Quantum LDPC codes with transversal non-Clifford gates: optimal parameters and existence

Determine the optimal parameters of quantum LDPC codes that support a transversally implementable non-Clifford gate, and ascertain whether asymptotically good quantum LDPC codes exist that support a transversally implementable non-Clifford gate.

Background

Quantum LDPC codes with asymptotically good parameters were only recently constructed, and the interaction between LDPC structure and transversal non-Clifford gates is largely unexplored.

Establishing optimal trade-offs and proving existence (or non-existence) of asymptotically good LDPC codes with transversal non-Clifford gates would significantly impact fault-tolerant quantum computing architectures.

References

Open Question 2: What are the optimal parameters of a quantum LDPC code supporting a transversally implementable non-Clifford gate? Does an asymptotically good quantum LDPC code supporting a transversally implementable non-Clifford gate exist?

— Constant-Overhead Magic State Distillation  (2408.07764 - Wills et al., 2024) in Section 1, Discussion and Future Directions

Indeed, constructing LDPC auxiliary codes that transversally realize the logical transversal $$ is an open problem.

— Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS Codes  (2608.19094 - Reddy et al., 19 Aug 2026) in Final paragraph of Section “CSS families realizing any addressable logical [Z-rotation] gate”