Grover's algorithm explained for children?

Question

Can you explain the mathematical details of Grover's algorithm for children?

Answer 1

This is the crypto stack exchange. And if there is one thing I learned in crypto, you always have to specify everything, especially the power of the participants.

Therefore, because you did not specify the knowledge of the children, I assume all children have graduated in computer science and quantum mechanics. Thus, they will easily understand the following:

1. Initialize the system to the uniform superposition over all states:

$$ |\psi_0\rangle = \frac{1}{\sqrt{N}} \sum_{i=0}^{N-1} |i\rangle, $$

Obviously $N$ is the search space.

2. Oracle for state $|\psi_0\rangle$ to invert $|w\rangle$:

\begin{equation} |\psi_1\rangle = U_w |\psi_0\rangle = \frac{1}{\sqrt{N}} \sum_{i=0}^{N-1} (-1)^{f(i)} |i\rangle, \end{equation}

where $f(i) = 1$ if $i=w$ or $f(i) = 0$

3. Amplify $|w\rangle$ with: \begin{equation} |\psi_2\rangle = S D |\psi_1\rangle = S \left(2 |\psi_1\rangle \langle \psi_1| - \mathbb{I} \right) |\psi_1\rangle, \end{equation} with: \begin{equation} S|i\rangle = (-1)^{i=w}|i\rangle. \end{equation} and: \begin{equation} D = 2|\psi_0\rangle\langle\psi_0| - I, \end{equation}

4. Iterate over 2. and 3. $O(\sqrt{N})$ times to increase probability. Trivial.

5. Measure quantum state. Everyone knows how that works.

"Yes, it really is that simple"

Edit: I forgot the relation to crypto. Just assume a cryptographic scheme as the black box function to which the Grovers algorithm is applied and brute force the key.

/s

Maybe this answers your question.