- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland
APP-260807-0000quantum-informationarXiv:2608.05688Release v1.0.0-arxivListed Aug 7, 2026
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Summary
The paper proves structure theorems for four nested groups of Clifford circuits that preserve a CSS code, each relevant to fault-tolerant logical operations. Every code-preserving Clifford circuit is a product of Z-diagonal circuits (S and CZ gates) and their X-basis analogues. Every element of the two-fold transversal group — generated by depth-one circuits of one- and two-qubit gates — factors into layers that are Z-diagonal, X-diagonal, or CNOT; consequently every transversal gate is a product of three transversal diagonal circuits (two, for connected non-self-dual codes). Every code-preserving automorphism circuit has a normal form: a Hadamard layer, a permutation, and two diagonal circuits. A further two-fold automorphism group, allowing a compensating qubit permutation, can realize strictly more logical gates than the two-fold transversal group.
As an application, the paper surveys 136 CSS codes with explicit certified generator data: 78 codes whose two-fold transversal circuits generate the full logical Clifford group (at distances up to 12), 58 mostly-QLDPC codes with exactly computed logical images (e.g., ≥460,800 for the gross code), and three new code families — bipartite grids, cut-complements, and quadrics — many of whose members realize the full logical Clifford group this way.