Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations under the condition R<1, where R measures the ratio of nonlinearity to dissipation using the ℓ2 norm. Here we develop an efficient quantum algorithm based on [1] for reaction-diffusion equations, a class of nonlinear partial differential equations (PDEs). To achieve this, we improve upon the Carleman linearization approach introduced in [1] to obtain a faster convergence rate under the condition RD<1, where RD measures the ratio of nonlinearity to dissipation using the ℓ∞ norm. Since RD is independent of the number of spatial grid points n while R increases with n, the criterion RD<1 is significantly milder than R<1 for high-dimensional systems and can stay convergent under grid refinement for approximating PDEs. As applications of our quantum algorithm we consider the Fisher-KPP and Allen-Cahn equations, which have interpretations in classical physics. In particular, we show how to estimate the mean square kinetic energy in the solution by postprocessing the quantum state that encodes it to extract derivative information.

1 aAn, Dong1 aFang, Di1 aJordan, Stephen1 aLiu, Jin-Peng1 aLow, Guang, Hao1 aWang, Jiasu uhttps://arxiv.org/abs/2205.0114101292nas a2200157 4500008004100000245012000041210006900161260000800230300000800238490000600246520080400252100001301056700001301069700001301082856003901095 2022 eng d00aTime-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation0 aTimedependent Hamiltonian Simulation of Highly Oscillatory Dynam capr a6900 v63 aWe propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

1 aAn, Dong1 aFang, Di1 aLin, Lin uhttps://arxiv.org/abs/2111.03103v2