Ludmila Botelho

# THE CODEBOOK

## Our Services

We offer a variety of services and plans tailored to business needs of any kind and of any size.

## Process

Research

What does the user need to understand quantum computers? What is relevant?

## 2.

Make a syllabus

What we expect from the module? Deliverables

- Exercises

- Code

## 3.

Build the page

Add the content to web-page: theory + coding challenges (codercises)

- Testing is important!

- Review

# ADDING NEW CONTENT

# Distance Measures

# DISTANCE MEASURES

(Taxicab metric)

(Euclidian distance)

# DISTANCE MEASURES

(Triangle inequality)

• $$d(p_1,p_1) = 0$$
• $$d(p_1,p_2) \geq 0$$
• $$d(p_1,p_2) = d(p_2,p_1)$$
• $$d(p_1,p_3) \leq d(p_1,p_2)+d(p_2,p_3)$$

(Symmetry)

(Symmetry)

(Isomorphism)

(Earth mover's distance)

(Hack week)

# A LIT BIT OF MATHEMATICS

# ?

# A LIT BIT OF MATHEMATICS

$$\frac{1}{4}$$

$$\frac{1}{4}$$

$$\frac{1}{4}$$

$$\frac{1}{4}$$

# A LIT BIT OF MATHEMATICS

Probabilities

# A LIT BIT OF MATHEMATICS

$$\frac{1}{4}$$

$$\frac{1}{4}$$

$$\frac{1}{4}$$

$$\frac{1}{4}$$

Probability vector

( $$\mathcal{l}_1$$-norm)

$$+$$

$$+$$

$$+$$

$$=$$

$$1$$

# A LIT BIT OF MATHEMATICS

$$\vert \psi \rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}$$

$$|\alpha|^2+|\beta|^2=1$$

($$\mathcal{l}_2$$-norm)

$$\alpha,\beta\in \mathbb{C}$$

# A LIT BIT OF MATHEMATICS

$$\rho = \rho^\dagger = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$$

$$a + b =1$$

$$\rho = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

$$\langle \psi \vert \rho \vert \psi \rangle \geq 0$$

(Hermitian)

(Positive semi-definite)

$$a, b \in[0,1]$$

# A LIT BIT OF MATHEMATICS

$$a_1$$

$$a_2$$

$$a_3$$

$$\vdots$$

$$a_n$$

$$b_1$$

$$b_2$$

$$b_3$$

$$b_n$$

$$\vdots$$

$$|a_1 - b_1|$$

+

$$|a_2 - b_2|$$

+

$$|a_3 - b_3|$$

+

$$\vdots$$

+

$$|a_n-b_n|$$

$$\rho = \begin{bmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ddots & \vdots\\ a_{1n} & \ldots & a_{nn} \end{bmatrix} = VDV^\dagger$$

$$D= \begin{bmatrix} d_{1} & & \\ & \ddots & \\ & & d_{n} \end{bmatrix}$$

Distance between vectors

# A LIT BIT OF MATHEMATICS

$$a_1$$

$$a_2$$

$$a_3$$

$$\vdots$$

$$a_n$$

$$b_1$$

$$b_2$$

$$b_3$$

$$b_n$$

$$\vdots$$

# A LIT BIT OF MATHEMATICS

$$a_1$$

$$a_2$$

$$a_3$$

$$\vdots$$

$$a_n$$

$$b_1$$

$$b_2$$

$$b_3$$

$$b_n$$

$$\vdots$$

Fidelity (or Bhattacharyya Coefficient)

# LEARNING BY DOING IT

## Exercises

# LEARNING BY DOING IT

## Exercises

def kolmogorov_distance(p, q):
"""Compute the Kolmogorov distance between two discrete probability distributions.

Args:
p (np.array[float]): A vector of probability distribution
q (np.array[float]): A vector of probability distribution.

Returns:
(float): A number that represents the trace distance
of the probability distributions.
"""

##################
##################

# CREATE A VECTOR x BASED ON p - q

# RETURN THE L1-NORM OF x DIVIDED BY 2
return 

## Codercises

# LEARNING BY DOING IT
def kolmogorov_distance(p, q):
"""Compute the Kolmogorov distance between two discrete probability distributions.

Args:
p (np.array[float]): A vector of probability distribution
q (np.array[float]): A vector of probability distribution.

Returns:
(float): A number that represents the trace distance
of the probability distributions.
"""

##################
##################

# CREATE A VECTOR x BASED ON p - q
x = p-q

# RETURN THE L1-NORM OF x DIVIDED BY 2
return np.sum(np.abs(x))/2
# LEARNING BY DOING IT

## D.1

Metrics and Norms in vector space

Kolmogorov distance

## D.2

Alternative measurement; Bhattacharyya coefficient

## D.3

Metrics and norms for quantum operators

Trace distance

Fidelity

(Classical)

(Quantum)

## D.1

Metrics and norms in vector space

Kolmogorov distance

## D.2

Alternative measurement: Bhattacharyya coefficient

## D.3

Metrics and norms for quantum operators

Trace distance

## N.3

Partial trace and purification

User Questions

Error correction module review

Teleportation Demo review

## N.4

The Bloch ball

Compute expectation value

Fidelity