On Optimality of CSS Codes for Transversal $T$

Understand

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate.

• Consequently, it is important to understand codes that implement such gates transversally.
• In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal $T$ and $T^{-1}$ gates preserve the codespace.
• Our Heisenberg perspective reduces this to a finite geometry problem that translates to the design of certain classical codes.