Description
Protection
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].
Encoding
Decoding
Notes
Parents
Children
- Matrix-model code — Matrix-model codes approximately protect against gauge-invariant errors in the large-mode limit.
- \(U(d)\)-covariant approximate erasure code
- W-state code — The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.
- Eigenstate thermalization hypothesis (ETH) code — ETH codes approximately protect against erasures in the thermodynamic limit.
- Matrix-product state (MPS) code — MPS codes approximately protect against erasures in the thermodynamic limit.
- Singleton-bound approaching AQECC
- Landau-level spin code — The Landau-level spin code approximately protects against rotational errors.
Cousins
- Renormalization group (RG) cat code — RG cat codes approximately protect against displacements that represent ultraviolet coherent operators.
- Qudit-into-oscillator code — Approximate QEC techniques of finding the entanglement fidelity can be adapted to bosonic codes with a finite-dimensional codespace [16].
- Square-lattice GKP code — Square-lattice GKP codes approximately protect against photon loss [16–18].
- Gottesman-Kitaev-Preskill (GKP) code — Approximate error-correction offered by GKP codes yields achievable rates that are a constant away from the capacity of Guassian loss channels [18–21].
- Covariant code — Normalizable constructions of infinite-dimensional \(G\)-covariant codes for continuous \(G\) are approximately error-correcting.
- Holographic code — Universal subspace approximate error correction is used to model black holes [22].
- Local Haar-random circuit qubit code
- \([[4,2,2]]\) CSS code — \([[4,1,2]]\) subcodes \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [1] and \(\{|\overline{01}\rangle,|\overline{11}\rangle\}\) [23] approximately correct a single amplitude damping error.
- Approximate secret-sharing code — Secret-sharing codes approximately correct errors on up to \(\lfloor (n-1)/2 \rfloor\) errors.
- GNU permutation-invariant code — GNU codes protect approximately against amplitude damping errors.
References
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- P. Hayden and G. Penington, “Learning the Alpha-bits of black holes”, Journal of High Energy Physics 2019, (2019) arXiv:1807.06041 DOI
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Page edit log
- Victor V. Albert (2023-07-24) — most recent
- Milan Tenn (2023-07-19)
- Milan Tenn (2023-07-11)
- Milan Tenn (2023-07-03)
- Milan Tenn (2023-06-28)
- Milan Tenn (2023-06-22)
- Milan Tenn (2023-06-20)
- Victor V. Albert (2022-08-12)
- Philippe Faist (2022-07-15)
- Victor V. Albert (2021-12-05)
- Manasi Shingane (2021-12-05)
Cite as:
“Approximate quantum error-correcting code (AQECC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/approximate_qecc