Computational complexity of time-dependent density functional theory

Abstract: Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving dynamical many-body problems in physics and chemistry. The mathematical foundations of TDDFT are established through the formal existence of a fictitious non-interacting system (known as the Kohn-Sham system), which can reproduce the one-electron reduced probability density of the actual system. We build upon these works and show that on the interior of the domain of existence, the Kohn-Sham system can be efficiently obtained given the time-dependent density. Since a quantum computer can efficiently produce such time-dependent densities, we present a polynomial time quantum algorithm to generate the time-dependent Kohn-Sham potential with controllable error bounds. As a consequence, in contrast to the known intractability result for ground state density functional theory (DFT), the computation of the necessary time-dependent potentials given the initial state is in the complexity class described by bounded error quantum computation in polynomial time (BQP).