Session 07 - CSS and QLDPC Codes

Status: in progress. This session continues Heptabase lesson 5f11785a-5fbb-4434-9c0b-504b0a8e0aca.

Prerequisites And Diagnostics

Before teaching, ask:

1. In a CSS code, what does H_XH_Z^T=0 check?
2. What is the difference between a stabilizer and a logical operator?
3. Why does a small-weight nontrivial logical operator define small distance?
4. What does it mean for a family to have k,d = Theta(n)?

If the learner answers 1-3 cleanly, compress logical operators and start at good parameters.

Lesson Scope

• Finish logical operators as quotient or coset representatives.
• Define [[n,k,d]] goodness and the LDPC tension.
• Explain why linear distance needs expansion.
• Introduce left-right Cayley complexes.
• State the quantum Tanner construction and the role of Theorems 17 and 18.
• Explain the anchor gap between diagonal expansion and stabilizer-presentation expansion.
• Briefly position hypergraph product, fiber bundle, Panteleev-Kalachev, and Leverrier-Zemor families.

Source Anchors

• quantum-tanner-expander-anchor.md
• quantum-tanner-diagonal-expansion-structure.md
• quantum-tanner-incidence-spectral-gap.md
• quantum-tanner-local-generator-blowup.md
• incidence-expansion-to-parity-tanner-expansion.md
• quantum-tanner-theorem17-parity-expander.md

Active Recall

• Explain C_X / C_Z^\perp in words.
• What three asymptotic properties make a QLDPC family "good" for this course?
• Why is expansion good for distance but bad for 2D routing?
• What expansion does the diagonal-graph theorem give, and what expansion does the lower-bound route need?

Next-Step Handoff

After this session, move to Session 08, where the code-side expansion demand is translated into syndrome-extraction circuits, SWAP routing, congestion, dilation, and CD(T_n,G).