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. 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. For bounded values of V-representability, we present a polynomial time quantum algorithm to generate the time-dependent Kohn–Sham potential with controllable error bounds.
Highlights
Time-dependent density functional theory (TDDFT) is predicated on the use of the time-dependent density as the fundamental variable and all observables and properties are functionals of the density
The crux of the theoretical foundations of TDDFT is an inverse map which has as inputs the density at all times and the initial state
Practical computational approaches to TDDFT rely on constructing the non-interacting time-dependent Kohn–Sham potential
Summary
To introduce TDDFT and its Kohn–Sham formalism, it is instructive to view the Schrödinger equation as a map [10]. The crux of the theoretical foundations of TDDFT is an inverse map which has as inputs the density at all times and the initial state It outputs the potential and the wave function at later times t,. Practical computational approaches to TDDFT rely on constructing the non-interacting time-dependent Kohn–Sham potential. While we discuss the computation of the full Kohn–Sham potential from a given external potential and initial density, we will not construct an explicit functional for the exchange-correlation potential. ΦN (t)], is an antisymmetric combination of single particle wave functions, φi (t), such that for all times t, the Kohn–Sham density, nKS (t) = 〈n〉 Φ(t) = ∑iN= 1|φi (t)|2, matches the interacting density 〈n〉 Ψ(t) If such a map exists, we call the system V-representable while implicitly referring to noninteracting VKS-representablity. Previous rigorous results by Farzanehpour and Tokatly [17] on lattice TDDFT are directly applicable to our quantum computational setting
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