This paper establishes the applicability of density functional theory methods to quantum computing systems. We show that ground state and time-dependent density functional theory can be applied to quantum computing systems by proving the Hohenberg-Kohn and Runge-Gross theorems for a fermionic representation of an $N$ qubit system. As a first demonstration of this approach, time-dependent density functional theory is used to determine the minimum energy gap $\Delta \left(N\right)$ arising when the quantum adiabatic evolution algorithm is used to solve instances of the nondeterministic-polynomial-complete problem MAXCUT. It is known that the computational efficiency of this algorithm is largely determined by the large-$N$ scaling behavior of $\Delta \left(N\right)$, and so determining this behavior is of fundamental significance. As density functional theory has been used to study quantum systems with $N\sim {10}^3$ interacting degrees of freedom, the approach introduced in this paper raises the realistic prospect of evaluating the gap $\Delta \left(N\right)$ for systems with $N\sim {10}^3$ qubits. Although the calculation of $\Delta \left(N\right)$ serves to illustrate how density functional theory methods can be applied to problems in quantum computing, the approach has a much broader range and shows promise as a means for determining the properties of very large quantum computing systems.

• Received 24 February 2009

DOI:https://doi.org/10.1103/PhysRevB.79.205117