Efficient quantum algorithm for solving differential equations with Fourier nonlinearity via Koopman linearization
Judd Katz, Gopikrishnan Muraleedharan, Abhijeet Alase
TL;DR
This work develops quantum algorithms for solving high-dimensional nonlinear ODEs with Fourier-type nonlinearity by employing Koopman linearization in a Fourier basis, thereby overcoming polynomial-only limitations of Carleman linearization. The method rescales the problem, lifts it to an infinite-dimensional linear system, truncates to a finite order, solves via a truncated Taylor-series discretization, and expresses the requested observable as an expectation value of a unitary, enabling efficient readout. Rigorous error bounds are derived both for the linearization truncation (in $p$-norms) and for Taylor truncation, with distinct analyses for dissipative and non-dissipative regimes, and the overall complexity is shown to scale polylogarithmically with the problem size under suitable conditions. The approach broadens the class of nonlinear ODEs amenable to quantum speedups and includes integrated classical readouts, potentially impacting applications in physics, engineering, and dynamical systems where nonpolynomial nonlinearities arise.
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.