Improved Quantum Codes with Transversal T Gates

This paper advances the theory of quantum error correction by constructing new families of quantum CSS codes that support transversal T gates. Transversal gates are highly desirable in fault-tolerant quantum computing because they are inherently fault-tolerant and low-overhead. While the Eastin-Knill theorem prohibits a universal transversal gate set, constructing codes with transversal non-Clifford gates (like the T gate) is a critical step toward efficient universal fault-tolerant quantum computation.

The author develops a framework based on divisible decreasing monomial codes. By puncturing these codes at a downward-closed set (a downset) on the Boolean hypercube, the author creates logical qubits while maintaining the necessary transversality properties. A key technical contribution is a closed-form expression for the distance of these punctured codes, which allows for precise optimization of code parameters. The author provides both an explicit construction using weighted Reed-Muller codes and a randomized construction that uses hypergraph-based protection to improve distance scaling.

The paper significantly broadens the achievable parameters for quantum codes with transversal T gates. Notably, it constructs the first families of such codes that achieve constant rate while maintaining growing distance. Furthermore, the author demonstrates that these codes can achieve a magic state distillation overhead exponent of zero, improving upon previous bounds. The results are presented as boundaries in the (dimension, distance) parameter space, showing improvements over existing explicit and non-explicit constructions.

These results provide new theoretical benchmarks for quantum code design. By relaxing the restriction to LDPC codes, the author demonstrates that higher performance can be achieved in terms of magic state distillation overhead. The new closed-form expression for the distance of punctured decreasing monomial codes is a general tool that may be useful for other researchers working on algebraic quantum codes and polar codes.