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codes/quantum/oscillators/coherent_state/cat_concatenated.yml

@@ -10,30 +10,32 @@ name: 'Concatenated cat code'
 introduced: '\cite{arxiv:1409.6759}'
 
 description: |
-  A concatenated code whose outer code is a cat code. Most examples encode physical qubits of an inner stabilizer codes into the two-component cat code.
+  A concatenated code whose outer code is a cat code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another cat outer code. Most examples encode physical qubits of an inner stabilizer code into the two-component cat code.
 
 protection: |
-  The cat code suppressed dephasing errors exponentially with the size of the coherent states, so the inner code (e.g., a quantum repetition code \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2212.11927}) can be highly biased toward one type of noise while still ensuring good performance.
+  The cat code suppresses dephasing errors exponentially with the size of its coherent states, so the inner code (e.g., a quantum repetition code \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2212.11927,arxiv:2212.11927}) can be highly biased toward one type of noise while still ensuring good performance.
+
+  A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.
 
 
 relations:
   parents:
     - code_id: qsc
   cousins:
     - code_id: quantum_repetition
-      detail: 'Cat codes have been concatenated with quantum repetition codes \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2012.04108,arxiv:2212.11927}.'
+      detail: 'Two-component cat codes have been concatenated with quantum repetition codes \cite{arxiv:1904.09474,arxiv:1905.00450,arxiv:2009.10756,arxiv:2012.04108,arxiv:2212.11927}.'
     - code_id: rotated_surface
       detail: 'Cat codes have been concatenated with rotated surface codes \cite{arxiv:2012.04108}.'
     - code_id: ldpc
       detail: 'Cat codes have been concatenated with LDPC codes (treated as qubit stabilizer codes) \cite{arxiv:2401.09541}.'
     - code_id: lhz
       detail: 'LHZ parity-codes have been concatenated with cat codes \cite{arxiv:2404.11332}.'
     - code_id: steane
-      detail: 'Steane codes have been concatenated with cat codes \cite{arxiv:0707.0327}.'
+      detail: 'Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) \cite{arxiv:0707.0327}.'
     - code_id: qubit_golay
-      detail: 'Quantum Golay codes have been concatenated with cat codes \cite{arxiv:0707.0327}.'
-
-
+      detail: 'Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) \cite{arxiv:0707.0327}.'
+    - code_id: self_correct
+      detail: 'A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.'
 
 
 # Begin Entry Meta Information

codes/quantum/oscillators/coherent_state/two-legged-cat.yml

@@ -66,10 +66,6 @@ relations:
       detail: 'Two-legged cat and quantum repetition codes can be thought of as classical codes because they protect against only one type of noise. Two-legged cat codes (quantum repetition) codes suppress cavity dephasing (bit-flip) noise exponentially with \(|\alpha|^2\) (\(n\)). The stability offered by cat codes has been linked to several favorable properties of phases of matter associated with the repetition-code Hamiltonian \cite{arxiv:1804.11293,arxiv:2008.02816}.'
     - code_id: coherent_state_c-q
       detail: 'Two-component cat codes can be thought of as coherent-state c-q codes because they protect against only one type of noise and thus only reliably store classical information.'
-    - code_id: self_correct
-      detail: 'A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.'
-    - code_id: cat_concatenated
-      detail: 'A concatenation of the repetition code with the two-component cat code is a candidate for a memory that may be self-correcting, but only in the limit of infinite energy per mode \cite{arxiv:2205.09767}.'
 
 
 # Begin Entry Meta Information

codes/quantum/oscillators/hybrid/hybrid_cat.yml

@@ -11,10 +11,10 @@ name: 'Hybrid cat code'
 introduced: '\cite{arxiv:1112.0825,arxiv:2401.00450}'
 
 description: |
-  A hybrid qubit-oscillator code admitting codewords that are tensor products of either a cat or coherent state and a single-qubit (photon polarization) state.
+  A hybrid qubit-oscillator code admitting codewords that are tensor products of a single-qubit (e.g., photon polarization) state with either a cat state or a coherent state.
    
