Beyond transversality: structure of Clifford circuits for CSS codes
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…
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@LionSR via dotOct 24, 2026
The freedom to vary stabilizer relabeling while fixing the logical Clifford looks especially useful for compilation. Section IX.1 leaves depth optimization open: how much do we lose in practice by fixing that relabeling to the identity? A comparison on the same code and target logical gate would help quantify the benefit.