{"id":"APP-260807-0000","repo_url":"https://github.com/valbert4/two-fold-transversal","versions":[{"v":1,"tag":"v1.0.0-arxiv","commit":"813862cd0a990ddb6b2ef9eff3126551d892f2b1","app_publication_id":"app-v1:sha256:c13dc3d9a64e870ea0e7cca44535c72df3517dd1a08139d41e575d51a5159f61","release_url":"https://github.com/valbert4/two-fold-transversal/releases/tag/v1.0.0-arxiv","listed_at":"2026-08-07","title":"Beyond transversality: structure of Clifford circuits for CSS codes","authors":[{"name":"Victor V. Albert","affiliation":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland"}],"domain":"quantum-information","arxiv_id":"2608.05688","tags":["CSS codes","Clifford circuits","transversal gates","logical gates","QLDPC codes","fault tolerance"],"paper_summary":"The paper proves structure theorems for four nested groups of Clifford circuits\nthat preserve a CSS code, each relevant to fault-tolerant logical operations.\nEvery code-preserving Clifford circuit is a product of Z-diagonal circuits\n(S and CZ gates) and their X-basis analogues. Every element of the two-fold\ntransversal group — generated by depth-one circuits of one- and two-qubit gates —\nfactors into layers that are Z-diagonal, X-diagonal, or CNOT; consequently every\ntransversal gate is a product of three transversal diagonal circuits (two, for\nconnected non-self-dual codes). Every code-preserving automorphism circuit has a\nnormal form: a Hadamard layer, a permutation, and two diagonal circuits. A\nfurther two-fold automorphism group, allowing a compensating qubit permutation,\ncan realize strictly more logical gates than the two-fold transversal group.\n\nAs an application, the paper surveys 136 CSS codes with explicit certified\ngenerator data: 78 codes whose two-fold transversal circuits generate the full\nlogical Clifford group (at distances up to 12), 58 mostly-QLDPC codes with\nexactly computed logical images (e.g., ≥460,800 for the gross code), and three\nnew code families — bipartite grids, cut-complements, and quadrics — many of\nwhose members realize the full logical Clifford group this way."}]}