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 descent. We numerically simulate the performance of iCANS for the variational quantum eigensolver and for variational quantum compiling, with and without noise. In all cases, and especially in the noisy case, iCANS tends to out-perform state-of-the-art optimizers for VHQCAs. We therefore believe this adaptive optimizer will be useful for realistic VHQCA implementations, where the number of measurements is limited.
Quantum chemistry is a near-term application for quantum computers. This application may be facilitated by variational quantum-classical algorithms (VQCAs), although a concern for VQCAs is the large number of measurements needed for convergence, especially for chemical accuracy. Here we introduce a strategy for reducing the number of measurements (i.e., shots) by randomly sampling operators $h_i$ from the overall Hamiltonian $H = \sum_i c_i h_i$. In particular, we employ weighted sampling, which is important when the $c_i$'s are highly non-uniform, as is typical in chemistry. We integrate this…
Resource frugal optimizer for quantum machine learning
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Leiden University（莱顿大学）；Universidad Autónoma de Madrid（马德里自治大学）；University of Warsaw（华沙大学）；Normal Computing Corporation（诺默尔计算公司）
Variational quantum algorithms, which have risen to prominence in the noisy intermediate-scale quantum setting, require the implementation of a stochastic optimizer on classical hardware. To date, most research has employed algorithms based on the stochastic gradient iteration as the stochastic classical optimizer. In this work we propose instead using stochastic optimization algorithms that yield stochastic processes emulating the dynamics of classical deterministic algorithms. This approach results in methods with theoretically superior worst-case iteration complexities, at the expense of gr…
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 highe…
Variational Hybrid Quantum Classical Algorithms (VHQCAs) are a class of quantum algorithms intended to run on noisy intermediate-scale quantum (NISQ) devices. These algorithms employ a parameterized quantum circuit (ansatz) and a quantum-classical feedback loop. A classical device is used to optimize the parameters in order to minimize a cost function that can be computed far more efficiently on a quantum device. The cost function is constructed such that finding the ansatz parameters that minimize its value, solves some problem of interest. We focus specifically on the Variational Quantum Lin…
Variational quantum algorithms (VQAs) offer a promising path toward using near-term quantum hardware for applications in academic and industrial research. These algorithms aim to find approximate solutions to quantum problems by optimizing a parametrized quantum circuit using a classical optimization algorithm. A successful VQA requires fast and reliable classical optimization algorithms. Understanding and optimizing how off-the-shelf optimization methods perform in this context is important for the future of the field. In this work, we study the performance of four commonly used gradient-free…
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 algorithms are a leading candidate for early applications on noisy intermediate-scale quantum computers. These algorithms depend on a classical optimization outer-loop that minimizes some function of a parameterized quantum circuit. In practice, finite sampling error and gate errors make this a stochastic optimization with unique challenges that must be addressed at the level of the optimizer. The sharp trade-off between precision and sampling time in conjunction with experimental constraints necessitates the development of new optimization strategies to minimize overall wa…
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…
Variational quantum algorithms (VQAs) are promising methods that leverage noisy quantum computers and classical computing techniques for practical applications. In VQAs, the classical optimizers such as gradient-based optimizers are utilized to adjust the parameters of the quantum circuit so that the objective function is minimized. However, they often suffer from the so-called vanishing gradient or barren plateau issue. On the other hand, the normalized gradient descent (NGD) method, which employs the normalized gradient vector to update the parameters, has been successfully utilized in sever…
Efficient classical optimizers are crucial in practical implementations of Variational Quantum Algorithms (VQAs). In particular, to make Stochastic Gradient Descent (SGD) resource efficient, adaptive strategies have been proposed to determine the number of measurement shots used to estimate the gradient. However, existing strategies overlook the overhead that occurs in each iteration. In terms of wall-clock runtime, significant circuit-switching and communication latency can slow the optimization process when using a large number of iterations. In terms of economic cost when using cloud servic…
