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Quantum Engineering Year 2: Advanced Quantum Science Month 35Day 972

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Year 2·Month 35·Week 3

Day 972: Expressibility & Trainability

Day 972 of 2,016~19 min read

Learning Objectives

•**Quantify expressibility** of parameterized quantum circuits
•**Explain the barren plateau phenomenon** and its mathematical origin
•**Identify conditions** that lead to barren plateaus
•**Apply mitigation strategies** for vanishing gradients
•**Design circuits** balancing expressibility and trainability
•**Analyze gradient landscapes** numerically and theoretically

Today's Schedule (7 hours)

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Year 2 Semester 2B Fault Tolerance Hardware Month 35 Advanced Algorithms - Week 139 QML Foundations Schedule Overview Learning Objectives Morning Session Theory 3 hours 1 Expressibility Covering Hilbert Space Motivation Formal Definition Computing Expressibility 2 Entangling Capability Definition Meyer-Wallach Measure 3 The Barren Plateau Phenomenon Discovery McClean et al 2018 Mathematical Origin Intuition Why Barren Plateaus Occur 4 Types of Barren Plateaus 41 Depth-Induced Barren Plateaus 42 Cost-Function-Induced Barren Plateaus 43 Noise-Induced Barren Plateaus 44 Entanglement-Induced Barren Plateaus 5 Detecting Barren Plateaus Variance of Gradients Practical Detection Algorithm 6 Mitigation Strategies 61 Parameter Initialization 62 Layer-wise Training 63 Local Cost Functions 64 Shallow Circuits 65 Problem-Inspired Anstze 66 Correlated Parameters 7 The Expressibility-Trainability Trade-off The Fundamental Dilemma Optimal Design Principles Afternoon Session Problem Solving 2 hours Worked Example 1 Computing Expressibility Worked Example 2 Gradient Variance Scaling Worked Example 3 Local vs Global Cost Functions Practice Problems Problem 1 Expressibility Calculation Direct Application Problem 2 Barren Plateau Onset Intermediate Problem 3 Cost Function Design Challenging Evening Session Computational Lab 2 hours Lab Analyzing and Mitigating Barren Plateaus Summary Key Formulas The Expressibility-Trainability Trade-off Key Takeaways Mitigation Strategies Summary Daily Checklist Preview Day 973

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