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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(N) arising when the quantum adiabatic evolution algorithm is used to solve instances of the NP-Complete problem MAXCUT. It is known that the computational efficiency of this algorithm is largely determined by the large-N scaling behavior of Delta(N), and so determining this behavior is of fundamental significance. As density functional theory has been used to study quantum systems with N ~ 1000 interacting degrees of freedom, the approach introduced in this paper raises the realistic prospect of evaluating the gap Delta(N) for systems with N ~ 1000 qubits. Although the calculation of Delta(N) 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.
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Time-Dependent Density Functional Theory (TDDFT) has recently been extended to describe many-body open quantum systems (OQS) evolving under non-unitary dynamics according to a quantum master equation. In the master equation approach, electronic excitation spectra are broadened and shifted due to relaxation and dephasing of the electronic degrees of freedom by the surrounding environment. In this paper, we develop a formulation of TDDFT linear-response theory (LR-TDDFT) for many-body electronic systems evolving under a master equation, yielding broadened excitation spectra. This is done by mapping an interacting open quantum system onto a non-interacting open Kohn-Sham system yielding the correct non-equilibrium density evolution. A pseudo-eigenvalue equation analogous to the Casida equations of usual LR-TDDFT is derived for the Redfield master equation, yielding complex energies and Lamb shifts. As a simple demonstration, we calculate the spectrum of a C$^{2+}$ atom in an optical resonator interacting with a bath of photons. The performance of an adiabatic exchange-correlation kernel is analyzed and a first-order frequency-dependent correction to the bare Kohn-Sham linewidth based on Gorling-Levy perturbation theory is calculated.
It is well known that the quantum Zeno effect can protect specific quantum states from decoherence by using projective measurements. Here we combine the theory of weak measurements with stabilizer quantum error correction and detection codes. We derive rigorous performance bounds which demonstrate that the Zeno effect can be used to protect appropriately encoded arbitrary states to arbitrary accuracy, while at the same time allowing for universal quantum computation or quantum control.
Based on a generalization of Hohenberg-Kohns theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix $gamma$ as a natural variable but still recovers quantum correlations in an exact way it is particularly well-suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying $v$-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals for this $N$-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. The respective gradient forces are found to diverge in the regime of Bose-Einstein condensation, $ abla_{gamma} mathcal{F} propto 1/sqrt{1-N_{mathrm{BEC}}/N}$, providing a natural explanation for the absence of complete BEC in nature.
Transport through an Anderson junction (two macroscopic electrodes coupled to an Anderson impurity) is dominated by a Kondo peak in the spectral function at zero temperature. The exact single-particle Kohn-Sham potential of density functional theory reproduces the linear transport exactly, despite the lack of a Kondo peak in its spectral function. Using Bethe ansatz techniques, we calculate this potential exactly for all coupling strengths, including the cross-over from mean-field behavior to charge quantization caused by the derivative discontinuity. A simple and accurate interpolation formula is also given.
We show that braidings of the metaplectic anyons $X_epsilon$ in $SO(3)_2=SU(2)_4$ with their total charge equal to the metaplectic mode $Y$ supplemented with measurements of the total charge of two metaplectic anyons are universal for quantum computation. We conjecture that similar universal computing models can be constructed for all metaplectic anyon systems $SO(p)_2$ for any odd prime $pgeq 5$. In order to prove universality, we find new conceptually appealing universal gate sets for qutrits and qupits.
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