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Quantum Engineering Year 2: Advanced Quantum Science Month 31Day 856

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Year 2·Month 31·Week 3

Day 856: Transversal Gates on CSS Codes

Day 856 of 2,016~20 min read

Learning Objectives

•**Prove** that X and Z are always transversal on CSS codes
•**Demonstrate** that CNOT between CSS code blocks is transversal
•**Identify** conditions under which Hadamard is transversal (self-dual codes)
•**Analyze** the S gate transversality for specific CSS codes
•**Explain** why T is never transversal on CSS codes
•**Characterize** the complete transversal gate set for common CSS codes

Today's Schedule (7 hours)

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Overview Schedule Learning Objectives The CSS Code Structure Review CSS Code Definition Logical Operators Theorem 1 Transversal X on CSS Codes Statement Proof Theorem 2 Transversal Z on CSS Codes Statement Proof Theorem 3 Transversal CNOT Between Code Blocks Statement Proof Theorem 4 Transversal Hadamard on Self-Dual CSS Codes Definition Self-Dual CSS Code Statement Proof Counterexample Non-Self-Dual Codes The S Gate Partial Transversality The Phase Gate Transversality Condition for S Steane Code Example Why T Is Never Transversal on CSS Codes The T Gate Theorem T Cannot Be Transversal on CSS Codes Deeper Reason Clifford Hierarchy Complete Transversal Gate Sets Steane Code 713 Surface Code n 1 d Color Code n 1 d 15-Qubit Reed-Muller Code 15 1 3 Worked Examples Example 1 Verify Transversal X on Steane Code Example 2 Why H is Not Transversal on Surface Code Example 3 Transversal CNOT Action on Logical States Practice Problems Level 1 Direct Application Level 2 Intermediate Level 3 Challenging Computational Lab Summary Key Formulas Main Takeaways Daily Checklist Preview Day 857

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