Quantum Information Theory

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Quantum Computing

Boolean Functions and Logic Gates

• Bits: The Basic Unit of Information in Classical Computers
• What Are Boolean Functions?
  • $\text{AND}$ Gate (logical conjunction)
  • $\text{OR}$ Gate (logical disjunction)
  • $\text{NOT}$ Gate (logical negation)
  • $\text{XOR}$ Gate (exclusive disjunction)
  • $\text{NAND}$ Gate (negated conjunction)
  • $\text{NOR}$ Gate (negated disjunction)
  • $\text{CNOT}$ Gate
  • Toffoli Gate ($\text{CCNOT}$ Gate)
  • Fredkin Gate ($\text{CSWAP}$ Gate)
• Cloning Function $\text{cl}$
• Projections and Injections $p_{i}$, $\imath_{i}$, $\jmath_{i}$
• Functionally Complete Sets and Universal Gates

Quantum Gates

• Qubits: The Basic Unit of Information in Quantum Computers
• Quantum Entanglement
• Quantum Gates and Quantum Circuits
• $1$-Qubit Gates $\mathbb{C}^{2} \to \mathbb{C}^{2}$
  • Pauli Gates $X$, $Y$, and $Z$
  • Hadamard Gate $H$
  • Phase Gate $R_{\theta}$
• $2$-Qubit Gates $(\mathbb{C}^{2})^{\otimes 2} \to (\mathbb{C}^{2})^{\otimes 2}$
  • Quantum $\text{CNOT}$ Gate
  • Swap Gate $\text{ex}$
• $3$-Qubit Gates $(\mathbb{C}^{2})^{\otimes 3} \to (\mathbb{C}^{2})^{\otimes 3}$
  • Quantum Toffoli Gate
  • Quantum Fredkin Gate
• Solovay–Kitaev Theorem
• No-Cloning Theorem

Quantum Circuits

• Bell Circuit
• Superdense Coding
• Quantum Teleportation
• Quantum Cryptography

Quantum Algorithms

• Balanced and Constant Functions
• Deutsch–Jozsa Algorithm
• Bernstein–Vazirani Algorithm
• Simon’s Algorithm
• Shor’s Algorithm
• Grover’s Algorithm

Information Theory

Classical Information Theory

• Definition of Information
• Shannon Entropy
  • Joint Entropy
  • Conditional Entropy
  • Relative Entropy (Kullback–Leibler Divergence)
  • Mutual Information
• Compression Rate, Encoding, and Decoding

Quantum Information Theory

• von Neumann Entropy

References

• Stephen M. Barnett, Quantum Information (2009)
• 김영훈·허재성, 양자 정보 이론 (2020)