Approximate quantum error-correcting code (AQECC)[17] 


Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery.


Many of the state fidelity conditions that hold exactly for (exact) QECCs can be shown to hold up to some error \(\epsilon\) for approximate QECCs.

Entanglement fidelity

Let \(f(\rho_1,\rho_2)\) be the fidelity between quantum states. Let entanglement fidelity between channels \(\mathcal{N}\) and \(\mathcal{M}\) be defined as \begin{align} F_{\rho}(\mathcal{N},\mathcal{M}) =f((\mathcal{N}\otimes\mathrm{id})\ket{\psi}\bra{\psi},(\mathcal{M}\otimes\mathrm{id})\ket{\psi}\bra{\psi})~, \tag*{(1)}\end{align} where \(\ket{\psi}\) is a purification of the mixed state \(\rho\). The worst-case entaglement fidelity is then defined as \begin{align} F(\mathcal{N},\mathcal{M})=\min_{\rho} F_{\rho}(\mathcal{N},\mathcal{M})~. \tag*{(2)}\end{align}

Now, based on the Bures distance and worst case entaglement fidelity, we define \begin{align} d(\mathcal{N},\mathcal{M})=\sqrt{1-F(\mathcal{N},\mathcal{M})} \tag*{(3)}\end{align} as a measure of distance between quantum channels [6].

Given some encoding map \(\mathcal{U}\) and some noise channel \(\mathcal{E}\), the code described by \(\mathcal{U}\) is \(\epsilon\)-correctable if there exists some quantum channel \(\mathcal{D}\) such that \begin{align} d(\mathcal{D}\mathcal{E}\mathcal{U}(\rho),\rho)\leq \epsilon \tag*{(4)}\end{align} for all logical states \(\rho\) [6].

Complementary channel formulation

Necessary and sufficient approximate error-correction conditions

Analogously to the Knill-Laflamme conditions for (exact) QECCs, there exist various formulations of necessary and sufficient conditions for approximate error correction to determine if some code is \(\epsilon\)-correctable under a noise channel.

Necessary and sufficient conditions for approximate error correction can also be expressed in terms of complementary channels. Given some code defined by projector \(\Pi = U U^\dagger\), \(\Pi\) is \(\epsilon\)-correctable with respect to some noise channel \(\mathcal{E}\) if \begin{align} \Pi E_i^{\dagger}E_j \Pi =\lambda_{ij}\Pi +\Pi B_{ij}\Pi ~, \tag*{(5)}\end{align} where \(\Lambda(\rho)=\mathrm{Tr}(\rho)\sum_{ij} \lambda_{ij}|i\rangle\langle j|\) is a density operator, \begin{align} (\Lambda+B)(\rho)=\Lambda(\rho)+\sum_{ij}\mathrm{Tr}(\rho B_{ij})|i\rangle\langle j| \tag*{(6)}\end{align} is the output state of the complementary noise channel \(\mathcal{E}^C = \Lambda+B\), and the Bures distance \(d(\Lambda+B,\Lambda)\le\epsilon\) [6].

In addition to the necessary and sufficient error correction conditions, there exist sufficient conditions for AQECCs. Given a noise channel \(\mathcal{U}(\rho)=\sum_{n} A_n \rho A_n^{\dagger}\) where \(\forall{n}\), \(A_n\) is a Kraus operator, and code projector \(\Pi \), express the following using polar decomposition, \(A_n \Pi =U_n \sqrt{\Pi A_n^{\dagger}A_n \Pi }\), and let \(p_n\) and \(p_n\lambda_n\) be the largest and smallest eigenvalues for \(\Pi A_n^{\dagger}A_n \Pi \). Then, we are guaranteed that if \begin{align}\Pi U_m^{\dagger}U_n \Pi =\delta_{mn} \Pi \land p_n(1-\lambda_n)\le O(f(\epsilon))\tag*{(7)}\end{align} we have a fidelity \(F \geq 1-O(f(\epsilon))\) after recovery [1].

Universal subspace AQECCs and alpha-bits

Universal subspace approximate error correction is a type of approximate error correction that quantifies protection of information stored in (strict) subspaces of a logical space.

