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 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., \( U_L\ket\perp = \ket\perp\).Primary Hierarchy
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
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