Application of a variational hybrid quantum-classical algorithm to heat conduction equation
arXiv:2207.14630 · doi:10.1063/5.0121778
Abstract
The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm (VQA) leads a promising direction for solving partial differential equations on Noisy Intermediate Scale Quantum (NISQ) devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still in an early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of VQLS implementation are discussed by various test instances of linear systems. Finally, the successful statevector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present algorithm by proof-of-concept results. In addition, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.
arXiv comment36 pages, 19 figures
References in corpus (23)
- Adam: A Method for Stochastic Optimization
- Quantum Computing in the NISQ era and beyond
- Supplementary information for "Quantum supremacy using a programmable superconducting processor"
- Quantum algorithm for solving linear systems of equations
- Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
- Synthesis of Quantum Logic Circuits
- Variational quantum algorithms for nonlinear problems
- PennyLane: Automatic differentiation of hybrid quantum-classical computations
- Quantum algorithm for linear differential equations with exponentially improved dependence on precision
- Solving nonlinear differential equations with differentiable quantum circuits
- Quantum algorithms and the finite element method
- Variational algorithms for linear algebra
- Quantum Algorithm for Simulating the Wave Equation
- A Comparison of Various Classical Optimizers for a Variational Quantum Linear Solver
- Hybrid quantum linear equation algorithm and its experimental test on IBM Quantum Experience
- Quantum Fast Poisson Solver: the algorithm and modular circuit design
- Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines
- A hybrid classical-quantum approach to solve the heat equation using quantum annealers
- Quantum vs. classical algorithms for solving the heat equation
- Quantum Finite Volume Method for Computational Fluid Dynamics with Classical Input and Output
- Towards Solving the Navier-Stokes Equation on Quantum Computers
- Hybrid algorithms to solve linear systems of equations with limited qubit resources
- Hybrid Quantum-Classical Algorithm for Hydrodynamics
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- Variational Quantum Simulation of Partial Differential Equations: Applications in Colloidal Transport
- Variational quantum algorithm for generalized eigenvalue problems and its application to the finite element method
- Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm
- Variational Quantum Crank-Nicolson and Method of Lines for the Solution of Initial Value Problems
- qlbm -- A Quantum Lattice Boltzmann Software Framework
- Quantum circuits for partial differential equations in Fourier space
- Resource-efficient quantum algorithm for linear systems of equations