Week 7 - my notes
Table of Contents
Week 7 (qubit by qubit)
Recap
Review:
- We saw the math of qubit states - vectors
- X - 180 rotation around the X-axis
- Z - 180 rotation around the Z-axis
- H - creates a superposition
- We have to normalize states - they must have a length of 1
- Phase is the sign between \(|0\rangle\) and \(|1\rangle\)
Math for Quantum: Gates
- The representation of the gates is done by using matrices.
- When applying a gate, the input state and output state (qubit) can be represented as vectors.
- Applying and X gate: \(|0\rangle\) X \(|1\rangle\)
- We can represent this X using a matrix.
- We use vectors to represent a quantum state \[\begin{bmatrix}0\\1\end{bmatrix}\]
- We use matrices to represent a quantum gate \[\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\]
The formula
Gate x Quantum State = New Quantum State
- Matrix x Vector = New vector
- E.g: (X gate matrix) (initial state) = final state
\[\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}0\\1\end{bmatrix}\]
- X gate: \[\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\]
- Z gate: \[\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}\]
- H gate:
\[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix}\]
- You don’t need to remember all of them, you can always look it up!
Multiplying matrices
- You start by multiplying the first line of the first matrix by the first columm of the second matrix
note I have already studied this subject (matrices) at school, so I apologize for not explaning each step clearly
\[\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}0*1 + 1*0\\1*1 + 0*0\end{bmatrix} =\begin{bmatrix}0\\1\end{bmatrix}\]
- Let’s start with a \(|1\rangle\) state:
\[\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix} = \begin{bmatrix}0*0 + 1*1\\1*0 + 0*1\end{bmatrix} =\begin{bmatrix}1\\0\end{bmatrix}\]
- Let’s try to apply a Z gate to a \(|0\rangle\) state:
\[\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}1*1 + 0*0\\0*1 + (-1)*0\end{bmatrix} =\begin{bmatrix}1\\0\end{bmatrix}\]
The math of Quantum Resources
Math for superposition:
- Apply the H gate to a \(|0\rangle\) input \[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix}1*1 + 1*0\\1*1 + (-1)*0\end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix} = |+\rangle\]
- Apply the H gate to a \(|1\rangle\) input \[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix}1*0 + 1*1\\1*0 + (-1)*1\end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1\\-1\end{bmatrix} = |-\rangle\]
Interference with two H gates:
What is the final state of a circuit with input \(|0\rangle\) and two H gates after it? \[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix}1*1 + 1*0\\1*1 + (-1)*0\end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix}\]
\[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix}1\\1\end{bmatrix} = \frac{1}{\sqrt{2}}*\frac{1}{\sqrt{2}} \begin{bmatrix}1*1 + 1*1\\1*1 + (-1)*1\end{bmatrix} = \frac{1}{2}\begin{bmatrix}2\\0\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}\]
- The example above is a interference! (more on this later)
- Just like our interferometer we made in Quantum Flytrap
- A beamsplitter is what we use to create an H gate in real life and in Quantum Flytrap, we put the example above to test.