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 gradients. Furthermore, it is not clear how "good" the minima are in terms of generalization and stability against perturbations of the data and there is, therefore, a need for tools to quantitatively study the convergence of the VQCs. In this work, we introduce a way to compute the Hessian of the loss function of VQCs and show how to characterize the loss landscape with it. The eigenvalues of the Hessian give information on the local curvature and we discuss how this information can be interpreted and compared to classical neural networks. We benchmark our results on several examples, starting with a simple analytic toy model to provide some intuition about the behavior of the Hessian, then going to bigger circuits, and also train VQCs on data. Finally, we show how the Hessian can be used to adjust the learning rate for faster convergence during the training of variational circuits.
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）
Energy Landscape Plummeting in Variational Quantum Eigensolver: Subspace Optimization, Non-iterative Corrections and Generator-informed Initialization for Improved Quantum Efficiency
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）
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…
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…
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…
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…
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…
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）
We propose the simulation of quantum-optical systems in the ultrastrong-coupling regime using a variational quantum algorithm. More precisely, we introduce a short-depth variational form to prepare the groundstate of the multimode Dicke model on a quantum processor and present proof-of-principle results obtained via cloud access to an IBM device. We moreover provide an algorithm for characterizing the groundstate by Wigner state tomography. Our work is a first step towards digital quantum simulation of quantum-optical systems with potential applications to the spin-boson, Kondo and Jahn-Teller…
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…
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）
We propose a theory of characterizing quantum circuits with qubit functional configurations. Any quantum circuit can be decomposed into alternating sequences of 1-qubit unitary gates and CNOT gates. Each CNOT sequence prepares the current quantum state into a layer of qubit functional configuration to specify the rule for the next 1-qubit unitary sequence on how to collectively modify the state vector entries. All the functional configuration layers on a quantum circuit define its type which can include many other circuits sharing the same configuration layers. Studying the functional configur…
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…
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）
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…
Exploring critical states of the quantum Rabi model via Hamiltonian variational ansätze
South China Normal University（华南师范大学）；Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education)（原子与亚原子结构及量子控制教育部重点实验室）；Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter（广东省物质结构与基本相互作用卓越基础研究中心）；School of Physics, South China Normal University（华南师范大学物理学院）；Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials（广东省量子工程与量子材料重点实验室）；Guangdong-Hong Kong Joint Laboratory of Quantum Matter（广东-香港量子物质联合实验室）；Frontier Research Institute for Physics, South China Normal University（华南师范大学前沿物理研究院）
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…
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…
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）
Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum many-body states in analog quantum simulation. We describe a protocol where spatial deformations of the many-body Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of Fermi-Hubbard models in quasi-1D geometries, fi…
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）
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…
