~

codes/eacq.yml

@@ -34,8 +34,8 @@ features:
 
 relations:
   cousins:
-    - code_id: oaecc
-      detail: 'EQ hybrid codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to operator-algebra QECCs when said subsystems are interpreted as noiseless physical subsystems.'
+    - code_id: hybridqecc
+      detail: 'EQ hybrid codes utilize additional ancillary subsystems in a pre-shared entangled state, but reduce to hybrid QECCs when said subsystems are interpreted as noiseless physical subsystems.'
 
 
 # Begin Entry Meta Information

codes/oaecc.yml

@@ -5,16 +5,16 @@
 
 code_id: oaecc
 
-name: 'Operator-algebra QECC'
+name: 'Operator-algebra QECC (OAQECC)'
 short_name: 'OAQECC'
 introduced: '\cite{arxiv:quant-ph/0608071,arxiv:0705.1574,arxiv:1005.0353,doi:10.1142/S0219749908003839,arxiv:2012.14001}'
 
 description: |-
-  A code that encompasses both subsystem codes and hybrid codes using an operator-algebraic framework.
+  A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, and hybrid codes using a unified operator-algebraic framework.
 
-  A simple example of an OAQECC encodes both a single logical qubit in a subsystem code and a single classical bit in some Hilbert space \(\mathsf{H}\).
-  A different quantum subsystem code \(\mathsf{A}_j\otimes\mathsf{B}_j\), with \(\mathsf{A}_j\) the logical qubit factor and \(\mathsf{B}_j\) the gauge qubit factor, is associated with each of the two values \(j\in\{0,1\}\) of the classical bit.
-  The corresponding decomposition is
+  A simple example of an OAQECC encodes a single logical qubit in a subsystem code and a single classical bit.
+  A different quantum subsystem code \(\mathsf{A}_j\otimes\mathsf{B}_j\), with \(\mathsf{A}_j\) the logical qubit factor, and \(\mathsf{B}_j\) the gauge qubit factor, is associated with each of the two values \(j\in\{0,1\}\) of the classical bit.
+  The corresponding decomposition of the Hilbert space \(\mathsf{H}\) is
   \begin{align}
     \mathsf{H}=(\mathsf{A}_{1}\otimes\mathsf{B}_{1})\oplus(\mathsf{A}_{2}\otimes\mathsf{B}_{2})\oplus\mathsf{C}^{\perp}~,
   \end{align}
@@ -23,20 +23,20 @@ description: |-
 
   More generally, the code is determined by a finite dimensional \(C^*\) algebra \(\mathcal{A}\) of operators on \(\mathsf{H}\).
   This \textit{logical algebra} induces a decomposition of the Hilbert space as
-  \begin{align}\mathsf{H} = \bigoplus_\gamma A_\gamma \otimes B_\gamma,\end{align}
+  \begin{align}\mathsf{H} = \bigoplus_\gamma \mathsf{A}_\gamma \otimes \mathsf{B}_\gamma,\end{align}
   with respect to which \(\mathcal{A}\) takes the form
-  \begin{align}\mathcal{A} = \bigoplus_\gamma I_\gamma \otimes \mathcal{L}(B_\gamma),\end{align}
-  where \(\mathcal{L}(B_\gamma)\) denotes the full set of linear maps on \(B_\gamma\).
+  \begin{align}\mathcal{A} = \bigoplus_\gamma I_\gamma \otimes \mathcal{L}(\mathsf{B}_\gamma),\end{align}
+  where \(\mathcal{L}(\mathsf{B}_\gamma)\) denotes the full set of linear maps on \(\mathsf{B}_\gamma\).
 
 protection: |
   Given an error operation \(\mathcal{E}\), one says that \(\mathcal{A}\) is \textit{correctable} for \(\mathcal{E}\) if there exists a recovery operation \(\mathcal{R}\) such that
-  \begin{align}P_{\mathcal{A}} (\mathcal{R} \circ \mathcal{E})^\dagger(X) P_{\mathcal{A}} = X\end{align} for all \(X \in \mathcal{A}\), where \(P_{\mathcal{A}}\) is the unit projection onto \(\mathcal{A}\).
+  \begin{align}\Pi_{\mathcal{A}} (\mathcal{R} \circ \mathcal{E})^\dagger(X) \Pi_{\mathcal{A}} = X\end{align} for all \(X \in \mathcal{A}\), where \(\Pi_{\mathcal{A}}\) is the unit projection onto \(\mathcal{A}\).
 
