[Quantum Information] Quantum gates

Quantum gates

Quantum gates

Universality: There exist minimal sets of gates that suffice to implement any classical Boolean circuits. E.g. AND and NOT.
Irreversible computing: The two-bit gates are essentially irresverible and non-invertable: e.g. \(a\oplus b\) of XOR gate
- loss the information
- because two input bits just flow into one output bit
Reversible computing: the number of inputs and outputs are equal
The three bits Toffeli-gate is a universal reversible logic gate: any reversible classical circuit can be built from Toffeli-gate only.
Wiki: logic gate

In quantum information processing, all unitary transformations are allowed, represents valid quantum operations.
time evolution \(U(t) = \text{exp}(-\frac{it}{h}H)\)
Example:
- single qubit rotation: \(H\propto\sigma, \sigma\in\{X,Y,Z\}\)
- e.g. \(H_j=E\cdot X_j\) on the \(j\)-th qubit
  \(H:\) hamiltonian
  \(E:\) energy
  \(X:\) \(X\)-rotation
- \(R_{X_j}(\alpha)=\text{exp}(-\frac{i\alpha}{2}X_j)\)
  \(\alpha\) is controlled by the strength \(E\) and the time \(t\)
  \(\frac{2E\cdot t}{h}\propto \alpha\)
- \(R_{X_j}\) can be simplified:
  use identity for operators \(A\), for which \(A^2 = \mathbb{I}\)
  \(\begin{aligned} e^{i\alpha A} &= \sum_{n=0}^{\infty}\frac{1}{n!}(i\alpha)^n A^n \\ &= (1-\frac{\alpha^2}{2}+...)\mathbb{I}+i(\alpha-\frac{1}{6}\alpha^3+...)A \\ &= \cos{(\alpha)}\cdot\mathbb{I}+i\sin{(\alpha)}\cdot A \end{aligned}\)
- Since \(\sigma^2=\mathbb{I}\)
  \(R_{\sigma}(\alpha)=\text{exp}(-\frac{i\alpha}{2}\sigma)= \cos{(\alpha)}\cdot\mathbb{I}-i\sin{(\alpha)}\cdot \sigma\)

classical: only NOT gate
quantum: many more (infinitely)
- e.g. rotation operations
Pauli gates can be generated from rotations, up to overall phase
\(R_{\sigma}(\pi)=(-i)\cdot\sigma\)
Hadamard gate
- \[ H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1 \\1 & -1\end{pmatrix} \]
phase gate
- \[ S=\begin{pmatrix}1&0 \\0 & i\end{pmatrix} \]
T-gate
- \[ T=\begin{pmatrix}1&0 \\0 & e^{i\pi/4}\end{pmatrix} \]

no correlations / entanglement can be created by a single qubit gate
we need two-qubit gate.

CNOT
- \[ CNOT = \begin{pmatrix} 1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0&0&0&1\\0&0&1&0\end{pmatrix} \]
controlled-U
- \(U\) applies \(\Leftrightarrow\) the first qubit is \(|1\rangle\)
creation of entanglement: \(C_1NOT_2\cdot H_1|\psi_{12}\rangle\)
Definition of entangled state: all pure states cannot be written as a product state, are entangled
\(\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) \neq(\alpha_1|0\rangle+\beta_1|1\rangle)(\alpha_2|0\rangle+\beta_2|1\rangle)\)