Quantum
Notation
- \(\mathbb{C}^{2^n}\) : complex vector space for an \(n\)-qubit state
- \(\lvert \psi \rangle\) : ket / state vector
- \(\langle \psi \rvert\) : bra / conjugate transpose of a ket
- \(\langle \phi \mid \psi \rangle\) : inner product between states
- \(\lvert \psi \rangle \otimes \lvert \phi \rangle\) : tensor product of states
- \(U^\dagger\) : adjoint (conjugate transpose) of \(U\)
- \(\Pr(x)\) : probability of observing outcome \(x\)
- \(x \oplus y\) : bitwise XOR
- \(a \bmod N\) : remainder of \(a\) modulo \(N\)
Complex Numbers
\[z = a + bi\]
\[\bar{z} = a - bi\]
\[\lvert z \rvert = \sqrt{a^2 + b^2}\]
\[e^{i\theta} = \cos \theta + i \sin \theta\]
State Notation
\[\lvert \psi \rangle\]
\[\langle \psi \rvert\]
\[\langle \phi \mid \psi \rangle\]
\[\lVert \lvert \psi \rangle \rVert^2 = \langle \psi \mid \psi \rangle\]
Qubit States
\[\lvert 0 \rangle =
\begin{bmatrix}
1 \\
0
\end{bmatrix},
\quad
\lvert 1 \rangle =
\begin{bmatrix}
0 \\
1
\end{bmatrix}\]
\[\lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle\]
\[\lvert \alpha \rvert^2 + \lvert \beta \rvert^2 = 1\]
Multi-Qubit Systems
\[\lvert \psi \rangle \otimes \lvert \phi \rangle\]
\[\lvert 00 \rangle = \lvert 0 \rangle \otimes \lvert 0 \rangle, \quad
\lvert 01 \rangle = \lvert 0 \rangle \otimes \lvert 1 \rangle\]
\[\lvert \psi \rangle = \sum_{x \in \{0,1\}^n} \alpha_x \lvert x \rangle\]
\[\sum_{x \in \{0,1\}^n} \lvert \alpha_x \rvert^2 = 1\]
Unitary Evolution
\[U^\dagger U = I\]
\[\lvert \psi' \rangle = U \lvert \psi \rangle\]
Measurement
\[\Pr(0) = \lvert \alpha \rvert^2, \quad \Pr(1) = \lvert \beta \rvert^2\]
\[\Pr(x) = \lvert \langle x \mid \psi \rangle \rvert^2\]
\[\lvert \psi \rangle \xrightarrow{\text{measure}} \frac{\lvert x \rangle \langle x \mid \psi \rangle}{\sqrt{\Pr(x)}}\]
Common Gates
\[X =
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}\]
\[Y =
\begin{bmatrix}
0 & -i \\
i & 0
\end{bmatrix}\]
\[Z =
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}\]
\[H = \frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}\]
\[S =
\begin{bmatrix}
1 & 0 \\
0 & i
\end{bmatrix}\]
\[T =
\begin{bmatrix}
1 & 0 \\
0 & e^{i\pi/4}
\end{bmatrix}\]
\[\mathrm{CNOT} =
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{bmatrix}\]
