[Quantum Information] Quantum gates

  Quantum gates

Notes from RWTH Aachen University course 
“Quantum Information” Summer semester 2020
professor: Müller, Markus

Quantum gates

Classical

• 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

Quantum

• 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 $\sigma^2=\mathbb{I}$
    $R_{\sigma}(\alpha)=\text{exp}(-\frac{i\alpha}{2}\sigma)= \cos{(\alpha)}\cdot\mathbb{I}-i\sin{(\alpha)}\cdot \sigma$

Single-qubit gate:

• 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.

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
  $\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) \neq(\alpha_1|0\rangle+\beta_1|1\rangle)(\alpha_2|0\rangle+\beta_2|1\rangle)$