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Using accurate quantum energy computations in nanotechnologic applications is usually very computationally intensive. That makes it difficult to apply in subsequent quantum simulation. In this paper, we present some preliminary results pertaining to stochastic methods for alleviating the numerical expense of quantum estimations. The initial information about the quantum energy originates from the Density Functional Theory. The determination of the parameters is performed by using methods stemming from machine learning. We survey the covariance method using marginal likelihood for the statistical simulation. More emphasis is put at the position of equilibrium where the total atomic energy attains its minimum. The originally intensive data can be reproduced efficiently without losing accuracy. A significant acceleration gain is perceived by using the proposed method.

Nanotechnology tends to be the technology of focus in this century [^{5}. Due to this significant acceleration gain, the preparation process shows to be worth calculating.

For a configuration of nuclei

and

and the volume of the atoms are very large in comparison to those of the electrons. Thus, the atoms move comparatively slower than the electrons. As a consequence, one treats the time-independent Hamiltonian operator with respect to a set of nuclei

The above three terms are related to the kinetic energy, the atom-electron interaction and the inter-electron interaction while

such that

in which

The anti-symmetrization operator

where

in which the Hartree potential is the inverse of the Poisson operator such as

For the local density approximation (LDA), the exchange energy density is expressed as

energy density are only known for some extreme cases. The external electrostatic field potential

The main improvement from LDA to GGA is that the exchange-correlation energy does not depend only on the total electron density but also on its gradient such as

depends on

In this section, we will survey the main points about kernel-based approximation which are relevant in the quantum approximation. The most general setup of that approximation in any dimension

in which

where

If

For the most applied cases where

Suppose

For a noisy data

Let us denote the sampling points by

We denote by

By taking the logarithm, the computation of the marginal likelihood is as follows

from which one obtains the log marginal likelihood

In the presence of noise

The first term

That is to say, for given

where those expressions are dependent on some set of hyper-parameters which we discuss below. The value of the generalized distance

in which

From the chain rule, one obtains

This is singular when

Working with

where

where

which is regular for all values of

In this section, we would like to report some results from computer simulation of the formerly described approach. First, we will present some results pertaining to general real valued multi-variate functions. That will be followed by some application in quantum simulation.

The former theoretical approach was implemented by using C/C++, BLAS/LAPACK and NLOPT. The BLAS packet is used for the fast vector operations. We use LAPACK for the linear operations such as Cholesky factorization and dense matrix solvers. We use NLOPT for the nonlinear operations for both the geometry optimization and the optimal hyper-parameters in the log marginal likelihood (25). NLOPT supports diverse nonlinear optimization operations [

As a first test, we consider the reconstruction of the function

nel-based approximation. Since that function does not present any special feature such as cusp or boundary layer or any special interesting region, we use only randomly generated points. The initial guess of the hyper-parameters is provided by the users. One considers the determination of the final hyper-parameters as an unconstrained nonlinear optimization. The results of the computations are collected in

For the application to nano-simulation, we consider the unit cell generated by three vectors

After a transformation

configuration

The first transformation consists of an isotropic one that corresponds to a unidimensional function which is a scaling. That transformation amounts to stretching and confining the unit cell equally in all

Table 1. Comparison w.r.t.. Error of the function and gradient values on for dimension.

Dimension | Error in | Number of data | ||||
---|---|---|---|---|---|---|

10 | 20 | 30 | 40 | 50 | ||

4 | Function | 1.4727e-01 | 2.4419e-02 | 2.3565e-02 | 1.6221e-02 | 1.4906e-02 |

Gradient | 4.7896e-01 | 1.1478e-01 | 1.0937e-01 | 8.6681e-02 | 7.7206e-02 | |

5 | Function | 9.0570e-02 | 3.1160e-02 | 2.7457e-02 | 2.6331e-02 | 1.8296e-02 |

Gradient | 3.8708e-01 | 1.3668e-01 | 1.2931e-01 | 1.2187e-01 | 9.8637e-02 | |

6 | Function | 1.0068e-01 | 3.5685e-02 | 2.7100e-02 | 2.0901e-02 | 1.8697e-02 |

Gradient | 3.8986e-01 | 1.5762e-01 | 1.2739e-01 | 1.1043e-01 | 1.0545e-01 |

This is the Voigt strain tensor which is usually encountered in anisotropic transformations such as elasticity. This general transformation corresponds to a six-variate function since the Voigt strain matrix is supposed to be symmetric. The isotropic and 2D/3D anisotropic transformations are ensured to be non-singular, provided that the diagonal parameters are strictly positive. The tensor transformation in (38) is non-singular as long as the parameters