-  Codewords of the coherent-state version \cite{arxiv:1112.0825} are \(|\alpha\rangle|+\rangle\) and \(|-\alpha\rangle|-\rangle\), i.e., hyper-entangled states of the polarization \(|\pm\rangle\) and occupation-number degrees of freedom of a photon, with the latter being in a coherent state \(|\pm\alpha\rangle\).
-  Codewords of a cat-state version \cite{arxiv:1712.10206,arxiv:2401.00450} are \((\left|\alpha\right\rangle +\left|-\alpha\right\rangle )|+\rangle\) and \((\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle )|-\rangle\)
+  Codewords of the coherent-state version \cite{arxiv:1112.0825} are \(|\alpha\rangle|+\rangle\) and \(|-\alpha\rangle|-\rangle\), i.e., hyper-entangled states \cite{doi:10.1080/09500349708231877} of the occupation-number and polarization degrees of freedom of a photon.
+  Codewords of the cat-state version \cite{arxiv:1712.10206,arxiv:2401.00450} are proportional to \((\left|\alpha\right\rangle +\left|-\alpha\right\rangle )|+\rangle\) and \((\left|i\alpha\right\rangle -\left|-i\alpha\right\rangle )|-\rangle\)
 
 features:
   fault_tolerance:
@@ -28,12 +28,12 @@ relations:
     - code_id: oscillators
       detail: 'The physical Hilbert space of a hybrid qubit-oscillator code contains at least one oscillator.'
   cousins:
-    - code_id: qudits_into_oscillators
-      detail: 'A hybrid qudit-oscillator code with \(n_1=0\) is a qudit-into-oscillator code.'
     - code_id: cat
       detail: 'Hybrid cat codewords consist of a bosonic mode in either coherent or cat states.'
     - code_id: rbh
       detail: 'Hybrid cat codes can be concatenated with RBH codes \cite{arxiv:2401.00450}.'
+    - code_id: oscillators_concatenated
+      detail: 'Hybrid cat codes can be concatenated with RBH codes \cite{arxiv:2401.00450}.'
 
 
 

codes/quantum/oscillators/stabilizer/hyperplane/cv_cluster_state.yml

@@ -20,9 +20,9 @@ description: |
   The exact analog cluster state is non-normalizable, so approximate constructs have to be considered.
 
   Analog cluster states are analog stabilizer states defined on a graph.
-  There is one nullifier \(\eta_j\) per graph vertex \(j\) of the form
+  There is one nullifier \(\hat{\eta}_j\) per graph vertex \(j\) of the form
   \begin{align}
-    \eta_j = \hat{p}_{j} - \sum_{k\in N(j)} V_{jk} \hat{x}_k~,
+    \hat{\eta}_j = \hat{p}_{j} - \sum_{k\in N(j)} V_{jk} \hat{x}_k~,
   \end{align}
   where the neighborhood \(N(j)\) is the set of vertices which share an edge with \(j\), and where \(V_{jk}\) is a weighed (real-valued) adjacency matrix of a graph \cite{arxiv:1912.06463}.
 

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

@@ -19,12 +19,11 @@ features:
 
 relations:
   parents:
-    - code_id: multimodegkp
+    - code_id: gkp_concatenated
+      detail: 'The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code \cite{arxiv:2201.12337}. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via \term{Construction A}.'
     - code_id: qudits_into_oscillators
   cousins:
     - code_id: dfour
-    - code_id: gkp_concatenated
-      detail: 'The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code \cite{arxiv:2201.12337}. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via \term{Construction A}.'
     - code_id: quantum_repetition
       detail: 'The \(D_4\) hyper-diamond GKP code can be seen as a concatenation of a rotated square-lattice GKP code with a repetition code \cite{arxiv:2201.12337}. This is related to the fact that the four-bit repetition code yields the \(D_4\) hyper-diamond lattice code via \term{Construction A}.'
 

codes/quantum/oscillators/stabilizer/lattice/gkp-cluster-state.yml

@@ -14,28 +14,27 @@ alternative_names:
   - 'Hybrid cluster-state code'
 
 description: |
-  This code can be thought of as a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. 
-  Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i \theta_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(\theta_{jk}\).
+  Cluster-state code can consists of a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. 
+  Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\).
   Provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol \cite{arxiv:1310.7596,arxiv:2010.02905,arxiv:1712.00294,arxiv:2104.03241}.
 
