Approximate quantum error-correcting code (AQECC)[1–7] 

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].