Banasthali Vidyapith（巴纳斯塔利大学）；Umm Al Quwain University（乌姆盖万大学）
In this paper, we have studied the performance and role of local optimizers in quantum variational circuits. We studied the performance of the two most popular optimizers and compared their results with some popular classical machine learning algorithms. The classical algorithms we used in our study are support vector machine (SVM), gradient boosting (GB), and random forest (RF). These were compared with a variational quantum classifier (VQC) using two sets of local optimizers viz AQGD and COBYLA. For experimenting with VQC, IBM Quantum Experience and IBM Qiskit was used while for classical ma…
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）
Variational quantum-classical hybrid algorithms are seen as a promising strategy for solving practical problems on quantum computers in the near term. While this approach reduces the number of qubits and operations required from the quantum machine, it places a heavy load on a classical optimizer. While often under-appreciated, the latter is a computationally hard task due to the barren plateau phenomenon in parameterized quantum circuits. The absence of guiding features like gradients renders conventional optimization strategies ineffective as the number of qubits increases. Here, we introduc…
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…
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…
Compact representations of fermionic Hamiltonians are necessary to perform calculations on quantum computers that lack error-correction. A fermionic system is typically defined within a subspace of fixed particle number and spin while unnecessary states are projected out of the Hilbert space. We provide a bijective mapping using combinatoric ranking to bijectively map fermion basis states to qubit basis states and express operators in the standard spin representation. We then evaluate compact mapping using the Variational Quantum Eigensolver (VQE) with the unitary coupled cluster singles and d…
For a large class of variational quantum circuits, we show how arbitrary-order derivatives can be analytically evaluated in terms of simple parameter-shift rules, i.e., by running the same circuit with different shifts of the parameters. As particular cases, we obtain parameter-shift rules for the Hessian of an expectation value and for the metric tensor of a variational state, both of which can be efficiently used to analytically implement second-order optimization algorithms on a quantum computer. We also consider the impact of statistical noise by studying the mean squared error of differen…
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 …
Numerical Optimization Strategies for the Variational Hamiltonian Ansatz in Noisy Quantum Environments
Gran Sasso Science Institute（格兰萨索科学研究所）；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）；Technical University of Munich（慕尼黑工业大学）；Fraunhofer Institute for Chemical Technology ICT（弗劳恩霍夫化学技术研究所）；Saarland University（萨尔兰大学）；University of L’Aquila（拉奎拉大学）
Rensselaer Polytechnic Institute（伦斯勒理工学院）；Department of Computer Science（计算机科学系）；Troy NY, USA（美国纽约州特洛伊市（此处仅为地址信息，根据提取规则应忽略，正确仅保留大学））
Quantum Variational Circuits (QVCs) are often claimed as one of the most potent uses of both near term and long term quantum hardware. The standard approaches to optimizing these circuits rely on a classical system to compute the new parameters at every optimization step. However, this process can be extremely challenging, due to the nature of navigating the exponentially scaling complex Hilbert space, barren plateaus, and the noise present in all foreseeable quantum hardware. Although a variety of optimization algorithms are employed in practice, there is often a lack of theoretical or empiri…
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…
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）
Combinatorial optimization on near-term quantum devices is a promising path to demonstrating quantum advantage. However, the capabilities of these devices are constrained by high noise or error rates. In this paper, we propose an iterative Layer VQE (L-VQE) approach, inspired by the Variational Quantum Eigensolver (VQE). We present a large-scale numerical study, simulating circuits with up to 40 qubits and 352 parameters, that demonstrates the potential of the proposed approach. We evaluate quantum optimization heuristics on the problem of detecting multiple communities in networks, for which …
Variational quantum algorithm for enhanced continuous variable optical phase sensing
Technical University of Denmark（丹麦技术大学）；University of Padova（帕多瓦大学）；Delft University of Technology（代尔夫特理工大学）；Harvard University（哈佛大学）；Freie Universität Berlin（柏林自由大学）
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…
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）
