Algebraic Coding Problems From Quantum Fault-Tolerance
summary
Shows that non-Clifford gates are handled by magic-state distillation and gives the algebraic condition (CSS-T) needed for a code to be preserved by transversal T: for every codeword x in Cx⊥ the space Cz⊥ must contain a self-dual code supported on supp(x). Classic examples are the 15→1 triorthogonal and Reed–Muller constructions (full-weight checks), which work but have poor n, k, d tradeoffs; currently no LDPC CSS-T constructions are known and finding one with growing k and d is the open problem.
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chapters (16)
- 0:00Introduction
- 1:11Quantum fault tolerance focus
- 2:56Quantum gates and noise models
- 4:47Logical circuits and fault tolerance
- 6:15Non-Clifford T gate challenges
- 7:39Quantum error correction basics
- 9:52LDPC codes in quantum error correction
- 10:56Surface codes and scaling limits
- 12:20Universal gate sets motivation
- 13:49Magic state distillation for T gates
- 16:20Triorthogonal code properties
- 17:27CSS-T problem definition
- 19:54Recent progress and divisibility
- 21:41Open problem: CSS-T code goals
- 24:54Q&A
- 26:46Workshop announcement
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