Quantum error-transmuting code (QETC)[1]

Protection

For a pair \(E=\{E_i\}\) and \(\overline{M} = \{\overline{m}_{\alpha}\}\) of physical and logical error sets, \(\mathsf{C}\) is a QETC if there exists a recovery operation \(\mathcal{R}\) such that for all physical noise channels \(\mathcal{E}\) with Kraus operators proportional to elements of \(\{E_i\}\), and density matrices \(\rho\) supported on \(\mathsf{C}\), we have: \begin{align} \mathcal{R} \circ \mathcal{E} (\rho) = \mathcal{M} (\rho). \tag*{(1)}\end{align} Here \(\mathcal{M}\) is a logical noise channel whose Kraus operators are proportional to elements of the cosets \(\{\mathcal{A}(\overline{m}_{\alpha})\mathsf{S}\} \subset \mathcal{G}_n\), where \begin{align} \mathcal{G}_k \xrightarrow[]{\mathcal{A}} \mathsf{N(S)}/\mathsf{S} \tag*{(2)}\end{align} is a choice of isomorphism, i.e. an identification of the logical Pauli group with elements of the physical Pauli group. \(\mathsf{N(S)}\) denotes the normaliser of \(\mathsf{S}\) in \(\mathcal{G}_n\).

Equivalently, this can be rephrased in terms of a generalization of the Knill-Laflamme conditions for error-correction. Label the cosets of \(\mathsf{N(S)}\) in \(\mathcal{G}_n\) by \(\mathfrak{n}\), and the physical errors in a given coset by \(E_{\mathfrak{n}} := \{E_{\mathfrak{n},i}\}\). Then \(\mathsf{C}\) is a QETC for a pair \(\{E, \overline{M}\}\) if and only if there exists an isomorphism \(\mathcal{A}: \overline{\mathcal{G}}_k \rightarrow \mathsf{N(S)}/\mathsf{S}\), such that for each \(\mathfrak{n}\) there exists a map: \begin{align} \pi_{\mathfrak{n}}: E_{\mathfrak{n}} \rightarrow \mathcal{A}(\overline{M}),\qquad \pi_{\mathfrak{n}}(E_{\mathfrak{n},i}) := \overline{m}_{\pi_{\mathfrak{n}},i} \tag*{(3)}\end{align} so that \(\forall\, E_{\mathfrak{n},i},\,E_{\mathfrak{n},j} \in E_{\mathfrak{n}}\): \begin{align} P E_{\mathfrak{n},i}^\dagger E_{\mathfrak{n},j} P = P{m}_{\pi_{\mathfrak{n}},i}^\dagger m_{\pi_{\mathfrak{n}},j}P. \tag*{(4)}\end{align} Here, \(P\) is the projector onto \(\mathsf{C}\), and \( m_{\pi_{\mathfrak{n}},i}\) is a choice of representative of \(\overline{m}_{\pi_{\mathfrak{n}},i}\) in \(\mathsf{N(S)}\).

For a QETC, the effective code distance \(d_{\text{eff}}\) is defined to be \(2w+1\), where \(w\) is the maximum weight such that the set of all errors with weight \(\leq w\) obey the generalised Knill-Laflamme conditions above. The code \(\mathsf{C}\) is thus able to transmute the set of all Pauli errors of weight \(\leq w\) to the admissible error set \(\overline{M}\).