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Quantum Engineering Year 2: Advanced Quantum Science Month 29Day 802

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Year 2·Month 29·Week 3

Day 802: Logical Operators on the Planar Code

Day 802 of 2,016~20 min read

Learning Objectives

•**Construct** logical X operators as strings from smooth to smooth boundary
•**Construct** logical Z operators as strings from rough to rough boundary
•**Explain** the string-net interpretation of logical operators
•**Prove** that deformable strings represent the same logical operator
•**Calculate** the minimum weight (code distance) from operator geometry
•**Describe** the complementary pairing between X and Z logical operators

Today's Schedule (7 hours)

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Month 29 Topological Codes Week 115 Surface Code Implementation Year 2 Advanced Quantum Science Schedule Overview Learning Objectives Morning Session Logical Operator Theory 3 hours 1 Review Boundaries and Logical Structure 2 Logical Z Rough-to-Rough Strings Construction Why This Works Minimum Weight 3 Logical X Smooth-to-Smooth Strings Construction Why This Works Minimum Weight 4 Anticommutation of Logical Operators Proof 5 String Deformation and Equivalence Classes Deformation Rules Equivalence Classes 6 The String-Net Picture Z-strings Electric Strings X-strings Magnetic Strings Topological Invariance 7 Multiple Representatives Counting for Logical Z Quantum Mechanics Connection Topological Protection The Homological Interpretation Anyonic Worldlines Afternoon Session Worked Examples 25 hours Example 1 All Minimum-Weight Logical Z Representatives Example 2 Verifying Commutation with Stabilizers Example 3 String Deformation Practice Problems Problem Set 802 Direct Application Intermediate Challenging Evening Session Computational Lab 15 hours Lab 802 Logical Operator Visualization and Verification Expected Output Summary Key Formulas Key Takeaways Daily Checklist Preview Day 803 References

Day 801Day 802 of 2,016Day 803