Eigenvalue estimation of differential operators with a quantum algorithm (11 pages)

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 x1,x2,...,xD with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy 1/N 2 is 2S+11+1/+Dln N qubits and ON 2S+11+1/ ln c N D gate operations, where N is the number of points to which each argument is discretized, and c are implementation dependent constants of O1. Optimal classical methods require N D bits and 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 D2S +11+1/. 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.