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 \(d_L\) 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.

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

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

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
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Zoo Code ID: w_state

Cite as:
“W-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/w_state
BibTeX:
@incollection{eczoo_w_state, title={W-state code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/w_state} }
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“W-state code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/w_state

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