Asymptotically Good Quantum Codes with Addressable and Transversal Non-Clifford Gates

TL;DR

This paper introduces a new family of asymptotically good quantum codes that support transversal non-Clifford gates, specifically the CCZ gate, enabling efficient quantum error correction and gate implementation.

Contribution

It constructs the first asymptotically good quantum codes supporting transversally addressable non-Clifford gates using algebraic geometry codes, improving previous constructions.

Findings

Supports transversally addressable CCZ gates on three logical qubits

Achieves asymptotically good parameters with positive rate and distance

Enables depth-one implementation of CCZ gates on logical qubits

Abstract

Constructing quantum codes with good parameters and useful transversal gates is a central problem in quantum error correction. In this paper, we continue our work in arXiv:2502.01864 and construct the first family of asymptotically good quantum codes (over qubits) supporting transversally addressable non-Clifford gates. More precisely, given any three logical qubits across one, two, or three codeblocks, the logical $\mathsf{CCZ}$ gate can be executed on those three logical qubits via a depth-one physical circuit of $\mathsf{CCZ}$ gates. This construction is based on the transitive, iso-orthogonal algebraic geometry codes constructed by Stichtenoth (IEEE Trans. Inf. Theory, 2006). This improves upon our construction from arXiv:2502.01864, which also supports transversally addressable $\mathsf{CCZ}$ gates and has inverse-polylogarithmic rate and relative distance.