Quantum LDPC Codes with Transversal CCZ Gates
summary
Constructs quantum LDPC codes that admit transversal CCZ by encoding qubits as level‑1 cochains, proving a cup product (via the Leibniz rule) produces Qx⊥‑invariant trilinear forms, and lifting 1D classical LDPC codes to 3D via tensor/ balanced product constructions. Gives explicit families: a Reed‑Solomon based family with k ≈ n^{1−ε} and d ≈ n^{1/3} up to polylog locality, and cites Guanyu Zhu’s LDPC family with k,d ≈ √n. Identifies the bottleneck: local codes need a multiplicative property (repetition or algebraic), so asymptotically good CCZ‑LDPC remains open.
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chapters (12)
- 0:00Transversal CCZ gates intro
- 1:55Quantum LDPC codes overview
- 5:28Topology of codes and CCZ gates
- 11:57State of the art in CCZ codes
- 16:05Coding theory for CCZ gates
- 22:282D toric code supports CZ proof
- 28:57Beyond 3D toric code
- 35:05Cochain complexes and cup product
- 43:04LDPC codes to cochain complexes
- 48:50Choosing local codes challenges
- 55:49Summary and open challenges
- 1:00:13Q&A trade-offs and Hadamard
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