A Comparison of Various Classical Optimizers for a Variational Quantum Linear Solver

Abstract

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 Linear Solver (VQLS), and examine the effect of several gradient-free and gradient-based classical optimizers on performance. We focus on both the average rate of convergence of the classical optimizers studied, as well as the distribution of their average termination cost values, and how these are affected by noise. Our work demonstrates that realistic noise levels on NISQ devices present a challenge to the optimization process. All classical optimizers appear to be very negatively affected by the presence of realistic noise. If noise levels are significantly improved, there may be a good reason for preferring gradient-based methods in the future, which performed better than the gradient-free methods with the only shot-noise present. The gradient-free optimizers, Simultaneous Perturbation Stochastic Approximation (SPSA) and Powell's method, and the gradient-based optimizers, AMSGrad and BFGS performed the best in the noisy simulation, and appear to be less affected by noise than the rest of the methods. SPSA appears to be the best performing method. COBYLA, Nelder-Mead and Conjugate-Gradient methods appear to be the most heavily affected by noise, with even slight noise levels significantly impacting their performance.

arXiv comment17 pages; pre-accepted version

References in corpus (15)

• Adam: A Method for Stochastic Optimization
• Quantum Computing in the NISQ era and beyond
• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• A variational eigenvalue solver on a quantum processor
• Quantum algorithm for solving linear systems of equations
• Quantum Circuit Learning
• On the Convergence of Adam and Beyond
• An approximate Fourier transform useful in quantum factoring
• Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer
• Performance of hybrid quantum/classical variational heuristics for combinatorial optimization
• Classical Optimizers for Noisy Intermediate-Scale Quantum Devices
• Using models to improve optimizers for variational quantum algorithms
• Quantum linear systems algorithms: a primer
• Near-term quantum algorithms for linear systems of equations
• Robust and efficient algorithms for high-dimensional black-box quantum optimization

Cited by in corpus (35)

• The Variational Quantum Eigensolver: a review of methods and best practices
• A Review on Quantum Approximate Optimization Algorithm and its Variants
• A case study of variational quantum algorithms for a job shop scheduling problem
• Benchmarking of Different Optimizers in the Variational Quantum Algorithms for Applications in Quantum Chemistry
• Nuclear shell-model simulation in digital quantum computers
• Solving hadron structures using the basis light-front quantization approach on quantum computers
• Application of a variational hybrid quantum-classical algorithm to heat conduction equation
• QAOA Performance in Noisy Devices: The Effect of Classical Optimizers and Ansatz Depth
• A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer
• An Empirical Comparison of Optimizers for Quantum Machine Learning with SPSA-based Gradients
• A stabilizer framework for Contextual Subspace VQE and the noncontextual projection ansatz
• Bridging Paradigms: Designing for HPC-Quantum Convergence
• Simulation of a Variational Quantum Perceptron using Grover's Algorithm
• Computing Graph Edit Distance with Algorithms on Quantum Devices
• A quantum computing concept for 1-D elastic wave simulation with exponential speedup
• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach
• Variational quantum computing for quantum simulation: principles, implementations, and challenges
• Quantum Algorithms for Geologic Fracture Networks
• Near Term Algorithms for Linear Systems of Equations
• A near-term quantum simulation of the transverse field Ising model hints at Glassy Dynamics
• Simulating thermodynamic properties of dinuclear metal complexes using Variational Quantum Algorithms
• Scalability Challenges in Variational Quantum Optimization under Stochastic Noise
• BCQQ: Batch-Constraint Quantum Q-Learning with Cyclic Data Re-uploading
• Reducing measurement costs by recycling the Hessian in adaptive variational quantum algorithms
• Efficient parameterised compilation for hybrid quantum programming
• Generalized Probabilistic Approximate Optimization Algorithm
• Exploring new variational quantum circuit ansatzes for solving $SU(2)$ matrix models
• Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations
• A Noise-Aware Scalable Subspace Classical Optimizer for the Quantum Approximate Optimization Algorithm
• A low-circuit-depth quantum computing approach to the nuclear shell model
• Quantum natural gradient with thermal-state initialization
• A quantum approach for optimal control
• Resource-efficient quantum algorithm for linear systems of equations
• Interpolation-based coordinate descent method for parameterized quantum circuits
• Nonlinear path-following via the asymptotic numerical method on a quantum processor