-  Equivalently, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if there exists a recovery operation \(\mathcal{R}\) such that for any \(\gamma\) and density operators \(\rho_\gamma,\sigma_\gamma\) supported on \(A_\gamma\) and \(B_\gamma\), respectively, there exists a state \(\tau_\gamma\) supported on \(A_\gamma\) such that
+  Equivalently, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if there exists a recovery operation \(\mathcal{R}\) such that for any \(\gamma\) and density operators \(\rho_\gamma,\sigma_\gamma\) supported on \(\mathsf{A}_\gamma\) and \(\mathsf{B}_\gamma\), respectively, there exists a state \(\tau_\gamma\) supported on \(A_\gamma\) such that
   \begin{align}(\mathcal{R} \circ \mathcal{E})(\rho_\gamma \otimes \sigma_\gamma) = \tau_\gamma \otimes \sigma_\gamma.\end{align}
 
   An algebraic condition for correctability can be given in terms of the Kraus operators \(E_j\) of \(\mathcal{E}\).
-  Indeed, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if \begin{align}P_{\mathcal{A}} E_j^\dagger E_k P_{\mathcal{A}} \in \mathcal{A}'\end{align}
+  Indeed, \(\mathcal{A}\) is correctable for \(\mathcal{E}\) if \begin{align}\Pi_{\mathcal{A}} E_j^\dagger E_k \Pi_{\mathcal{A}} \in \mathcal{A}'\end{align}
   for all \(j,k\), where \(\mathcal{A}'\) is the commutant of \(\mathcal{A}\).
 
   Tradeoffs between error correction and privacy have been studied \cite{arxiv:1811.10425}.

codes/quantum/hybridqecc.yml

@@ -12,14 +12,13 @@ introduced: '\cite{arxiv:quant-ph/0203105,arXiv:quant-ph/0311131,arxiv:quant-ph/
 
 description: |
   A quantum code which encodes both quantum and classical information.
-  Usually denoted as \(((n,K:M))\) or \(((n,K:M,d))\), where \(K\) is the dimension of the underlying quantum code, where \(M\) is the size of the classical code, and where \(d\) is the distance.
 
   A simple example of a hybrid QECC encodes a single qubit and a single classical bit.
   A different quantum code \(\mathsf{C}_j\) is associated with each of the two values \(j\in\{0,1\}\) of the classical bit.
   The error words corresponding to correctable errors must satisfy the \term{Knill-Laflamme conditions} for each code \cite[Eq. (3)]{arxiv:1701.06963}, and error words beloning to different codes must be orthogonal to each other \cite[Eq. (4)]{arxiv:1701.06963}.
-  The corresponding decomposition of the hilbert space is
+  The corresponding decomposition of the Hilbert space \(\mathsf{H}\) is
   \begin{align}
-    \mathsf{C}_{1}\oplus\mathsf{C}_{2}\oplus\mathsf{C}^{\perp}~,
+    \mathsf{H} = \mathsf{C}_{1}\oplus\mathsf{C}_{2}\oplus\mathsf{C}^{\perp}~,
   \end{align}
   where \(\mathsf{C}^\perp\) is the combined error space of both codes.
 
@@ -33,7 +32,7 @@ features:
 relations:
   parents:
     - code_id: oaecc
-      detail: 'An OAQECC which has no gauge qubits but has a block structure that corresponds to a classical code is a hybrid QECC.'
+      detail: 'An OAQECC which has no gauge structure (e.g., gauge qubits) but has a block structure that corresponds to a classical code is a hybrid QECC.'
     - code_id: quantum_into_quantum
   cousins:
     - code_id: qecc

codes/quantum/oecc.yml

@@ -33,7 +33,7 @@ notes:
 relations:
   parents:
     - code_id: oaecc
-      detail: 'An OAQECC which has gauge qubits but no block structure is a subsystem QECC.'
+      detail: 'An OAQECC which has gauge structure (e.g., gauge qubits) but no block structure is a subsystem QECC.'
     - code_id: quantum_into_quantum
   cousins:
     - code_id: qecc

codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml

@@ -16,6 +16,8 @@ description: |
 
   The centralizer for the stabilizer group within the displacement operators case can be identified with the symplectic dual lattice \({\mathcal{L}}^{\perp}\) (i.e. all points in \(\mathbb{R}^{2n}\) that have integer symplectic inner product with all points in \({\mathcal{L}}\) ), such that logical operations are identified with the dual quotients \({\mathcal{L}}^{\perp}/{\mathcal{L}}\). The size of this dual quotient is the determinant of the Gram matrix, yielding the logical dimension \(d=\sqrt{\| \det{A}\|}\) \cite{arxiv:quant-ph/0008040}.
 
