Efficient quantum algorithm for solving differential equations with Fourier nonlinearity via Koopman linearization

Abstract

Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original nonlinear ODE into a higher-dimensional linear ODE using a linearization technique, most commonly Carleman linearization. Existing works restrict their analysis to ODEs where the nonlinearities are polynomial functions of the dependent variables, significantly limiting their applicability. In this work we construct an efficient quantum algorithm for solving ODEs with `Fourier' nonlinear terms expressible as $d{\bf u}/dt = G_0 + G_1 e^{i{\bf u}}$, where ${\bf u}$ denotes a vector of $n$ complex variables evolving with $t$, $G_0$ is an $n$-dimensional complex vector, $G_1$ is an $n \times n$ complex matrix and $e^{i{\bf u}}$ denotes the vector with entries $\{e^{iu_j}\}$. To tackle the Fourier nonlinear term, which is not expressible as a finite sum of polynomials of ${\bf u}$, our algorithm employs a generalization of the Carleman linearization technique known as Koopman linearization. We also make other methodological advances towards relaxing the stringent dissipativity condition required for efficient solution extraction and towards integrated readout of classical quantities from the solution state. Our results open avenues to the development of efficient quantum algorithms for a significantly wider class of high-dimensional nonlinear ODEs, thereby broadening the scope of their applications.