Universal transversal gates

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.