+  The space of all single-mode GKP codes is the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed \cite{arXiv:2407.03270}.
+
 protection: 'The level of protection against displacement errors is quantified by the Euclidean code distance \(\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2\) \cite{arxiv:2109.14645}.'
 
 features:
@@ -33,7 +35,9 @@ features:
     - 'By applying GKP error correction to Gaussian input states, computational universality can be achieved without additional non-Gaussian elements  \cite{arxiv:1903.00012}.'
 
   fault_tolerance:
-    - 'Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors \cite{arxiv:quant-ph/0008040}.'
+    - 'Logical Clifford operations are given by Gaussian unitaries, which map bounded-size errors to bounded-size errors \cite{arxiv:quant-ph/0008040}.
+    For single-mode GKP codes, these operations correspond to non-trivial loops in the space of all single-mode GKP codes (the moduli space of elliptic curves, i.e., the three sphere with a trefoil knot removed) \cite{arXiv:2407.03270}.
+    Such gates provide another example of monodromy under the particular notion of parallel transport introduced in Ref. \cite{arxiv:1309.7062}.'
 
   decoders:
     - 'The MLD decoder for Gaussian displacement errors is realized by evaluating a lattice theta function, and in general the decision can be approximated by either solving (approximating) the closest vector problem (CVP) \cite{doi:10.1109/TIT.2002.800499} (a.k.a. closest lattice point problem) or by using other effective iterative schemes when, e.g., the lattice represents a concatenated GKP code \cite{arxiv:1810.00047,arxiv:1908.03579,arxiv:2109.14645,arxiv:2111.07029}.

codes/quantum/properties/qecc.yml

@@ -68,7 +68,7 @@ relations:
     - code_id: approximate_qecc
       detail: 'QAECCs correcting a noise channel exactly reduce to QECCs.'
     - code_id: oaecc
-      detail: 'An OAQECC which has no gauge qubits and no block structure is a QECC.'
+      detail: 'An OAQECC which has no gauge degrees of freedom (e.g., gauge qubits) and no block structure is a QECC.'
   cousins:
     - code_id: ecc
       detail: 'Quantum information cannot be copied using a linear process \cite{doi:10.1038/299802a0}, so one cannot send several copies of a quantum state through a channel as can be done for classical information.

codes/quantum/qubits/hybrid_qubits_into_qubits.yml → codes/quantum/qubits/oa/hybrid_qubits_into_qubits.yml

@@ -24,10 +24,13 @@ protection: |
 
 relations:
   parents:
+    - code_id: oa_qubits_into_qubits
+      detail: 'An OA qubit code which has no gauge qubits but has a block structure that corresponds to a classical code is a hybrid qubit code.'
     - code_id: hybridqecc
   cousins:
     - code_id: qubits_into_qubits
-      detail: 'Any qubit code can be converted into a hybrid qubit code by using some its qubits to store only classical information \cite{arxiv:0802.2414}.'
+      detail: 'A hybrid qubit code storing no classical information reduces to a qubit code.
+      Conversely, any qubit code can be converted into a hybrid qubit code by using some its qubits to store only classical information \cite{arxiv:0802.2414}.'
     - code_id: qubit_classical_into_quantum
       detail: 'A hybrid qubit code storing no quantum information reduces to a qubit c-q code.'
 

codes/quantum/qubits/oa_stabilizer/hybrid_stabilizer.yml → codes/quantum/qubits/oa/hybrid_stabilizer.yml

@@ -15,10 +15,8 @@ description: |
   Usually denoted as \([[n,k:c]]\) or \([[n,k:c,d]]\), where \(k\) (\(c\)) is the number of encoded qubits (classical bits), and where \(d\) is the distance.
 
   The algebraic structure of a hybrid stabilizer code is the same as that of a USt code whose cosets are indexed by a linear binary code:
-  both codes utilize codewords of an inner \([[n,k]]\) qubit stabilizer code \(\mathsf{C}\) and its cosets \(t \mathsf{C}\) by \(2^c\) Pauli strings \(t\), with the coset index \(t\) corresponding to a codeword of the outer \([n,c]\) linear binary code.
-  However, the hybrid stabilizer code does not utilize superpositions of codewords of \(t \mathsf{C}\) and \(t^{\prime} \mathsf{C}\) for \(t \neq t^{\prime}\).
-  As such, a hybrid stabilizer code can be thought of as a particular USt code some of whose qubits are used for storing classical information.
-
+  both codes utilize codewords of an inner \([[n,k]]\) qubit stabilizer code \(\mathsf{C}\) and its cosets \(t \mathsf{C}\), where the \(2^c\) Pauli strings \(t\) correspond to the outer \([n,c]\) linear binary code.
+  However, the hybrid stabilizer code does not utilize superpositions of codewords of \(t \mathsf{C}\) and \(t^{\prime} \mathsf{C}\) for \(t \neq t^{\prime}\) since the different coset blocks correspond to classical codewords.
 
