Towards local testability for quantum coding
summary
Construction produces an LDPC quantum code (hemi-cube) that encodes one logical qubit with minimum distance distance ~√n and local-testability/soundness scaling as 1/log^2 n. Key tools are an explicit quotient of the n-cube by the repetition code and algorithmic filling inequalities (send syndromes along coordinate directions) to bound syndrome-to-error weight, with checks of logarithmic weight. A generalization takes a quotient by classical code and gives k = binom(p+k-1,p) logical qubits (local testability conjectured).
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chapters (15)
- 0:00Introduction
- 0:20Vocabulary
- 1:35Current status
- 3:30Constraints
- 6:16Classical codes
- 6:51Quantum LDPC codes
- 10:56Soundness
- 12:26Locality stable codes
- 18:20Not locally testable codes
- 20:53Local testability construction
- 23:23Real projective plane
- 24:29Discrete topology
- 25:47Minimum distances
- 26:24Local testability
- 30:35Summary
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