-  Logical Clifford gates are performed on the cluster state via a combination of linear-optical gates and homodyne measurements on subsets of vertices \cite{arxiv:quant-ph/0605198,arxiv:0903.3233}. Magic-state distillation is required for universal computation. GKP error correction can be naturally combined with CV measurement-based protocols since the performance of both is quantified by a squeezing parameter.
 
 features:
   encoders:
     - 'Initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i \theta_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(\theta_{jk}\).'
   general_gates:
+    - 'Logical Clifford gates are performed on the cluster state via a combination of linear-optical gates and homodyne measurements on subsets of vertices \cite{arxiv:quant-ph/0605198,arxiv:0903.3233}. Magic-state distillation is required for universal computation.'
     - 'Single-mode logical Clifford gates can be performed using Gaussian operations and measurements on a 1D GKP cluster state, while two-mode logical Clifford gates require a 2D cluster state. Magic-state distillation using photon-counting can be used for a non-Clifford logical \(\pi/8\) gate.'
     - 'Gate teleportation and error correction can be performed without active squeezing \cite{arxiv:2008.12791}.'
+  decoders:
+    - 'GKP error correction can be naturally combined with CV MBQC protocols since the performance of both is quantified by a squeezing parameter \cite{arxiv:1310.7596}.'
   threshold:
     - 'A lower bound on the squeezing required to obtain a particular error rate can be formulated in terms of the displacement noise strength. This in turn determines how much squeezing is required in order to be below threshold for a particular concatenated code. A threshold of \(10^{-6}\) yields a required squeezing of 20.5 dB \cite{arxiv:1310.7596}. Anti-squeezing does not affect the threshold \cite{arxiv:1903.02162}.'
 
-  fault_tolerance:
-    - 'First encoding demonstrating the possibility of fault-tolerant measurement-based computation with analog cluster states. A fault-tolerance threshold can be achieved by concatenating existing fault-tolerant schemes for qubit-based cluster-state encodings with the GKP code \cite{arxiv:1310.7596}.'
-
 relations:
   parents:
-    - code_id: quantum_lattice
-      detail: 'A GKP CV-cluster-state code can be created by initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i \theta_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(\theta_{jk}\).'
+    - code_id: gkp-stabilizer
+      detail: 'A GKP CV-cluster-state code can be created by initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\).'
     - code_id: qudits_into_oscillators
   cousins:
     - code_id: cv_cluster_state

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

@@ -10,7 +10,7 @@ name: 'Concatenated GKP code'
 introduced: '\cite{arxiv:1706.03011}'
 
 description: |
-  A concatenated code whose outer code is a GKP code.
+  A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code.
 
 protection: |
   The analog syndrome information of the outer GKP code can improve protection of the inner code. As an example, concatenating a three-qubit quantum repetition code with GKP codes can correct some two-bit-flip errors \cite{arxiv:1706.03011}.

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

@@ -10,7 +10,7 @@ name: 'GKP-surface code'
 introduced: '\cite{arxiv:1712.00294,arxiv:1810.00047}'
 
 description: |
-  A concatenated code whose outer code is a GKP code and whose inner code is a toric code \cite{arxiv:1810.00047}, surface code \cite{arxiv:1712.00294}, rotated surface code \cite{arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994}, or XZZX surface code \cite{arxiv:2207.04383}.
+  A concatenated code whose outer code is a GKP code and whose inner code is a toric surface code \cite{arxiv:1810.00047}, rotated surface code \cite{arxiv:1712.00294,arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994,arxiv:2303.04702}, or XZZX surface code \cite{arxiv:2207.04383}.
 
