Quantum Computing

4.

Quantum Gates and Circuits

4.1.

Single-Qubit Gates

4.1.1.

4.1.1.1.

Pauli-X (NOT) Gate

4.1.1.1.1.

Matrix Representation

4.1.1.1.2.

Action on Bloch Sphere

4.1.1.1.3.

Bit Flip Operation

4.1.1.2.

4.1.1.2.1.

Matrix Representation

4.1.1.2.2.

Action on Bloch Sphere

4.1.1.2.3.

Combined Bit and Phase Flip

4.1.1.3.

4.1.1.3.1.

Matrix Representation

4.1.1.3.2.

Action on Bloch Sphere

4.1.1.3.3.

Phase Flip Operation

4.1.2.

4.1.2.1.

Matrix Representation

4.1.2.2.

Creation of Superposition

4.1.2.3.

Basis Change Operation

4.1.3.

4.1.3.1.

Phase Gate (S Gate)

4.1.3.1.1.

Matrix Representation

4.1.3.1.2.

Phase Shift Operation

4.1.3.2.

4.1.3.2.1.

Matrix Representation

4.1.3.2.2.

Role in Universal Gate Sets

4.1.3.3.

General Phase Gate

4.1.4.

4.1.4.1.

4.1.4.2.

4.1.4.3.

4.1.4.4.

Euler Decomposition

4.1.4.5.

Arbitrary Single-Qubit Rotations

4.1.5.

4.2.

Multi-Qubit Gates

4.2.1.

The Controlled-NOT (CNOT) Gate

4.2.1.1.

Matrix Representation

4.2.1.2.

Entanglement Generation

4.2.1.3.

Control and Target Qubits

4.2.2.

The Controlled-Z (CZ) Gate

4.2.2.1.

Matrix Representation

4.2.2.2.

Phase Flip Operation

4.2.2.3.

Symmetric Nature

4.2.3.

4.2.3.1.

Matrix Representation

4.2.3.2.

Swapping Qubit States

4.2.3.3.

Decomposition into CNOTs

4.2.4.

The Toffoli (CCNOT) Gate

4.2.4.1.

Matrix Representation

4.2.4.2.

Universality for Classical Computation

4.2.4.3.

Reversible Classical Gates

4.2.5.

Controlled-U Gates

4.2.5.1.

General Controlled Operations

4.2.5.2.

Decomposition into Basic Gates

4.2.5.3.

Multi-Control Gates

4.2.6.

4.3.

Quantum Circuits

4.3.1.

Circuit Diagrams and Notation

4.3.1.1.

Standard Symbols

4.3.1.2.

Reading and Constructing Circuits

4.3.1.3.

Wire Conventions

4.3.2.

Reversibility of Quantum Gates

4.3.2.1.

Unitary Evolution

4.3.2.2.

Implications for Computation

4.3.2.3.

Inverse Operations

4.3.3.

Universal Quantum Gate Sets

4.3.3.1.

Solovay-Kitaev Theorem

4.3.3.2.

Minimal Universal Sets

4.3.3.3.

Approximation of Arbitrary Gates

4.3.4.

Circuit Depth and Width

4.3.5.

Parallel vs Sequential Operations

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5. Core Quantum Phenomena for Computation