An Efficient Decomposition of the Carleman Linearized Burgers' Equation

TL;DR

This paper introduces a novel polylogarithmic decomposition method for efficiently loading the Carleman linearized Burgers' equation onto a quantum computer, enabling accurate solutions with reduced resource requirements.

Contribution

The paper presents the first explicit polylogarithmic data loading method for Carleman linearized systems, combining linearization, matrix decomposition, and quantum algorithms.

Findings

Matrix decomposition scales polylogarithmically with system size.

Numerical simulations demonstrate accurate quantum solutions.

Resource estimates show reduced gate complexity for implementation.

Abstract

Herein, we present a polylogarithmic decomposition method to load the matrix from the linearized 1-dimensional Burgers' equation onto a quantum computer. First, we use the Carleman linearization method to map the nonlinear Burgers' equation into an infinite linear system of equations, which is subsequently truncated to order $\alpha$. This new finite linear system is then embedded into a larger system of equations with the key property that its matrix can be decomposed into a linear combination of $\mathcal{O}(\log n_t + \alpha^2\log n_x)$ terms for $n_t$ time steps and $n_x$ spatial grid points. While the terms in this linear combination are not unitary, each can be implemented using a simple block encoding procedure. A numerical simulation is performed by combining our approach with the variational quantuam linear solver demonstrating that accurate solutions are possible. Finally, a resource estimate shows that the upper bound of the Clifford and T gate counts scale like $\mathcal{O}(\alpha(\log n_x)^2)$ and $\mathcal{O}((\log n_x)^2)$, respectively. This is therefore the first explicit polylogarithmic data loading method with respect to $n_x$ and $n_t$ for a Carleman linearized system.