W-state code[1]

Encodes a quantum state of a \(d_L\)-dimensional Hilbert space into \(n\) physical quantum systems, each associated with a Hilbert space of dimension \(d_L+1\). The encoding resembles the structure of the W state [2]: \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.

This code enables universal quantum computation with transversal gates. 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 [3][1] rather than an exact error-correcting code.