Beyond transversality: structure of Clifford circuits for CSS codes

Victor V. Albert — Joint Center for Quantum Information and Computer Science, NIST/University of Maryland

Structure theorems for the Clifford circuits that preserve a CSS code: every code-preserving Clifford circuit factors into Z- and X-diagonal circuits; two-fold transversal (depth-one two-local) circuits factor into diagonal and CNOT layers; automorphism circuits have a Hadamard–permutation–diagonal normal form. Applied to 218 certified codes: 173 codes with at most 120 qubits, and 45 larger ones, realize the full logical Clifford group through two-fold transversal circuits alone.

arXiv: 2608.05688. This release, v2.0.0-arxiv, matches the second arXiv version; for the first version use the tag v1.0.0-arxiv.

Talk to this paper

This paper is published with an AI agent (Agentic Publication Protocol). Clone this repo and open it in an AI coding agent to ask questions, check claims, and explore the proofs and data.

Claude Code: clone and open — it reads AGENTS.md automatically. Or use /load-paper https://github.com/valbert4/two-fold-transversal.

Codex or other agents: clone and open — any agent that reads AGENTS.md picks up the paper context.

What is new in version 2

• New full Clifford codes. The survey grows from 78 to 218 full codes (173 in Table I, 45 larger ones in the records), including a census of codes invariant under rank-3 permutation groups and the bipartite-grid family up to 120 qubits. Every one of the 218 is certified from scratch.
• Relation to Holmes et al. Section V explains how their single-layer logical reach relates to the fixed-matching groups characterized here.
• The fixed-matching layer theorem (Thm. D.1) is also formalized in an external Lean development, Axiomatic-AI/CliffordCSS, which the paper cites; it is not part of this repository.

Checking the paper's claims

Authoritative figure/table map and Quick claim checks (six commands, seconds to minutes each): code/figure-reproduction/README.md. Highlights:

• python3 code/figure-reproduction/check_tables_and_claims.py — 1,247 cross-checks of both code tables, the stated numbers and both certificates against the data.
• python3 code/figure-reproduction/gen_table1.py — confirms Table I is exactly what the data generate.
• python3 code/figure-reproduction/fig_fourclasses_verify.py — re-derives everything drawn in Fig. 1.
• python3 code/certificate/verify_certificate.py — validates the Local Reduction Lemma certificate (independent rebuild: code/certificate/full_reduction.py).
• python3 code/two-fold-aut/certify.py — the [[9,3,2]] two-fold-automorphism claim, complete 945-matching enumeration (~4 min).

Reproducing results

Setup (Python, GAP, TeX) and runtimes: environment/README.md. The paper builds with cd paper && bash compile.sh.

Re-certifying all 218 full codes from scratch is documented in data/certify/README.md (bash data/certify/run_certification.sh): about 3 hours on 20 cores, with GAP and its recog and genss packages. All certificates are committed, so nothing requires it.

Data

data/README.md documents the two datasets (218 full codes and 58 QLDPC codes, with depth-one generator strings and logical-image data): schema, gate-string grammar, sources, provenance, and verification.

Citation

@misc{albert2026beyond,
  title         = {Beyond transversality: structure of {C}lifford circuits for {CSS} codes},
  author        = {Albert, Victor V.},
  year          = {2026},
  eprint        = {2608.05688},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph},
  url           = {https://arxiv.org/abs/2608.05688},
  note          = {Code and data: \url{https://github.com/valbert4/two-fold-transversal/releases/tag/v2.0.0-arxiv}}
}

License

See LICENSE: MIT for code, CC-BY-4.0 for paper text, data, and documentation.