W-state code[1] 

Approximate block quantum code whose encoding resembles the structure of the W state [2]. This code enables universal quantum computation with transversal gates.

The encoding is of a \(d_L\)-dimensional Hilbert space into \(n\) physical quantum systems, each associated with a Hilbert space of dimension \(d_L+1\): \begin{align} \ket\psi \to \frac{1}{\sqrt{n}}\bigl(\ket{\psi\perp\perp\ldots} + \ket{\perp\psi\perp\ldots} + \cdots + \ket{\perp\perp\ldots\psi}\bigr)\ , \tag*{(1)}\end{align} where on each physical system, \(\ket\perp\) denotes the \((d_L+1)\)-th basis state, and \(\ket\psi\) is encoded using the first \(d_L\) basis states.

Indeed, to apply any logical unitary \(U\) it suffices to apply \(U\) on each physical system, where the unitary is taken to act nontrivially only on the \(d_L\) first basis states of each system. Universal computation with transversal gates does not violate the Eastin-Knill theorem because this code is an approximate error-correcting code [1,3] rather than an exact error-correcting code.