A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer

Abstract

Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE -- the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices -- especially when considering large system sizes and more complicated non-linear PDEs that are required in order to be competitive with classical high-end solvers.

arXiv comment22 pages, 18 figures, 2 tables

References in corpus (19)

• A variational eigenvalue solver on a quantum processor
• Quantum algorithm for solving linear systems of equations
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• DGM: A deep learning algorithm for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• Error mitigation for short-depth quantum circuits
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• Scalable mitigation of measurement errors on quantum computers
• Quantum algorithms and the finite element method
• Sequential minimal optimization for quantum-classical hybrid algorithms
• Variational Quantum State Eigensolver
• Variational Quantum algorithm for Poisson equation
• Avoiding barren plateaus using classical shadows
• A Comparison of Various Classical Optimizers for a Variational Quantum Linear Solver
• Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information
• Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation
• Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
• Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation

Cited by in corpus (10)

• Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits
• Incompressible Navier-Stokes solve on noisy quantum hardware via a hybrid quantum-classical scheme
• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach
• A variational quantum algorithm for tackling multi-dimensional Poisson equations with inhomogeneous boundary conditions
• Quantum Dynamics Simulation of the Advection-Diffusion Equation
• Quantum circuits for partial differential equations in Fourier space
• An Efficient Decomposition of the Carleman Linearized Burgers' Equation
• Nonlinear path-following via the asymptotic numerical method on a quantum processor
• Performance evaluation of variational quantum eigensolver and quantum dynamics algorithms on the advection-diffusion equation
• A variational quantum eigensolver tailored to multi-band tight-binding simulations of electronic structures