Description
Encodes quantum information in a \(K\)-dimensional (logical) subspace of an \(N\)-dimensional (physical) Hilbert space such that it is possible to recover said information from errors. The logical subspace is spanned by a basis comprised of code basis states or codewords.Protection
Denoting Hilbert spaces by the letter \(\mathsf{H}\), a finite-dimensional quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) and a set of correctable errors \({\cal{E}}:\mathsf{H}_N\to\mathsf{H}_M\) with the following property: there exists a quantum operation \({\cal{D}}:\mathsf{H}_M\to \mathsf{H}_K\) such that for all \(E\in\cal{E}\) and states \(|\psi\rangle\in\mathsf{H}_{K}\), \begin{align} {\cal D}(EU|\psi\rangle\langle\psi|U^{\dagger}E^{\dagger})=c(E,|\psi\rangle)|\psi\rangle\langle\psi|\tag*{(1)}\end{align} for some constant \(c\) [1]. A code is said to protect against or correct the errors \(\mathcal{E}\).
Knill-Laflamme error-correction conditions
Equivalently, correction capability is determined by of the quantum error-correction conditions, which may admit infinite terms due to non-normalizability of ideal code states. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code.
Knill-Laflamme conditions: The Knill-Laflamme error-correction conditions [2–4][5; Thm. 10.1] are necessary and sufficient conditions for a code to successfully correct a set of errors in a finite-dimensional Hilbert space. A code (defined by a partial isometry \(U\)) with code space projector \(\Pi = U U^\dagger\) can correct a set of errors \(\{ E_j \}\) if and only if \begin{align} \Pi E_i^\dagger E_j \Pi = c_{ij}\, \Pi\qquad\text{for all \(i,j\),} \tag*{(2)}\end{align} where the QEC matrix elements \(c_{ij}\) are arbitrary complex numbers.
The Knill-Laflamme conditions can alternatively be expressed in terms of the complementary channel, in an entropic information-theoretic way via a data processing inequality [6–10], or can be interpreted thermodynamically [11]. They motivate higher-rank numerical ranges, which are generalizations of the numerical range of an operator [12–14]. They have been extended to sequences of multiple errors and rounds of correction [15].
Degeneracy: A code is degenerate with respect to a noise model if different errors map code states to the same error subspace. For a linearly independent error set \(\cal{E}\), degeneracy is equivalent to \(\text{rank}(c_{ij}) < |\cal{E}|\) [1].
Correctability of quantum channels
From now on, we use \(\mathcal{E}\) to denote a noise channel constructed out of the set of errors \(E\) and let \(\mathcal{U}(\cdot)=U(\cdot)U^\dagger\) be the superoperator corresponding to the partial encdoing isometry \(U\). A noise channel is correctable if there exists a recovery channel \(\mathcal{D}\) such that \begin{align} \mathcal{D}\mathcal{E}\mathcal{U}(\rho)=\rho \tag*{(3)}\end{align} for all logical states \(\rho\).
The above is equivalent to the fidelity between \(\rho\) and \(\mathcal{D}\mathcal{E}\mathcal{U}(\rho)\) being one for any notion of distance between quantum states. In particular, we can consider a scenario where we send only one part of an entangled state through a channel and determine whether the entanglement has been preserved during transmission. Using the notion of entanglement fidelity, a quantum channel \(\mathcal{E}\) is exactly correctable iff there exists a quantum channel \(\mathcal{D}\) such that \begin{align} (\mathcal{D}\mathcal{E}\mathcal{U}\otimes\mathrm{id})(\ket{\psi}\bra{\psi})=\ket{\psi}\bra{\psi} \tag*{(4)}\end{align} for all states \(\rho\) and their corresponding purifications \(\ket{\psi}\) (i.e., states \(\ket{\psi}\) such that \(\text{Tr}_{2}(|\psi\rangle\langle\psi|)=\rho\)).
The above entanglement fidelity condition can be alternatively expressed using complementary channels.
Complementary channel: A complementary channel \(\mathcal{E}^C\) is obtained from a channel \(\mathcal{E}\) that acts on a system by interpreting the channel as coming from a unitary operation acting on a larger system-environment tensor-product space (i.e., performing an isometric extension) and then tracing out the system factor (instead of the second environmental factor) [16; Sec. 5.2.2]. A noise channel \({\cal E}(\cdot)=\sum_{j}E_{j}(\cdot)E_{j}^{\dagger}\) admits a complementary channel of the form \begin{align} {\cal E}^{C}(\cdot)=\sum_{j,k}\text{Tr}\{E_{j}(\cdot)E_{k}^{\dagger}\}|j\rangle\langle k|~. \tag*{(5)}\end{align}
A channel \(\mathcal{E}\) is correctable if \(\mathcal{E}^C(\rho)=\rho_0\mathrm{Tr}(\rho)\) for some constant state \(\rho_0\), which is equivalent to the Knill-Laflamme conditions [17,18]. The logical and physical dimensions are related to the channel rank for non-degenerate codes via the quantum packing bound [19].
Decoding
The operation \(\cal{D}\) in the definition of this code is called the decoder. However, the term decoder can sometimes be used for the unencoder \(\cal{U}\) (i.e., the inverse of the encoder), which does not correct errors.There are several recovery maps which work for noise that is not exactly correctable; see AQECC entry.QECCs are useful [20] for the mean king's measurement problem [21].Protection can be implemented via autonomous QEC (a.k.a. continuous or continuous-time QEC) [22–26] via, e.g., reservoir engineering [27]; see review [28]. There are analogues of the Knill-Laflamme conditions for autonomous QEC [29,30], and it has been adapted to non-Markovian noise [31]. Information-theoretic bounds have been derived for open-loop control [32].Code Capacity Threshold
Coherent information of the state under the action of a noise channel can be used to estimate the optimal threshold [33].Primary Hierarchy
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Page edit log
- Victor V. Albert (2023-07-24) — most recent
- Milan Tenn (2023-07-17)
- Milan Tenn (2023-07-07)
- Milan Tenn (2023-07-03)
- Milan Tenn (2023-06-28)
- Victor V. Albert (2022-07-15)
- Victor V. Albert (2022-03-18)
- Victor V. Albert (2021-12-09)
Cite as:
“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qecc_finite
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/qecc_finite.yml.