Eigenvalue estimation of differential operators with a quantum algorithm

We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions psi(x(1),x(2),...,x(D)) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy Theta(1/N-2) is Theta((2(S+1)(1+1/nu)+D)ln N) qubits and O(N(2(S+1)(1+1/nu))ln(c) N-D) gate operations, where N is the number of points to which each argument is discretized, nu and c are implementation dependent constants of O(1). Optimal classical methods require Theta(N-D) bits and Omega(N-D) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D > 2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.