Approximate operator-algebra error-correcting code[1,2] 


Code encoding quantum and/or classical information that approximately corrects against noise affecting operators forming an algebra.


Given an algebra \(\mathcal{A}\), \(\mathcal{A}\) is \(\epsilon\)-correctable under noise channel \(\mathcal{N}\) if there exists some quantum channel \(\mathcal{R}\) such that \begin{align} ||(\mathcal{R}\circ\mathcal{N})-P_{\mathcal{A}}||_{\diamond}\leq\epsilon~, \tag*{(1)}\end{align} where \(P_{\mathcal{A}}\) is the projector onto algebra \(A\) and we use the diamond norm \(\diamond\) [3].

Let the minimal error for some algebra \(\mathcal{A}\) under noise channel \(\mathcal{N}\) be \begin{align} \epsilon_{\mathcal{A}}=\min_{\mathcal{R}} ||\mathcal{R}\circ\mathcal{N}-P_{\mathcal{A}}||_{\diamond}~. \tag*{(2)}\end{align} Let \(\delta_{\mathcal{A}}=||\mathcal{N}^C-\mathcal{N}^C\circ P_{\mathcal{A}'}||_{\diamond}\) for commutant \(\mathcal{A}'\) of algebra \(\mathcal{A}\) and complementary channel \(\mathcal{N}^C\) of noise channel \(\mathcal{N}\). Then [1], \begin{align} \delta_{\mathcal{A}}^2/4\leq \epsilon_{\mathcal{A}}\leq 2\delta_{\mathcal{A}}^{1/2}~. \tag*{(3)}\end{align}

Complementary channel formulation

Given projector \(\mathcal{P}_{\mathcal{A}}\) onto algebra \(\mathcal{A}\) and noise channel \(\mathcal{N}\). Using worst-case entanglement fidelity \(F\), we can quantify approximate operator algebra error correction as \begin{align}F(R\mathcal{N},\mathcal{P}_{\mathcal{A}})\geq 1-\epsilon\tag*{(4)}\end{align} for some small \(\epsilon\) and recovery channel \(R\).

This is equivalent to considering the complementary channel \(\mathcal{N}^C\) with projector \(\mathcal{P}_{\mathcal{A}'}\) onto the commutant \(\mathcal{A}'\) of \(\mathcal{A}\) such that \begin{align} F(\mathcal{N},R'\mathcal{P}_{\mathcal{A}'})\geq 1-\epsilon \tag*{(5)}\end{align} for that same value of \(\epsilon\) and some channel \(R'\).

This formulation acts as a necessary and sufficient condition for approximate operator-algebra QECC and has been generalized to scenarios where there are restrictions on locality [4]. When \(\epsilon=0\) we can derive the standard correctability conditions for operator algebras [2].




  • Holographic code — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [5,6].


C. Bény, “Conditions for the approximate correction of algebras”, (2009) arXiv:0907.4207
C. Bény and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels”, Physical Review Letters 104, (2010) arXiv:0907.5391 DOI
D. Aharonov, A. Kitaev, and N. Nisan, “Quantum Circuits with Mixed States”, (1998) arXiv:quant-ph/9806029
C. Bény, Z. Zimborás, and F. Pastawski, “Approximate recovery with locality and symmetry constraints”, (2019) arXiv:1806.10324
A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015) arXiv:1411.7041 DOI
F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
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Zoo Code ID: approximate_oaecc

Cite as:
“Approximate operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_approximate_oaecc, title={Approximate operator-algebra error-correcting code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Approximate operator-algebra error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.