Given a subspace of a Hilbert space \(\mathsf{S}\) of dimension \(d\), noise channel \(\mathcal{E}\), and encoding \(\mathcal{U}\), we define the subspace as an \(\alpha\)-dit with error \(\epsilon\) if, for all subspaces \(\tilde{\mathsf{S}}\) of dimension less than or equal to \(d^{\alpha}+1\), there exists some channel \(\tilde{\mathcal{D}}\) such that \begin{align}||(\tilde{\mathcal{D}}\circ \mathcal{E}\circ \mathcal{U})|\psi\rangle-|\psi\rangle||_1\leq \epsilon\tag*{(8)}\end{align} for all \(|\psi\rangle\in \tilde{\mathsf{S}}\) [7].

Generalizing the notion of quantum information transmission and capacity of (exact) QECCs, one can achieve an \(\alpha\)-bit transmission rate \(r\) quantum channel \(\mathcal{E}\) iff, for sufficiently large \(d\) and \(n\), and for all \(\epsilon>0\), the channel \(\mathcal{E}^{\otimes n}\) is able to transmit \begin{align}\lceil \frac{n r}{\log(d)} \rceil\quad \textup{\(\alpha\)-dits}\tag*{(9)}\end{align} with total error \(\epsilon\) across those \(\alpha\)-dits. The \(\alpha\)-bit capacity \(Q\) of \(\mathcal{E}\) is defined as the supremum of achievable transmission rates [7].


Given a decoder, an encoding that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [3,8] (see also Ref. [9]).Variational quantum circuit encoder [10].


Given an encoding, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [3,8] (see also Ref. [9]).The Petz recovery map (a.k.a. the transpose map) [11,12], a quantum channel determined by the codespace and noise channel, recovers information perfectly for strictly correctable noise and yields an infidelity of recovery that is at most twice away from the infidelity of the best possible recovery [13]. The infidelity of a modified Petz recovery map can be bounded using relative entropies between uncorrupted and corrupted code states on countably infinite Hilbert spaces [14].


See review [15].





D. W. Leung et al., “Approximate quantum error correction can lead to better codes”, Physical Review A 56, 2567 (1997) arXiv:quant-ph/9704002 DOI
A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997) DOI
M. Reimpell and R. F. Werner, “Iterative Optimization of Quantum Error Correcting Codes”, Physical Review Letters 94, (2005) arXiv:quant-ph/0307138 DOI
C. Crepeau, D. Gottesman, and A. Smith, “Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes”, (2005) arXiv:quant-ph/0503139
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
P. Hayden and G. Penington, “Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit”, Communications in Mathematical Physics 374, 369 (2020) arXiv:1706.09434 DOI
K. Audenaert and B. De Moor, “Optimizing completely positive maps using semidefinite programming”, Physical Review A 65, (2002) arXiv:quant-ph/0109155 DOI
A. S. Fletcher, “Channel-Adapted Quantum Error Correction”, (2007) arXiv:0706.3400
C. Cao et al., “Quantum variational learning for quantum error-correcting codes”, Quantum 6, 828 (2022) arXiv:2204.03560 DOI
D. Petz, “Sufficient subalgebras and the relative entropy of states of a von Neumann algebra”, Communications in Mathematical Physics 105, 123 (1986) DOI
D. PETZ, “SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS”, The Quarterly Journal of Mathematics 39, 97 (1988) DOI
H. Barnum and E. Knill, “Reversing quantum dynamics with near-optimal quantum and classical fidelity”, (2000) arXiv:quant-ph/0004088
M. Junge et al., “Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy”, Annales Henri Poincaré 19, 2955 (2018) arXiv:1509.07127 DOI
A. Jayashankar and P. Mandayam, “Quantum Error Correction: Noise-adapted Techniques and Applications”, (2022) arXiv:2208.00365
V. V. Albert et al., “Performance and structure of single-mode bosonic codes”, Physical Review A 97, (2018) arXiv:1708.05010 DOI
B. M. Terhal and D. Weigand, “Encoding a qubit into a cavity mode in circuit QED using phase estimation”, Physical Review A 93, (2016) arXiv:1506.05033 DOI
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
K. Sharma et al., “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New Journal of Physics 20, 063025 (2018) arXiv:1708.07257 DOI
M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018) arXiv:1801.04731 DOI
P. Hayden and G. Penington, “Learning the Alpha-bits of black holes”, Journal of High Energy Physics 2019, (2019) arXiv:1807.06041 DOI
G. Alber et al., “Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes”, Physical Review Letters 86, 4402 (2001) arXiv:quant-ph/0103042 DOI
Page edit log

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: approximate_qecc

Cite as:
“Approximate quantum error-correcting code (AQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
  title={Approximate quantum error-correcting code (AQECC)},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Approximate quantum error-correcting code (AQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.