{add_to_codebook}

Ludmila Botelho

# THE CODEBOOK

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Process

Research

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

1.

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

New Module!

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)

What about quantum states?

What about quantum states?

# 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.
    """

    ##################
    # YOUR CODE HERE #
    ##################

    # 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.
    """
    
    ##################
    # YOUR CODE HERE #
    ##################
    
    # 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

Codercises

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

D.4

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

D.4

Fidelity

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