Universal transversal gates

TL;DR

This paper establishes a necessary and sufficient condition for quantum codes to have universal transversal gates, overcoming the Eastin-Knill no-go theorem in certain error models, and introduces a new code construction with improved error correction capabilities.

Contribution

It provides a general condition for universal transversal gates and presents a novel code construction that corrects local and correlated errors more effectively.

Findings

The Eastin-Knill theorem does not hold for all error models.

A new code construction reduces logical error probability from  to .

The universality condition determines the existence of universal gate sets for quantum codes.

Abstract

A long-standing challenge in quantum error correction is the infeasibility of universal transversal gates, as shown by the Eastin-Knill theorem. We obtain a necessary and sufficient condition for a quantum code to have universal transversal gates and show that the Eastin-Knill no-go result is a special case that does not hold for a general error model. We present a code construction using $n$ $d$-dimensional systems that changes the logical error probability from a lower bound $\Omega (1/n\log d)$ to an upper bound $\mathcal O (1/n d)$ and allows exact correction of both local and correlated errors. Our universality condition determines the existence of a universal gate set for any quantum error-correcting code.