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arXiv:0809.1170 · Physical Review B

Density functional theory and quantum computation

Frank Gaitan, Franco Nori

Published May 26, 2009• 18 citations
arXivHTMLPDF

Abstract

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 $\ensuremath{\Delta}(N)$ 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 $\ensuremath{\Delta}(N)$, and so determining this behavior is of fundamental significance. As density functional theory has been used to study quantum systems with $N\ensuremath{\sim}{10}^{3}$ interacting degrees of freedom, the approach introduced in this paper raises the realistic prospect of evaluating the gap $\ensuremath{\Delta}(N)$ for systems with $N\ensuremath{\sim}{10}^{3}$ qubits. Although the calculation of $\ensuremath{\Delta}(N)$ 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.

DOI
10.1103/physrevb.79.205117

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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 NNN qubit system. As a first demonstration of this approach, time-dependent density functional theory is used to determine the minimum energy gap \ensuremathΔ(N)\ensuremath{\Delta}(N)\ensuremathΔ(N) 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-NNN scaling behavior of \ensuremathΔ(N)\ensuremath{\Delta}(N)\ensuremathΔ(N), and so determining this behavior is of fundamental significance. As density functional theory has been used to study quantum systems with N\ensuremath∼103N\ensuremath{\sim}{10}^{3}N\ensuremath∼103 interacting degrees of freedom, the approach introduced in this paper raises the realistic prospect of evaluating the gap \ensuremathΔ(N)\ensuremath{\Delta}(N)\ensuremathΔ(N) for systems with N\ensuremath∼103N\ensuremath{\sim}{10}^{3}N\ensuremath∼103 qubits. Although the calculation of \ensuremathΔ(N)\ensuremath{\Delta}(N)\ensuremathΔ(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.
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.
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.
First-principles electronic structure theory explains properties of atoms, molecules and solids from underlying physical principles without input from empirical parameters. Time-dependent density functional theory (TDDFT) has emerged as arguably the most widely used first-principles method for describing the time-dependent quantum mechanics of many-electron systems. In this thesis, we will show how the fundamental principles of TDDFT can be extended and applied in two novel directions: The theory of open quantum systems (OQS) and quantum computation (QC). In the first part of this thesis, we prove theorems that establish the foundations of TDDFT for open quantum systems (OQS-TDDFT). OQS-TDDFT allows for a first principles description of non-equilibrium systems, in which the electronic degrees of freedom undergo relaxation and decoherence due to coupling with a thermal environment, such as a vibrational or photon bath. We then discuss properties of functionals in OQS-TDDFT and investigate how these differ from functionals in conventional TDDFT using an exactly solvable model system. Next, we formulate OQS-TDDFT in the linear-response regime, which gives access to environmentally broadened excitation spectra. Lastly, we present a hybrid approach in which TDDFT can be used to construct master equations from first-principles for describing energy transfer in condensed phase systems. In the second part of this thesis, we prove that the theorems of TDDFT can be extended to a class of qubit Hamiltonians that are universal for quantum computation. TDDFT applied to universal Hamiltonians implies that single-qubit expectation values can be used as the basic variables in quantum computation and information theory, rather than wavefunctions. This offers the possibility of simplifying computations by using the principles of TDDFT similar to how it is applied in electronic structure theory. Lastly, we discuss a related result; the computational complexity of TDDFT.
While quantum computers have shown significant promise for electronic structure calculations, their potential to accelerate density functional theory (DFT) calculations remains unclear. In this work, we present a qubit-efficient encoding scheme for wavefunctions in Kohn--Sham (KS) DFT, together with a quantum algorithm that computes all occupied orbitals simultaneously. We further show that our algorithm is particularly well suited to the Harris functional, enabling the total energy to be evaluated with a potential exponential speedup over classical approaches by entirely avoiding the costly readout of the electronic density. In addition, we propose a second method for achieving self-consistent DFT calculations using multiple copies of the wavefunction, which likewise circumvents density readout. The applicability of our algorithms is demonstrated through several numerical examples, and their efficiency is compared with that of existing approaches.
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.