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.