Eigenvalue estimation of differential operators with a quantum algorithm

We demonstrate how linear differential operators could be emulated by a\nquantum processor, should one ever be built, using the Abrams-Lloyd algorithm.\nGiven a linear differential operator of order 2S, acting on functions\npsi(x_1,x_2,...,x_D) with D arguments, the computational cost required to\nestimate a low order eigenvalue to accuracy Theta(1/N^2) is\nTheta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c)\ngate operations, where N is the number of points to which each argument is\ndiscretized, nu and c are implementation dependent constants of O(1). Optimal\nclassical methods require Theta(N^D) bits and Omega(N^D) gate operations to\nperform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby\nleads to exponential reduction in memory and polynomial reduction in gate\noperations, provided the domain has sufficiently large dimension D >\n2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy\nestimation of two or more particles can in principle be performed with fewer\nquantum mechanical gates than classical gates.\n