QIC Lab 3 Solved

Computational Quantum Physics Week 3
Due on Week 5

Slides template Starting from this week, it is mandatory to submit a report of the exercises in the presentation format (slides). The maximum length is six slides, including:

1. First slide: heading with your name, the date, exercise number and course name and year. 2. Different sections, such as (if applicable)

* Theory: Explain very briefly the theory you have based your solution on.
* Code development: Introduce strategies, tests, and report debugging problems, compilations

options.
* Results: Present data and explain your results.

(a) Upload the slide report in Moodle under the correspondent exercise.
(b) File names must include your name, exercise number and codewords SLIDES, and code. Example:

Ex3-Rossi-SLIDES.pdf
Exercise 1: Scaling of the matrix-matrix multiplication

Consider the program developed in Exercise 3 of Week 1 (matrix-matrix multiplication).

1. (a)  Write a python script that changes N between two values Nmin and Nmax, and launches the program.
2. (b)  Store the results of the time needed in different files depending on the multiplication method used.
3. (c)  Fit the scaling of the time needed for different methods as a function of the input size. Consider

   the biggest possible difference between Nmin and Nmax.

4. (d)  Plot the results for the different multiplication methods.

2: Eigenproblem
(a) Diagonalize A and store the N eigenvalues λi in crescent order.

Exercise
Consider a random Hermitian matrix A of size N.

(b) Compute the normalized spacings between eigenvalues

̄
si = ∆λi/∆λ where

̄
and ∆λ is the average ∆λi.

∆λi = λi+1 − λi,

Exercise 3: Random Matrix Theory
Study P(s), the distribution of the si defined in the previous exercise, accumulating values of si from different random matrices of size at least N = 1000.

1. (a)  Compute P (s) for a random HERMITIAN matrix.
2. (b)  Compute P (s) for a DIAGONAL matrix with random real entries.
3. (c)  Fit the corresponding distributions with the function:

P (s) = asα exp(−bsβ ) Hint: if necessary neglect the first matrix eigenvalue.

and report α, β, a, b.

Page 1 of 1