Week 14 - my notes

Week 14 - Math of Two-Qubit Circuits
Resources

Week 14 - Math of Two-Qubit Circuits

Recap

Vector and matrices

X gate: \[\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\]
Z gate: \[\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}\]
H gate: \[\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1\\1 & -1\end{bmatrix}\]

Multi-Qubit circuits

You can have single-qubit gates (ones that are applied to one qubit only)
You can have controlled gates (applied to multipled qubits at a time)

Introduction to 2-Qubit circuits & indexing

For individual circuits, writing a state in ket notation is super easy: \(|0\rangle\) or \(|1\rangle\)
For 2-qubit circuits, just put those together: \(|01\rangle\) or \(|10\rangle\). You may also find \(|0\rangle|1\rangle\) or \(|1\rangle|0\rangle\)
The index to be used is first (at left) q1 and at right q0: \(|q1q0\rangle\). This is known as right indexing.

Vector for multi-qubits states

The upper part is the amount of \(|0\rangle\) contribution
The lower part is the amount of \(|1\rangle\) contribution
As an example, a \(|0\rangle\) state:

\[\begin{bmatrix}1\\0\end{bmatrix}\]

We will have four parts now. Representing the components of \(|00\rangle\), \(|01\rangle\), \(|10\rangle\), and \(|11\rangle\)
As an example, a \(|10\rangle\) state:

\[\begin{bmatrix}0\\0\\1\\0\end{bmatrix}\]

Applying gates to Multi-Qubit circuits

For a single qubit: (Matrix) * (Initial state) = final state
The math can be the same, but the matrix will be huge.. so let’s try to predict it

Predicting final states

Find the ket representation of the final state by applying the gates in the circuit
Fill in the vector based on the elements in the ket

Math of the CX gate

CX gate also has a matrix representation:

\[\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{bmatrix}\]

Lets apply this gate to a \(|00\rangle\) state

\[\begin{bmatrix}1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\end{bmatrix}\cdot \begin{bmatrix}1\\0\\0\\0\end{bmatrix}=\begin{bmatrix}1\\0\\0\\0\end{bmatrix}\]

The Math of Entanglement

It generally is \[|q1q2\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\]
You can see the sight of dependence in there

Tensor Product Introduction

Could you combine two separete vectors to get a 4x1 vector?
Ofc, we call this mathematical operation a tensor product

Example

Qubit 0 is in state \(|q0\rangle\) = \(|1\rangle\) = \[\begin{bmatrix}0\\1\end{bmatrix}\]
Qubit 1 is in state \(|q1\rangle\) = \(|0\rangle\) = \[\begin{bmatrix}1\\0\end{bmatrix}\]
The math basically is:

\(|q1q0\rangle\) = \(|q1\rangle\otimes |q0\rangle\) = \(|0\rangle\otimes |1\rangle\) = \[\begin{bmatrix}1\\0\end{bmatrix}\otimes \begin{bmatrix}0\\1\end{bmatrix}\]

Resources

Tensor products for qubit states and gates https://docs.microsoft.com/en-us/azure/quantum/concepts-multiple-qubits

Linear algebra of tensor products https://www.math3ma.com/blog/the-tensor-product-demystified