Complexity of Time-dependent density functional theory
Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving fermionic 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 previous 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. We introduce a V-representability parameter which diverges at the boundary of the existence domain and serves to quantify the numerical difficulty of constructing the Kohn-Sham potential. The present work, in addition to contributing to on-going research about the foundations of TDDFT, is the latest application of quantum computational complexity theory to a growing list of problems in the physics and chemistry community.