Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly
K. Sai Mineesh Reddy, Navin Kashyap
TL;DR
This work tackles the problem of realizing fault-tolerant logical Clifford operations via transversal gates in CSS codes, and investigates the prospects for transversal T gates. It develops a framework based on puncturing and doubling of classical divisible codes to construct asymptotically good CSS codes in which physical transversal Z-rotations enact corresponding logical Z-rotations and, for suitable parameters, realize the full logical Clifford group. It additionally advances the CSS–T code theory by characterizing when transversal T implements the identity or a logical transversal T (or S†), proving the necessity but not sufficiency of C2 ∗ C1 ⊆ C1^⊥, and showing how doubling can yield CSS–T codes with different logical actions. The results establish a path to asymptotically good CSS codes supporting fault-tolerant Clifford operations, while highlighting open challenges for achieving transversal T realizations and LDPC-compatible constructions, with implications for fault-tolerant quantum computation and magic-state approaches.
Abstract
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group. Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.