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

Given a noise channel \(\mathcal{E}\), there exists a recovery channel \(\mathcal{D}\) such that \(F(\mathcal{D}\mathcal{E},\mathrm{id})=1-\epsilon\) iff there exists some \(\mathcal{D}'\) such that for complementary channel \(\mathcal{E}^C\), \(F(\mathcal{E}^C,\mathcal{D}')=1-\epsilon\) and \(D'(\rho)=\rho_0\) for some fixed \(\rho_0\). Note that \(F\) denotes worst case entaglement fidelity between channels.

We can generalize this by replacing \(\mathrm{id}\) with some channel \(M\). Given noise channel \(\mathcal{E}\), there exists a recovery channel \(\mathcal{D}\) such that \(F(\mathcal{D}\mathcal{E},\mathcal{M})=1-\epsilon\) iff there exists some \(\mathcal{D}'\) such that for complementary channel \(\mathcal{E}^C\), \(F(\mathcal{E}^C,\mathcal{D}'\mathcal{M}^C)=1-\epsilon\) [6].

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]. An alternative measure, the AQEC relative entropy, measures the relative entropy between \(\Lambda + B\) and \(\Lambda\) [8].

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}\left\lceil \frac{n r}{\log(d)} \right\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].

Circuit complexity

One can relate robustness of an approximate quantum code to the quantum circuit complexity [912] of creating states in the codespace. For a family of block codes, scaling as order \(O(k/n)\) of a code parameter called the subsystem variance characterizes the transition between code subspaces with low and high circuit complexity [13].


An extension of the BPT bound to approximate codes is done in Ref. [14].


Given a decoder, an encoding that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [3,15] (see also Ref. [16]).Variational quantum circuit encoder [17].


Given an encoding, a decoder that yields the optimal entanglement fidelity can be obtained by solving a semi-definite program [3,15] (see also Ref. [16]).The Petz recovery map (a.k.a. the transpose map) [18,19], 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 [20]. The infidelity of a modified Petz recovery map under erasure can be bounded using the conditional mutual information [14,21,22]. More generally, the fidelity can be expressed as a function of the Knill-Laflamme conditions [23; Thm. 1].


See review [24].





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“Approximate quantum error-correcting code (AQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_approximate_qecc, title={Approximate quantum error-correcting code (AQECC)}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Approximate quantum error-correcting code (AQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.