Chapter 21: Try1
Quantum Computing - Expanded Lecture Summaries
Lecture 3: Bloch Sphere, Eigenstates, Eigenvectors, Projection Operator
Key Concepts
Pauli-X Gate Matrix
The Pauli-X gate matrix is:
\[
X = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
Hadamard Gate Matrix
The Hadamard gate matrix is:
\[
H = \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\]
Pauli-X Gate Identity
The Pauli-X (NOT) gate satisfies the identity:
$$ X = HZH $$ where \(H\) is the Hadamard gate.
The Pauli-X gate matrix representation:
\[
X = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
Hadamard Transform
The Hadamard gate creates superposition states:
\[
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle), \quad H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)
\]
Matrix representation:
\[
H = \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\]
Pauli-Z Gate Identity
The Pauli-Z gate affects the phase of a qubit:
\[
Z|0\rangle = |0\rangle, \quad Z|1\rangle = -|1\rangle
\]
Matrix representation:
\[
Z = \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
\]
Projection Operator
The projection operator for a qubit basis state:
\[
P_0 = |0\rangle\langle0| = \begin{bmatrix}
1 & 0 \\
0 & 0
\end{bmatrix}, \quad P_1 = |1\rangle\langle1| = \begin{bmatrix}
0 & 0 \\
0 & 1
\end{bmatrix}
\]
Two-Qubit States and Matrices
A general two-qubit state:
\[
|\psi\rangle = c_{00}|00\rangle + c_{01}|01\rangle + c_{10}|10\rangle + c_{11}|11\rangle
\]
Matrix representation:
\[
|\psi\rangle = \begin{bmatrix}
c_{00} \\
c_{01} \\
c_{10} \\
c_{11}
\end{bmatrix}
\]
Toffoli Gate (CCNOT)
A three-qubit gate that flips the target qubit if both control qubits are \( |1\rangle \).
Matrix representation:
\[
\text{Toffoli} = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
\end{bmatrix}
\]