# W-state code[1]

## Description

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.

## Protection

The W state code is an approximate error-correcting code. Intuitively, if a subsystem is lost to the environment, the environment only gains access to \(\ket\psi\) with probability of order \(1/n\). Under a single located erasure, the worst-cast entanglement infidelity of the W state code can be upper bound as \begin{align} \epsilon_{\mathrm{worst}} \leq \frac{\sqrt{2} + d_L}{\sqrt{n}}\ . \tag*{(2)}\end{align}

In contrast to the Eigenstate thermalization hypothesis (ETH) code, the W state code does not saturate the scaling \(1/n\) in worst-case entanglement infidelity which is known to be optimal for covariant approximate error-correcting codes [1].

## Transversal Gates

## Parents

- Covariant code — The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.
- Approximate quantum error-correcting code (AQECC) — The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.

## References

- [1]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [2]
- W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways”, Physical Review A 62, (2000) arXiv:quant-ph/0005115 DOI
- [3]
- P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021) arXiv:1709.04471 DOI

## Page edit log

- Victor V. Albert (2022-08-18) — most recent
- Philippe Faist (2022-08-18)

## Cite as:

“W-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/w_state