Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations

TL;DR

This paper introduces a novel variational quantum algorithm using Lagrange polynomial encoding and derivative circuits to efficiently solve differential equations, demonstrating reduced gate complexity and accurate solutions for classical DE problems.

Contribution

The paper presents two new architectures of a variational quantum algorithm that improve efficiency and solution quality for differential equations using polynomial encoding and derivative circuits.

Findings

Reduced gate complexity compared to previous algorithms

Accurate solutions for classical differential equations

Demonstrated on mass-spring and Poisson equations

Abstract

Differential equations (DEs) serve as the cornerstone for a wide range of scientific endeavors, their solutions weaving through the core of diverse fields such as structural engineering, fluid dynamics, and financial modeling. DEs are notoriously hard to solve, due to their intricate nature, and finding solutions to DEs often exceeds the capabilities of traditional computational approaches. Recent advances in quantum computing have triggered a growing interest from researchers for the design of quantum algorithms for solving DEs. In this work, we introduce two different architectures of a novel variational quantum algorithm (VQA) with Lagrange polynomial encoding in combination with derivative quantum circuits using the Hadamard test differentiation to approximate the solution of DEs. To demonstrate the potential of our new VQA, two well-known ordinary differential equations are used: the damped mass-spring system from a given initial condition and the Poisson equation for periodic, Dirichlet, and Neumann boundary conditions. It is shown that the proposed new VQA has a reduced gate complexity compared to previous variational quantum algorithms, for a similar or better quality of the solution.