Efficient Quantum Solution of Differential Equations

We demonstrate how differential equations can be mapped onto a quantum mechanical system, and can thus be emulated by a quantum processor. We specify the class of differential equations for which computation can be achieved with an exponential reduction in memory space required, and a power law reduction in time resources compared to classical simulation. For example, we find Maxwell's electromagnetic equations can be emulated with logarithmic cost in the total number of grid points, and a time speed up of O(N^3) compared to a classical finite difference scheme, in calculating electromagnetic mode frequencies of resonant structures, where N is the total number of grid points along a Cartesian axis. Numerically, a problem with 10^16 grid points would require only ~70 logical qubits, and could be computed in only ~10^7 iterative steps, while the classical solution would require ~10^21 iterative steps.