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Quantum Computing: A Complete Guide

by Dr. Eleanor Rieffel & Wolfgang Polak

Quantum Bits and Quantum Gates

Single Qubit Gates

Quantum gates are operations that manipulate qubits. Unlike classical gates, quantum gates are reversible and represented by unitary matrices.

Pauli Gates

  1. X Gate (NOT gate):

Flips to and vice versa.

  1. Y Gate:

  1. Z Gate:

Adds a phase of to .

Hadamard Gate

Creates superposition:

Phase Gates

  1. S Gate:

Phase gate with rotation.

  1. T Gate:

Phase gate with rotation.

Multi-Qubit Gates

CNOT Gate (Controlled-NOT)

Flips the target qubit if the control qubit is .

Controlled-U Gates

Any single-qubit gate U can be made controlled:

Universal Gate Sets

A set of gates is universal if any quantum operation can be approximated using them. Common universal sets:

  1. {H, T, CNOT} - Clifford + T
  2. {H, S, CNOT} - Clifford gates (need additional non-Clifford gate)
  3. {Toffoli gate} - Universal for classical reversible computation

Quantum Circuits

Quantum algorithms are represented as quantum circuits:

  1. Wires represent qubits
  2. Boxes represent gates
  3. Time flows from left to right

Example: Bell state preparation

This circuit creates the entangled Bell state: