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Quantum Engineering Year 1: Quantum Mechanics Core Month 21Day 569

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Year 1·Month 21·Week 2

Day 569: Controlled Gates (CZ, Controlled-U)

Day 569 of 2,016~15 min read

Learning Objectives

•**Implement the CZ gate** and understand its symmetry
•**Construct controlled-U gates** for arbitrary single-qubit U
•**Relate CNOT and CZ** through basis transformation
•**Apply the ABC decomposition** for controlled gates
•**Build multi-controlled gates** (Toffoli) conceptually
•**Understand the control qubit formalism** mathematically

Today's Schedule (7 hours)

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On this page

1 The Controlled-Z CZ Gate 2 CZ Matrix 3 CZ Symmetry 4 CNOT-CZ Relationship 5 General Controlled-U Gates 6 Controlled-U Circuit Notation 7 ABC Decomposition 8 Finding ABC Decomposition 9 Multi-Controlled Gates 10 Control Qubit Formalism 11 Native Implementations Quantum Computing Connection

Day 568Day 569 of 2,016Day 570