Improved quantum algorithms for linear and nonlinear differential equations

Hari Krovi

Riverlane Research, Cambridge, MA

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We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). 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. In [1], a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices. The algorithm here is also exponentially faster than the bounds derived in [1] for certain classes of diagonalizable matrices.

Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in [2]). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.

Differential equations are an important part of many physics models from high-energy physics to fluid dynamics and plasma physics. There are several quantum algorithms that solve differential equations by producing a quantum state proportional to the solution. These quantum algorithms, however, are applicable only to certain types of differential equations. Specifically, for linear ODEs, they impose conditions such as normality or diagonalizability on the matrix $A$ encoding the linear ODE. This work develops quantum algorithms that can be applied to a substantially larger class of linear and nonlinear ordinary differential equations. We remove the condition of diagonalizability and replace it with one that has been studied in the theory of stability of differential equations, namely the norm of the exponential of the matrix $A$. This can then be used to give a quantum algorithm that applies to larger class of nonlinear differential equations as well.

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