigan（密歇根大学计算发现与工程研究所、电气工程和计算机科学系以及统计学系）
Landscape Analysis of Excited States Calculation over Quantum Computers
Fudan University（复旦大学）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学现代应用数学上海市重点实验室）；Key Laboratory of Computational Physical Sciences, Ministry of Education（教育部计算物理科学重点实验室）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Duke University（杜克大学）
Taylor University（泰勒大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Variational training of parameterized quantum circuits (PQCs) underpins many workflows employed on near-term noisy intermediate scale quantum (NISQ) devices. It is a hybrid quantum-classical approach that minimizes an associated cost function in order to train a parameterized ansatz. In this paper we adapt the qualitative loss landscape characterization for neural networks introduced in \cite{goodfellow2014qualitatively,li2017visualizing} and tests for connectivity used in \cite{draxler2018essentially} to study the loss landscape features in PQC training. We present results for PQCs trained on…
Variational quantum algorithms are poised to have significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, causing suboptimal solutions due to convergence to local, rather than global, minima. In this work, we introduce a variational quantum algorithm that uses classical Markov chain Monte Carlo techniques to provably converge to global minima. These performance gaurantees are derived from the ergodicity of our algorithm's state…
Laziness, Barren Plateau, and Noise in Machine Learning
Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普利兹克分子工程学院）；Chicago Quantum Exchange（芝加哥量子交易所）；Kadanoff Center for Theoretical Physics, The University of Chicago（芝加哥大学卡达诺夫理论物理中心）；qBraid Co.（qBraid公司）
Variational quantum algorithms are proposed to solve relevant computational problems on near term quantum devices. Popular versions are variational quantum eigensolvers and quantum ap- proximate optimization algorithms that solve ground state problems from quantum chemistry and binary optimization problems, respectively. They are based on the idea of using a classical computer to train a parameterized quantum circuit. We show that the corresponding classical optimization problems are NP-hard. Moreover, the hardness is robust in the sense that, for every polynomial time algorithm, there are ins…
Variational quantum computing for quantum simulation: principles, implementations, and challenges
QuIIN - Quantum Industrial Innovation, Centro de Competência Embrapii Cimatec. SENAI CIMATEC（SENAI CIMATEC）；Universidade SENAI CIMATEC（SENAI CIMATEC大学）；Centro de Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia（巴伊亚联邦西大学）
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 …
Institute for Advanced Study（高等研究院）；Samsung Research（三星研究院）；Korea Institute for Advanced Study（韩国高等研究院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）
We explore the effectiveness of variational quantum circuits in simulating the ground states of quantum many-body Hamiltonians. We show that generic high-depth circuits, performing a sequence of layer unitaries of the same form, can accurately approximate the desired states. We demonstrate their universal success by using two Hamiltonian systems with very different properties: the transverse field Ising model and the Sachdev-Ye-Kitaev model. The energy landscape of the high-depth circuits has a proper structure for the gradient-based optimization, i.e. the presence of local extrema -- near any…
