12/26/2023 0 Comments Nonlinear difference equationThese quantum algorithms, however, are applicable only to certain types of differential equations. There are several quantum algorithms that solve differential equations by producing a quantum state proportional to the solution. Popular summaryÄifferential equations are an important part of many physics models from high-energy physics to fluid dynamics and plasma physics. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas Liu et al., (2021) and Xue et al., (2021) additionally require normality. This kind of logarithmic dependence on error has also been achieved by Xue et al., (2021), but only for homogeneous nonlinear equations. First, we obtain an exponentially better dependence on error. The improvement over that result is two-fold. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in Liu et al., (2021)). The algorithm here is also exponentially faster than the bounds derived in Berry et al., (2017) for certain classes of diagonalizable matrices. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. In Berry et al., (2017), a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE).
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