We compare the BFGS optimizer, ADAM and Natural Gradient Descent (NatGrad) in the context of Variational Quantum Eigensolvers (VQEs). We systematically analyze their performance on the QAOA ansatz for the Transverse Field Ising Model (TFIM) as well as on overparametrized circuits with the ability to break the symmetry of the Hamiltonian. The BFGS algorithm is frequently unable to find a global minimum for systems beyond about 20 spins and ADAM easily gets trapped in local minima. On the other hand, NatGrad shows stable performance on all considered system sizes, albeit at a significantly higher cost per epoch. In sharp contrast to most classical gradient based learning, the performance of all optimizers is found to decrease upon seemingly benign overparametrization of the ansatz class, with BFGS and ADAM failing more often and more severely than NatGrad. Additional tests for the Heisenberg XXZ model corroborate the accuracy problems of BFGS in high dimensions, but they reveal some shortcomings of NatGrad as well. Our results suggest that great care needs to be taken in the choice of gradient based optimizers and the parametrization for VQEs.
Variational quantum eigensolver (VQE), which attracts attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called ansatz. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the ansatz, it has been difficult to analyze the performance of optimizers for the VQE systematically. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achi…
Variational-Quantum-Eigensolver(VQE) method on a quantum computer is a well-known hybrid algorithm to solve the eigenstates and eigenvalues that uses both quantum and classical computers. This method has the potential to solve quantum chemical simulation including polymer and complex optimization problems that are never able to be solved in a realistic time. Though they are many papers on VQE, there are many hurdles before practical application. Therefore, we tried to evaluate VQE methods with Genetic Algorithms(GA). In this paper, we propose the VQE method with GA. We selected ground and exci…
The variational quantum eigensolver is a hybrid algorithm composed of quantum state driving and classical parameter optimization, for finding the ground state of a given Hamiltonian. The natural gradient method is an optimization method taking into account the geometric structure of the parameter space. Very recently, Stokes et al. developed the general method for employing the natural gradient for the variational quantum eigensolver. This paper gives some simple case-studies of this optimization method, to see in detail how the natural gradient optimizer makes use of the geometric property to…
Statistical Benchmarking of Optimization Methods for Variational Quantum Eigensolver under Quantum Noise
IT4Innovations National Supercomputing Center, VSB - Technical University of Ostrava（IT4Innovations国家超级计算中心，俄斯特拉发理工大学）；Gran Sasso Science Institute（格兰萨索科学研究所）；Technical University of Munich（慕尼黑工业大学）；VSB-Technical University of Ostrava（俄斯特拉发理工大学）；Marine Research Institute, Klaipeda University（克莱佩达大学海洋研究所）；Université Montpellier, CNRS, ENSCM（蒙彼利埃大学，法国国家科学研究中心，国立高等化学学院）；Università dell’Aquila（阿奎拉大学）
Optimizing Unitary Coupled Cluster Wave Functions on Quantum Hardware: Error Bound and Resource-Efficient Optimizer
Eviden Quantum Lab（Eviden量子实验室）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；École Polytechnique, I.P. Paris（巴黎理工学院高等师范学校）；MajuLab, CNRS–UCA-SU-NUS-NTU International Joint Research Laboratory（MajuLab，CNRS-南法大学-苏黎世联邦理工学院-新加坡国立大学-南洋理工大学国际联合研究实验室）
Quantum computers are expected to be highly beneficial for chemistry simulations, promising significant improvements in accuracy and speed. The most prominent algorithm for chemistry simulations on NISQ devices is the Variational Quantum Eigensolver (VQE). It is a hybrid quantum-classical algorithm which calculates the ground state energy of a Hamiltonian based on parametrized quantum circuits, while a classical optimizer is used to find optimal parameter values. However, quantum hardware is affected by noise, and it needs to be understood to which extent it can degrade the performance of the …
Quantum simulation of quantum chemistry is one of the most compelling applications of quantum computing. It is of particular importance in areas ranging from materials science, biochemistry and condensed matter physics. Here, we propose a full quantum eigensolver (FQE) algorithm to calculate the molecular ground energies and electronic structures using quantum gradient descent. Compared to existing classical-quantum hybrid methods such as variational quantum eigensolver (VQE), our method removes the classical optimizer and performs all the calculations on a quantum computer with faster converg…
Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimizer Methods
National Tsing Hua University（国立清华大学）；University of Science, Vietnam National University（越南国立大学科学大学）；Forschungszentrum Jülich（于利希研究中心）；Phenikaa University（Phenikaa 大学）
We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. Using a massively parallel universal quantum computer simulator, we observe that a low-depth-circuit ansatz advantageously exploits the efficiently preparable Néel initial state, avoids potential barren plateaus, and works for both one- and two-dimensional lattices. The analysis reflects the decisive ingredients required for a simulation by comparing different ansätze, initial parameters, and gradient-based versu…
Variational hybrid quantum-classical algorithms (VHQCAs) have the potential to be useful in the era of near-term quantum computing. However, recently there has been concern regarding the number of measurements needed for convergence of VHQCAs. Here, we address this concern by investigating the classical optimizer in VHQCAs. We introduce a novel optimizer called individual Coupled Adaptive Number of Shots (iCANS). This adaptive optimizer frugally selects the number of measurements (i.e., number of shots) both for a given iteration and for a given partial derivative in a stochastic gradient desc…
We present a collection of optimizers tuned for usage on Noisy Intermediate-Scale Quantum (NISQ) devices. Optimizers have a range of applications in quantum computing, including the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization (QAOA) algorithms. They are also used for calibration tasks, hyperparameter tuning, in machine learning, etc. We analyze the efficiency and effectiveness of different optimizers in a VQE case study. VQE is a hybrid algorithm, with a classical minimizer step driving the next evaluation on the quantum processor. While most results to date conc…
Bias Analysis and Regularization of Sequential Minimal Optimization in Variational Quantum Eigensolvers
Berlin Institute for the Foundations of Learning and Data BIFOLD（柏林学习与数据基础研究所）；Technische Universität Berlin（柏林工业大学）；Padova University（帕多瓦大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器中心）；Computation-Based Science and Technology Research Center, The Cyprus Institute（塞浦路斯研究所计算科学与技术研究中心）；RIKEN AIP（理化学研究所先进智能计算中心）
Variational quantum algorithms aim at harnessing the power of noisy intermediate-scale quantum computers, by using a classical optimizer to train a parameterized quantum circuit to solve tractable quantum problems. The variational quantum eigensolver is one of the aforementioned algorithms designed to determine the ground-state of many-body Hamiltonians. Here, we apply the variational quantum eigensolver to study the ground-state properties of $N$-component fermions. With such knowledge, we study the persistent current of interacting SU($N$) fermions, which is employed to reliably map out the …
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values which do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated vi…
The development of quantum algorithms to solve quantum chemistry problems has offered a promising new paradigm of performing computer simulations at the scale of atoms and molecules. Although majority of the research so far has focused on designing quantum algorithms to compute ground and excited state energies and forces, it is useful to run different simulation tasks, such as geometry optimization, with these algorithms as subroutines. Towards this end, we have created an interface for the Variational Quantum Eigensolver based molecular Energy (VQE-E) and molecular Force (VQE-F) code to the …
Efficient Operator Selection and Warm-Start Strategy for Excitations in Variational Quantum Eigensolvers
ICAMS, Ruhr University Bochum（鲁尔大学波鸿校区计算物理与材料科学研究所）；German Aerospace Center (DLR), Institute for Frontier Materials on Earth and in Space（德国航空航天中心前沿地球与空间材料研究所）
By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to …
Probing the Ground State of the Antiferromagnetic Heisenberg Model on the Kagome Lattice using Geometrically Informed Variational Quantum Eigensolver
Constantine Quantum Technologies（君士坦丁量子技术）；Frères Mentouri University Constantine 1（费雷斯门图里大学君士坦丁1）；Laboratoire de Physique Mathématique et Subatomique（数学与亚原子物理实验室）；Theoretical Physics Laboratory（理论物理实验室）
Diffusion-Enhanced Optimization of Variational Quantum Eigensolver for General Hamiltonians
