SAP Project:

High-Performance Algorithms for Scalable Spin-Qubit Circuits with Quantum Dots

Jean-Pierre Leburton

Beckman Institute for Advanced Studies and Technology

University of Illinois at Urbana-Champaign

Material Associated with this Project: News article 1 | News article 2 | Final Report

Research Objectives
SCIENTIFIC GOALS
One of the greatest scientific and engineering challenges of this decade is the realization of a quantum computer. In this context, a solid-state system is highly desirable because of its compactness, scalability and compatibility with existing semiconductor technology. In quantum computing, the basic information unit is a quantum bit or qubit, i.e. a physical object that can be represented as a superposition of two basis states in a 2D Hilbert space. For this purpose, the use of spin states (S) rather than charge-states as qubits in semiconductor materials is relatively appealing because of their relative insensitivity to electric noise in the device environment.

For single spin operations, carrier confinement is a major issue: The recent demonstration of the analogue of Hund's rule involving single electron spin in "artificial atoms" [1], and fully tunable Kondo effects in semiconductor quantum dots (QDs) [2] has placed this objective within experimental reach. The ability to manipulate the spin S of electrons by combining gate electrodes and magnetic fields provides the necessary ingredients for controlling spin-qubit operations in nanoscale devices. This scenario has the advantage of relying on established semiconductor fabrication techniques, while semiconductor materials enjoy long spin coherence times, which is of utmost importance for preserving quantum information during many qubit operations [3, 4].

There are presently several proposals for realizing a Control-Not (C-NOT) gate, which is the fundamental circuit element for quantum computation with spins in semiconductors [5]: Among them, the Loss-DiVincenzo scheme is based on the manipulation of electron spins in coupled quantum dots [6]. The variant, advocated by Kane, uses the nuclear spin of isolated phosphorus dopant atoms as a qubit in a nanoscale silicon field-effect devise [3]. Both proposals are based on the electric control of a singlet (S=0)-triplet (S=1) transition through quantum mechanical exchange interaction amongst electrons by external electric field in confined nanostructures.

However, the experimental realization of a solid-state quantum device remains a challenge for which the interplay between device geometry and electrostatics, material parameters and spin physics should be integrated into a realistic scheme. Consequently, there is a need for comprehensive high-performance computer tools capable of simulating quantum operation within the device environment while describing the microscopic reality of quantum effects in nanostructures.

COMPUTATIONAL GOALS AND METHODS
Our computational approach relies on a 3D self-consistent Poisson-Kohn-Sham scheme based on a spin-dependent density functional theory (DFT) with local spin density approximation (LSDA). This code is particularly well suited to model microscopic quantum many-body phenomena within the device environment. The electronic states and eigenlevels E of the electron system are obtained by solving Kohn-Sham equations twice - once for electrons with spins up () and then for electrons with spin down () [7]:

where Hamiltonian is given by

Here M is the electrons mass, A is the vector potential corresponding to the uniform magnetic field B, g and are the Lande factor and effective Bohr magneton, is the LSDA exchange-correlation potential [8], and is the electrostatic potential determined from the solution of the 3D Poisson equation:

where charge density is equal to . Here is the permittivity of the material, p(r) is the hole concentration , n(r) the total electron concentration, are the ionized donor and acceptor concentrations, respectively. At equilibrium, the electron concentrations in the quantum dots for each spin are calculated from the wave functions obtained from the Kohn-Sham equations, i.e., . Outside the active dot region, Thomas-Fermi distribution is used in a first approximation. Hence, the electron density is locally a function of the position of the conduction band edge with respect to the Fermi level as opposed to electrons in the active region (quantum dots) that are calculated from the occupation of the quantized 0D eigenstates for which the wave function decays to zero at the device boundaries. The various gate voltages determine the boundary conditions for the Poisson equation. Specifically, Dirichlet conditions are imposed on the top and bottom surfaces of the structure. For the lateral surfaces vanishing electric fields (von Neumann boundary conditions) or periodic boundary conditions are assumed [9].