 relations:
   parents:
@@ -28,13 +26,14 @@ relations:
       detail: 'An \([[n,k:c,d]]\) hybrid stabilizer code is an \(((n,2^k:2^c,d))\) hybrid qubit code.'
   cousins:
     - code_id: qubit_stabilizer
-      detail: 'Any qubit stabilizer code can be converted into a hybrid stabilizer code by using some its qubits to store only classical information \cite{arxiv:0802.2414}.'
+      detail: 'A hybrid stabilizer code storing no classical information reduces to a qubit stabilizer code.
+      Conversely, any qubit stabilizer code can be converted into a hybrid stabilizer code by using some its qubits to store only classical information \cite{arxiv:0802.2414}.'
     - code_id: non_stabilizer
       detail: 'The algebraic structure of a hybrid stabilizer code is the same as that of a USt code whose cosets are indexed by a linear binary code \cite{arxiv:0802.2414}.'
     - code_id: shor_nine
       detail: 'The Shor code can be modified to store three additional classical bits to yield a \([[9,1:3,3]]\) hybrid stabilizer code \cite{arxiv:0802.2414}.'
     - code_id: iceberg
-      detail: 'The \([[2m+1,2m+2:1,2]]\) hybrid stabilizer code \cite{arxiv:1911.12260} (extendable to modular qudits \cite{arxiv:2002.11075}) is closely related to the \([[2m,2m-2,2]]\) qubit stabilizer code.'
+      detail: 'The \([[2m+1,2m+2:1,2]]\) hybrid stabilizer code \cite{arxiv:1911.12260} (extendable to modular qudits \cite{arxiv:2002.11075}) is closely related to the \([[2m,2m-2,2]]\) error-detecting code.'
     - code_id: stab_4_2_2
       detail: 'The \([[4,2,2]]\) codewords can be modified by signs to yield a \([[4,1:1,2]]\) hybrid stabilizer code \cite{arxiv:1806.03702}.'
     - code_id: subsystem_stabilizer

codes/quantum/qubits/oa_stabilizer/qubit_stabilizer_oaqecc.yml → codes/quantum/qubits/oa/qubit_stabilizer_oaqecc.yml

@@ -28,7 +28,7 @@ protection: |-
 
 relations:
   parents:
-    - code_id: oaecc
+    - code_id: oa_qubits_into_qubits
   cousins:
     - code_id: bacon_shor
       detail: 'The OA qubit stabilizer formalism yields hybrid Bacon-Shor codes \cite{arxiv:2304.11442}.'

codes/quantum/qubits/oa_qubits_into_qubits.yml

@@ -0,0 +1,24 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: oa_qubits_into_qubits
+
+name: 'Operator-algebra (OA) qubit code'
+short_name: 'OA qubit'
+
+description: |-
+  An OA code defined on \(n\) qubits.
+
+relations:
+  parents:
+    - code_id: oaecc
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-07-05'

codes/quantum/qubits/qubits_into_qubits.yml

@@ -132,6 +132,8 @@ notes:
 
 relations:
   parents:
+    - code_id: oa_qubits_into_qubits
+      detail: 'An OA qubit code which has no gauge qubits and no block structure is a qubit code.'
     - code_id: qudits_into_qudits
       detail: 'Modular-qudit quantum codes for \(q=2\) correspond to qubit codes.'
     - code_id: galois_into_galois

codes/quantum/qubits/subsystem_qubits_into_qubits.yml

@@ -18,6 +18,8 @@ description: |
 
 relations:
   parents:
+    - code_id: oa_qubits_into_qubits
+      detail: 'An OA qubit code which has gauge qubits but no block structure is a subsystem qubit code.'
     - code_id: subsystem_qudits_into_qudits
       detail: 'Subsystem modular-qudit codes reduce to subsystem qubit codes for qudit dimension \(q=2\).'
     - code_id: subsystem_galois_into_galois

codetree/kingdoms.yml

@@ -44,9 +44,8 @@ kingdoms_by_domain_id:
       name: 'Qubit Kingdom'
       root_codes:
         - code_id: qubits_into_qubits
-        - code_id: hybrid_qubits_into_qubits
         - code_id: subsystem_qubits_into_qubits
-        - code_id: qubit_stabilizer_oaqecc
+        - code_id: oa_qubits_into_qubits
         - code_id: ea_qubits_into_qubits
         - code_id: qetc