Quantum Computing Revision Sheet
Fundamental Notation and Concepts
Qubit Representation
- A qubit can be represented in the Dirac (Bra-Ket) notation as a linear combination of basis states: \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\) Where \(\alpha\) and \(\beta\) are complex numbers satisfying \(|\alpha|^2 + |\beta|^2 = 1\)
Basis States
\[
|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
\]
Quantum Gates
Pauli Gates
X Gate (NOT Gate)
- Matrix Representation:
\[
X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
\]
- Key Properties:
- \(X|0\rangle = |1\rangle\)
- \(X|1\rangle = |0\rangle\)
- Equivalent to classical NOT gate
- Rotates the qubit state by \(\pi\) radians around the X-axis
Z Gate
- Matrix Representation:
\[
Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
\]
- Key Properties:
- \(Z|0\rangle = |0\rangle\)
- \(Z|1\rangle = -|1\rangle\)
- Rotates the qubit state by \(\pi\) radians around the Z-axis
Multi-Qubit Gates
CNOT (Controlled-NOT) Gate
- Full Matrix Representation (4x4 Matrix for Two-Qubit System):
\[
CNOT = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{bmatrix}
\]
-
Alternative Names:
-
Controlled-X Gate
-
Toffoli Gate (when generalized to multiple control qubits)
-
Truth Table:
| Control | Target | Result |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
- Flips the target qubit only if the control qubit is in state \(|1\rangle\)
Hadamard Gate
\[
H = \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}$
\]
Key Operations
- On Basis States:
- \(H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\)
- \(H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)\)
Useful Properties
- Creates equal probability of measuring \(|0\rangle\) or \(|1\rangle\)
- Reversible: \(H \cdot H = I\) (Identity matrix)
- Crucial in creating quantum superposition
- Enables quantum parallelism by creating superposition states
Geometric Interpretation
- Rotates the qubit state by \(\pi\) radians around the X+Z axis of the Bloch sphere
- Transforms pure basis states into equal superposition states
Bell States
Definition
Bell states are maximally entangled two-qubit states:
- \(|B_{00}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\)
- \(|B_{01}\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)\)
- \(|B_{11}\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)\)
- \(|B_{10}\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)\)
Key Characteristics
- Represent the four maximally entangled two-qubit states
- Symmetric superpositions
- Each state has equal probability of measuring \(|0\rangle\) or \(|1\rangle\)
Useful Quantum Computing Relationships
Measurement Probabilities
- For a qubit \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\):
- Probability of measuring \(|0\rangle\): \(|\alpha|^2\)
- Probability of measuring \(|1\rangle\): \(|\beta|^2\)
Gate Composition
- Quantum gates are reversible
- Most quantum gates are unitary matrices
- Composition of quantum gates is done through matrix multiplication
Note: This revision sheet provides a high-level overview. Always supplement with detailed textbook study and practical problem-solving.