 
 
@@ -35,7 +35,7 @@ relations:
     - code_id: toric
       detail: 'GKP codes have been concatenated with toric codes \cite{arXiv:1810.00047}.'
     - code_id: rotated_surface
-      detail: 'GKP codes have been concatenated with rotated surface codes \cite{arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994}.'
+      detail: 'GKP codes have been concatenated with rotated surface codes \cite{arxiv:1712.00294,arxiv:1908.03579,arxiv:2101.03014,arxiv:2103.06994,arxiv:2303.04702}.'
     - code_id: xzzx
       detail: 'GKP codes have been concatenated with XZZX surface codes \cite{arxiv:2207.04383}.'
     - code_id: asymmetric_qecc

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

@@ -44,7 +44,7 @@ relations:
       detail: 'Quantum lattice codewords can be written as superpositions of coherent states lying on a lattice in phase space \cite{arxiv:quant-ph/0008040,arxiv:1708.05010}.'
   cousins:
     - code_id: points_into_lattices
-      detail: 'Quantum lattice codes can be thought of as quantum lattice codes because they store information in quantum superpositions of points on a lattice in quantum phase space.'
+      detail: 'Quantum lattice codes can be thought of as quantum analogues of lattices because they store information in quantum superpositions of points on a lattice in quantum phase space.'
     - code_id: css
       detail: 'Quantum lattice codes defined on rectangular lattices are CSS codes.
       There is no known relation to chain complexes for such codes.

codes/quantum/qubits/qubit_concatenated.yml

@@ -10,9 +10,7 @@ logical: qudits
 name: 'Concatenated qubit code'
 
 description: |
-  A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of a possibly non-qubit inner code and another qubit outer code.
-
-  A combination of two qubit codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the inner code consists of the logical subspace of the outer code.
+  A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another qubit outer code.
   An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code.
 
   Concatenating an \(((n,2,d))\) qubit code can be done recursively, with the \(r\)\textit{th level} of concatenation yielding an \(((n^r,2,d^r))\) code.

codes/quantum/qudits_galois/ea_stabilizer/ea_galois_stabilizer.yml

@@ -23,9 +23,9 @@ relations:
     - code_id: ea_galois_into_galois
   cousins:
     - code_id: galois_stabilizer
-      detail: 'Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes \cite{arxiv:1105.5872}\cite[Thm. 10]{arxiv:2010.07902}.'
+      detail: 'EA Galois-qudit stabilizer codes utilize additional ancillary Galois-qudits in a pre-shared entangled state, but reduce to Galois-qudit stabilizer codes when said qudits are interpreted as noiseless physical qudits. Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes \cite{arxiv:1105.5872}\cite[Thm. 10]{arxiv:2010.07902}.'
     - code_id: galois_grs
-      detail: 'Galios-qudit GRS codes can be used to construct EA Galois-qudit stabilizer codes \cite{arxiv:1606.00134,doi:10.1007/s11128-021-03028-w}.'
+      detail: 'Galois-qudit GRS codes can be used to construct EA Galois-qudit stabilizer codes \cite{arxiv:1606.00134,doi:10.1007/s11128-021-03028-w}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits_galois/ea_stabilizer/ea_quantum_lcd.yml

@@ -19,10 +19,8 @@ features:
 
 relations:
   parents:
-    - code_id: ea_galois_into_galois
+    - code_id: ea_galois_stabilizer
   cousins:
-    - code_id: galois_stabilizer
-      detail: 'Pure Galois-qudit codes can be used to make EA QECCs \cite{arxiv:1105.5872}\cite[Thm. 10]{arxiv:2010.07902}.'
     - code_id: lcd
     - code_id: maximal_entanglement_galois_stabilizer
       detail: 'Asymptotically good maximal-entanglement EA Galois-qudit stabilizer codes can be constructed from LCD codes \cite{arxiv:1606.00134}.'