Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

The current understanding of magnetohydrodynamic (MHD) turbulence envisions turbulent eddies which are anisotropic in all three directions. In the plane perpendicular to the local mean magnetic field, this implies that such eddies become current-sheetlike structures at small scales. We analyze the role of magnetic reconnection in these structures and conclude that reconnection becomes important at a scale, where S L is the outer-scale (L) Lundquist number and λ is the smallest of the fieldperpendicular eddy dimensions. This scale is larger than the scale set by the resistive diffusion of eddies, therefore implying a fundamentally different route to energy dissipation than that predicted by the Kolmogorov-like phenomenology. In particular, our analysis predicts the existence of the subinertial, reconnection interval of MHD turbulence, with the estimated scaling of the Fourier energy spectrum⊥ , where k ⊥ is the wave number perpendicular to the local mean magnetic field. The same calculation is also performed for high (perpendicular) magnetic Prandtl number plasmas (Pm), where the reconnection scale is found to be λ=L ∼ S Introduction.-Turbulence is a defining feature of magnetized plasmas in space and astrophysical environments, which are almost invariably characterized by very large Reynolds numbers. The solar wind [1], the interstellar medium [2,3], and accretion disks [4,5] are prominent examples of plasmas dominated by turbulence, where its detailed understanding is almost certainly key to addressing long-standing puzzles such as electron-ion energy partition, cosmic ray acceleration, magnetic dynamo action, and momentum transport.Weak collisionality implies that kinetic plasma physics is required to fully describe turbulence in many such environments [6]. However, turbulent motions at scales ranging from the system size to the ion kinetic scales, an interval which spans many orders of magnitude, should be accurately described by magnetohydrodynamics (MHD).The current theoretical understanding of MHD turbulence largely rests on the ideas that were put forth by Kolmogorov and others to describe turbulence in neutral fluids (the K41 theory of turbulence [7]), and then adapted to magnetized plasmas by Iroshnikov and Kraichnan [8,9] and, later, Goldreich and Sridhar (GS95) [10]. Very briefly, one considers energy injection at some large scale L (the forcing, or outer, scale), which then cascades to smaller scales through the inertial range where, by definition, dissipation is negligible and throughout which, therefore, energy is conserved. At the bottom of the cascade is the dissipation range, where the gradients in the flow are sufficiently large for the dissipation to be efficient.Turbulence in magnetized plasmas fundamentally differs from that in neutral fluids due to the intrinsic anisotropy