W-state code

W-state code [1]

Description

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 first $d_L$ 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.

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 $O(1/n)$. Under a single located erasure, the worst-case entanglement infidelity of the W state code can be upper bounded 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].

Encoding

There are complexity-theoretic bounds on $W$-state preparation [4].

Transversal and Permutation-Based Gates

All logical gates can be implemented transversally. The logical unitary $U_L$ can be performed with the physical unitary $U_L\otimes U_L\otimes\cdots\otimes U_L$, where on the physical space $U_L$ is taken to act trivially on $\ket\perp$, i.e., $ U_L\ket\perp = \ket\perp$.

Cousin

Eigenstate thermalization hypothesis (ETH) code— The W-state is not a unique ground state of any local Hamiltonian [5].

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Block quantum code

Covariant block quantum code

Parents

Covariant block quantum code

The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.

Permutation-invariant (PI) codeCyclic quantum QECC Quantum

Approximate quantum error-correcting code (AQECC)Quantum

The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.

Monomial stabilizer codeHamiltonian-based QECC Quantum

The W-state codespace is the joint $+1$ eigenspace of a finite pure monomial group. The group is generated by SWAPs of the physical subsystems together with the diagonal operator $\bar{\omega}\Lambda^{\otimes n}$, where $\Lambda$ imprints the phase $\omega=e^{2\pi i/n}$ on every basis state except $|\perp\rangle$. The diagonal operator restricts supports to states with exactly one non-$\perp$ tensor factor, and the SWAP orbits of such states are labeled by the $d_L$ logical basis states. This construction generalizes the monomial stabilizer description of the W state and of Dicke states [6].

W-state code

References

[1]
P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “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, S. Nezami, S. Popescu, and G. Salton, “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021) arXiv:1709.04471 DOI
[4]
J. Yi, R. Liu, and Z. Li, “Lovász Meets Lieb-Schultz-Mattis: Complexity in Approximate Quantum Error Correction”, (2025) arXiv:2510.04453
[5]
L. Gioia and R. Thorngren, “$W$ state is not the unique ground state of any local Hamiltonian”, (2024) arXiv:2310.10716
[6]
M. V. den Nest, “A monomial matrix formalism to describe quantum many-body states”, New Journal of Physics 13, 123004 (2011) arXiv:1108.0531 DOI

Page edit log

Victor V. Albert (2026-08-26) — most recent
Victor V. Albert (2026-06-08)
Victor V. Albert (2022-08-18)
Philippe Faist (2022-08-18)

Cite as:

“W-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/w_state, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/covariant/w_state.yml.