School of Physics, Beihang University（北京航空航天大学物理学院）；School of Physics, Beijing Institute of Technology（北京理工大学物理学院）；School of Instrument Science and Opto-Electronics Engineering, Beijing Information Science and Technology University（北京信息科技大学仪器科学与光电工程学院）
Prior-Informed Adaptive Shifts for Sequential Minimal Optimization in Variational Quantum Eigensolvers
Berlin Institute for the Foundations of Learning and Data BIFOLD（柏林学习数据基础研究所）；Technische Universität Berlin（柏林工业大学）；Padova University（帕多瓦大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步辐射中心）；Computation-Based Science and Technology Research Center, The Cyprus Institute（塞浦路斯研究院计算科学与技术研究中心）；RIKEN AIP（理化学研究所）
We perform a systematic investigation of variational forms (wave function Ansätze), to determine the ground state energies and properties of two-dimensional model fermionic systems on triangular lattices (with and without periodic boundary conditions), using the Variational Quantum Eigensolver (VQE) algorithm. In particular, we focus on the nature of the entangler blocks which provide the most efficient convergence to the system ground state inasmuch as they use the minimal number of gate operations, which is key for the implementation of this algorithm in NISQ computers. Using the concurrence…
Energy Landscape Plummeting in Variational Quantum Eigensolver: Subspace Optimization, Non-iterative Corrections and Generator-informed Initialization for Improved Quantum Efficiency
The iterative quantum phase estimation algorithm, applied to calculating the ground state energies of quantum chemical systems, is theoretically appealing in its wide scope of being able to handle both weakly and strongly correlated regimes. However, the goodness of the initial state that is sent as an input to the algorithm could strongly decide the quality of the results obtained. In this work, we generate the initial state by using the classical-quantum hybrid variational quantum eigensolver algorithm with unitary coupled cluster ansatz. We apply the procedure to obtain the ground state ene…
Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum machines. VQE aims to approximate the lowest eigenvalue of an exponentially sized matrix in polynomial time. It minimizes quantum resource requirements both by co-processing with a classical processor and by structuring computation into many subproblems. Each quantum subproblem involves a separate state preparation terminated by the measurement of one Pauli string. However, the number of such Pauli strings scales as $N^4$ for typical problems of interest--a daunting growth rate that poses a serious li…
Variational Quantum optimization algorithms, such as the Variational Quantum Eigensolver (VQE) or the Quantum Approximate Optimization Algorithm (QAOA), are among the most studied quantum algorithms. In our work, we evaluate and improve an algorithm based on VQE, which uses exponentially fewer qubits compared to the QAOA. We highlight the numerical instabilities generated by encoding the problem into the variational ansatz and propose a classical optimization procedure to find the ground-state of the ansatz in less iterations with a better or similar objective. Furthermore, we compare classica…
The variational quantum eigensolver is one of the most promising approaches for performing chemistry simulations using noisy intermediate-scale quantum (NISQ) processors. The efficiency of this algorithm depends crucially on the ability to prepare multi-qubit trial states on the quantum processor that either include, or at least closely approximate, the actual energy eigenstates of the problem being simulated while avoiding states that have little overlap with them. Symmetries play a central role in determining the best trial states. Here, we present efficient state preparation circuits that r…
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for current and near-term quantum devices. Despite its initial success, there is a lack of understanding involving several of its key aspects. There are problems with VQE that forbid a favourable scaling towards quantum advantage. In order to alleviate the problems, we propose and extensively test a quantum annealing inspired heuristic that supplements VQE. The improved VQE enables an efficient initial state preparation mechanism, in a recursive manner, for a quasi-dynamical unitary evolution. We conduct…
Benchmarking VQE Configurations: Architectures, Initializations, and Optimizers for Silicon Ground State Energy
eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE（纽约大学阿布扎赫（NYUAD））；Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE（纽约大学阿布扎赫研究机构）