for a molecular or bulk configuration

very dense points in the neighborhood of the optimal value

known as VNL (Virtual NanoLab). For given nuclei coordinates

the energy

the isosurface of the electron density function in

the composites which are

ing Face Centered Cubic as space group symmetry. The appropriate parameter values are obtained from the American Mineralogist Crystal Structure Database [

Accumulated sampling points at the geometric optimum: (a) Unidimensional; (b)Tridimensional; (c) Increasing the dimensions

Electronic density for the DFT simulation of composite germanium/silicon

The preparation step consists of a geometry optimization and the determination of the kernel approximation. The duration of the Gaussian kernel is dominated by the DFT computations related to the point samples. The numbers of point samples are 70, 105, 200 and 250 for isotropic, 2D anisotropic, 3D anisotropic and Voigt tensor respectively. The stochastic computation as described in section 2.2 is very fast. In fact, the application of stochastic simulation is at most 2 percent of the whole preparation. All the computations were performed with the DFT basis unpolarized single _{3}/mmc. The second one admits a Face Centered Cubic lattice possessing the space group Fd3m. For both configurations, the bond lengths are obtained from the American Mineralogist Crystal Structure Database [

DFT vs. kernel approximation for two silicon configurations: (a) 45 sampling points; (b) 225 sampling points

. Preparation overhead and acceleration gain

CONFIG. | DIM | PREPARATION | EVALUATIONS | |||
---|---|---|---|---|---|---|

Geom. opt. | Kernel setup | Direct DFT | Kernel | Ratio | ||

BCC Ge | 1D | 4.791 mn | 58.788 mn | 89.116 mn | 0.0238 sc | 4.45e-06 |

2D | 8.632 mn | 78.076 mn | 105.952 mn | 0.1257 sc | 1.97e-05 | |

3D | 13.353 mn | 134.019 mn | 182.194 mn | 0.5181 sc | 4.73e-05 | |

6D | 58.238 mn | 6.986 hr | 8.346 hr | 0.9142 sc | 3.04e-05 | |

Ge_{0.75}Si_{0.25} | 1D | 57.627 mn | 128.618 mn | 172.634 mn | 0.0272 sc | 2.62e-06 |

2D | 67.437 mn | 176.058 mn | 229.228 mn | 0.0776 sc | 5.64e-06 | |

3D | 139.958 mn | 276.281 mn | 363.696 mn | 0.7345 sc | 3.36e-05 | |

6D | 7.493 hr | 107.328 hr | 143.634 hr | 0.8964 sc | 1.73e-06 | |

Ge_{0.50}Si_{0.50} | 1D | 33.119 mn | 137.369 mn | 205.144 mn | 0.0225 sc | 1.82e-06 |

2D | 79.887 mn | 161.299 mn | 212.237 mn | 0.0710 sc | 5.57e-06 | |

3D | 104.440 mn | 279.807 mn | 377.709 mn | 0.4704 sc | 2.07e-05 | |

6D | 7.896 hr | 117.347 hr | 133.213 hr | 0.8978 sc | 1.87e-06 | |

Ge_{0.25}Si_{0.75} | 1D | 43.675 mn | 144.608 mn | 201.094 mn | 0.0193 sc | 1.59e-06 |

2D | 71.893 mn | 186.089 mn | 233.715 mn | 0.0807 sc | 5.75e-06 | |

3D | 159.021 mn | 395.511 mn | 498.650 mn | 0.5624 sc | 1.87e-05 | |

6D | 7.962 hr | 89.853 hr | 110.410 hr | 0.8994 sc | 1.35e-04 |

while 225 are used for the second one. One can observe that the values on the diagonals become more precise as more sampling points are used. Additional sampling points can be used if a better accuracy is desired with the costs of having longer preparation overhead.