Classical Coding Problem from Transversal $T$ Gates

Abstract

Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical $T$ and $T^{-1}$ gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal $T$, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal $T$ via physical transversal $T$. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal $T$. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle $Z$-rotations in arXiv:1910.09333.

This is a shorter version of arXiv:1910.09333. 5 pages main text. Presented at ISIT 2020. Comments welcome!

References in corpus (6)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Magic state distillation with low overhead
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Universal transversal gates with color codes - a simplified approach
• Diagonal gates in the Clifford hierarchy
• On the logical operators of quantum codes

Cited by in corpus (5)

• Structure of CSS and CSS-T Quantum Codes
• An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance
• Bhattacharyya parameter of monomials codes for the Binary Erasure Channel: from pointwise to average reliability
• CSS-T Codes from Reed Muller Codes
• Entanglement-assisted Quantum Reed-Muller Tensor Product Codes