However, in the present realization of our code, there are several bottlenecks:

1. Kohn-Sham equation above has magnetic field B in it (which is used for spin alignment and rotation in quantum C-NOT gate). This forces us to deal with complex matrices, and thus, slows down the overall code performance considerable. This is especially noticeable for situations requiring consideration of large electron number (N ~ 100) in the quantum dot system.

2. Solution of the Poisson equation is also (sometimes) a challenge as the number of grid points is ~10⁶ at present, and is likely to increase even more, when the simulations of systems consisting of many quantum dots (required for demonstration of a scalable quantum computer) become of an essence.

3. Another interesting problem arises when we attempt to study the relaxation of spin in time. This is also very important for quantum computation on quantum dots because of the spin decoherence issue in GaAs/AIGaAs nanostructures [6]. In this case, we have to solve a time-dependant Kohn-Sham equation within the framework of the so-called time-dependent density functional theory (TD-DFT):

The solution of such an equation in real time requires high-performance implementation of the existing code on the parallel platform.

4. In order to design a structure with particular parameters, e.g., a structure with large singlet-triplet energy separation, a large volume of data has to be analyzed. Hence, as the system's size and electron number increases, data management and its visualization also become important.

POTENTIAL BENEFITS
The successful implementation of the code on the parallel platform will allow us not only to speed up the overall performance, but also address such problems as computer-assisted design of the scalable-spin-qubit circuits based on quantum dots. We can also study large systems of quantum dots containing a sizable number of electrons in order to understand various physical phenomena such as charge redistribution and electron localization in coupled dots at high magnetic fields, formation and reconstruction of quantum Hall edges in extended quantum dots systems, and so on.

COMPUTATIONAL APPROACH
For simulation, the device is mapped onto a 3D non-uniform mesh that is necessary to simulate relatively "large" device features , and yet resolve nanometer-sized details in quantum dots area. Both the Poisson and Kohn-Sham equations are discretized using the finite element method. At present, the self-consistent system of donor charges with the electron charges and spins is solved iteratively using the Newton-Raphson method. The Kohn-Sham equation is solved using subspace iteration method based on Raleigh-Ritz analysis while the Poisson equation is solved by using a conjugate gradient method [10].

This approach is adequate for small- and intermediate-sized systems. However, successful description of larger systems calls for more sophisticated numerical approaches. We plan to implement PETSc package for Poisson solver and try various preconditioners for the conjugate gradient method in order to evaluate their performance for our particular problem. For the eigenvalue problem (that is, the solution of the Kohn-Sham equation) more sophisticated algorithms such as Davidson method with various types of filtering [11] or other similar approaches (PARPACK) will be used. The code will be parallelized and its performance will be evaluated on various clusters at NCSA.

ACCOMPLISHMENTS AND SIGNIFICANCE
In addition to the quantum dot simulations, fast and efficient Poisson and /or eigenvalue solvers can also be applied to other scientific problems such as simulation of the isolated phosphorus dopant atoms as qubits in a nonscale silicon-based field-effect device (Kane's proposal for quantum computation). Recent proposals for DNA (or other molecules) transport through a nanopore also rely on solving the Poisson equation in order to calculate DNA's feedback (potential distribution) in the surrounding materials. Different molecules will produce different potentials, and such a device can be used as a sensor to discern and analyze various types of molecules passing through the nanopore.

PUBLICATIONS
1. S. Tarucha, et al., Phys. Rev. Lett. 77, 3613 (1996).

2. S. Sasaki, et al., Nature 405, 764 (2000).

3. B. Kane, Nature 393, 133 (1998).

4. J.M. Kikkawa and D.D. Awschalom, Phys. Rev. Lett. 80, 4313 (1998).

5. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, 2002.

6. D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).

7. A. Thean and J.P. Leburton, J. Appl. Phys. 89, 2808 (2001).

8. J.P. Perdew and A. Zunger, Phys. Rev. B. 23, 5048 (1981).

9. D. Jovanovis and J.P. Leburton, Phys. Rev. B. 49, 7474 (1994).

10. P. Matagne and J.P. Leburton, Phys. Rev. B. 65, 235323 (2002).

11. Y. Saad, Numerical Methods for Large Eigenvalues Problems, Manchester Univ. Press, 1994.