cat_repetition, coherent_state_repetition

codes/quantum/oscillators/coherent_state/cat/cat.yml → codes/quantum/oscillators/qsc/cat/cat.yml

@@ -56,7 +56,7 @@ relations:
       detail: 'Cat codes are group representation codes with \(G\) being a cyclic group \cite{arxiv:2306.11621}.'
   cousins:
     - code_id: number_phase
-      detail: 'In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code \cite{arxiv:1901.08071}.'
+      detail: 'In the limit as \(N,S \to \infty\), phase measurement in the cat code has vanishing variance, just like in a number-phase code \cite{arxiv:1901.08071}. Conversely, a cat code can be thought of as an appropriately regularized number-phase code.'
 
 # - code_id: coherent_constellation
 #   detail: 'Cat-code codewords are constructed using a coherent-state constellation that forms the cyclic group \(\mathbb{Z}_{2S+2}\).'

codes/quantum/oscillators/coherent_state/cat/cat_concatenated.yml → codes/quantum/oscillators/qsc/cat/cat_concatenated.yml

@@ -10,23 +10,16 @@ name: 'Concatenated cat code'
 introduced: '\cite{arxiv:1409.6759}'
 
 description: |
-  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.
+  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 in its cat-state basis.
 
 protection: |
   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}) 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}.
-
-realizations:
-  - 'Superconducting circuit devices: a repetition code out of two-component cat qubits has been realized for distances 3 and 5 \cite{arxiv:2409.13025}.'
-
 
 relations:
   parents:
     - code_id: qsc
   cousins:
-    - code_id: quantum_repetition
-      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
@@ -37,8 +30,6 @@ relations:
       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: '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/qsc/cat/cat_repetition.yml

@@ -0,0 +1,45 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: cat_repetition
+physical: oscillators
+logical: qubits
+
+name: 'Cat-repetition code'
+introduced: '\cite{arxiv:1904.09474,arxiv:1905.00450}'
+
+description: |
+  A concatenated \(n\)-mode code whose outer code is a quantum repetition code and whose inner code is the cat code in its cat basis. 
+  
+  A basis of codewords for the two-component case,
+  \begin{align}
+    |\overline{\pm}\rangle\propto\left(\left|\alpha\right\rangle \pm\left|-\alpha\right\rangle \right)^{\otimes n}
+  \end{align}
+  for any complex \(\alpha\).
+
+
+protection: |
+  The cat-repetition code on a 2D mode lattice 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}.
+
+realizations:
+  - 'Superconducting circuit devices: a repetition code out of two-component cat qubits has been realized for distances 3 and 5 \cite{arxiv:2409.13025}.'
+
+
+relations:
+  parents:
+    - code_id: cat_concatenated
+  cousins:
+    - code_id: quantum_repetition
+      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: self_correct
+      detail: 'The cat-repetition code on a 2D mode lattice 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
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-12-06'

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

@@ -59,6 +59,10 @@ relations:
       detail: 'The cat code reduces to its two-component version for \(S=0\).'
     - code_id: squeezed_cat
       detail: 'The squeezed cat code reduces to the two-component cat code when there is no squeezing.'
+    - code_id: cat_repetition
+      detail: 'The cat-repetition code for \(n=1\) reduces to the two-component cat code.'
+    - code_id: coherent_state_repetition
+      detail: 'The coherent-state repetition code for \(n=1\) reduces to the two-component cat code.'
   cousins:
     - code_id: hamiltonian
       detail: 'The two-legged cat code forms the ground-state subspace of a Kerr Hamiltonian \cite{arxiv:1605.09408}.'

codes/quantum/oscillators/tiger/coherent_state_repetition.yml

@@ -0,0 +1,38 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: coherent_state_repetition
+physical: oscillators
+logical: qubits
+
+name: 'Coherent-state repetition code'
+introduced: '\cite{arxiv:quant-ph/0109077,arxiv:quant-ph/0306004}'
+
+description: |
+  A concatenated \(n\)-mode code (for odd \(n\)) whose outer code is a quantum repetition code and whose inner code is the cat code in its coherent-state basis. 
+  
+  A basis of codewords for the two-component case is
+  \begin{align}
+    |\overline{\pm}\rangle\propto\left|\pm\alpha\right\rangle ^{\otimes n}
+  \end{align}
+  for any complex \(\alpha\).
+
+
+relations:
+  parents:
+    - code_id: cat_concatenated
+    - code_id: tiger
+      detail: 'The coherent-state repetition code is a tiger code whose matrix \(G\) is a generator matrix of the repetition code (over the integers), and whose matrix \(H\) is zero \cite{arxiv::2411.09668}.'
+  cousins:
+    - code_id: cat_repetition
+      detail: 'The cat (coherent-state) repetition code is a concatenation whose inner code is the cat code in its cat (coherent-state) basis.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-12-06'

codes/quantum/oscillators/tiger/tiger.yml

@@ -0,0 +1,58 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: tiger
+physical: oscillators
+
+name: 'Tiger code'
+introduced: '\cite{arxiv:2411.09668}'
+
+
+description: |
+  A CSS-like multi-mode bosonic non-stabilizer code that generalizes the pair-cat code and whose syndromes are linear combinations of occupation-number operators.
+  
+  A tiger code is defined for a pair of integer matrices, \(G\) and \(H\), satisfying a homological constraint \(GH^{\text{T}} = 0\). 
+  Stabilizer-like operators of the code are either linear combinations of occupation-number operators defined by rows of \(H\), or products of annihilation/creation operators whose powers are defined by rows of \(G\).
+
+  The structure of the logical space is determined from the homology of the integer chain complex defined by \(G\) and \(H\). The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension.
+
+  Codewords are coherent states projected into the subspace defined by the \(H\)-induced constraint, and their corresponding normalizations are Gelfand-Kapranov-Zelevinsky hypergeometric functions \cite{doi:10.1007/BF01078777,doi:10.1215/S0012-7094-94-07313-4}.
+  Codewords can be finitely or infinitely supported in Fock space, depending on the \(H\)-induced constraint.
+  When written in terms of coherent states, codewords are orbits of a set of fiducial coherent states (determined by the aforementioned homology calculation) under a group of tensor-product rotations generated by \(H\).
+  Therefore, codewords consist of continuous but compact coherent-state constellations. 
+
+  Using multi-index notation, a projected coherent state can be written in two ways,
+  \begin{align}
+    |\boldsymbol{\alpha}\rangle_{\boldsymbol{\Delta}}^{H}&\propto\int d\boldsymbol{\phi}e^{i\boldsymbol{\phi}(H\hat{\mathbf{n}}-\boldsymbol{\Delta})}|\boldsymbol{\alpha}\rangle\\&\propto\sum_{H\mathbf{n}=\boldsymbol{\Delta}}\frac{\boldsymbol{\alpha}^{\mathbf{n}}}{\sqrt{\mathbf{n}!}}|\mathbf{n}\rangle~,
+  \end{align}
+  where \(\boldsymbol{\alpha}\) is a complex vector, \(\boldsymbol{\Delta}\) is an integer vector, and \(\boldsymbol{\phi}\) is a vector of phases iterating over the elements of the group generated by \(H\).
+  Tiger codewords are of the above form, and their phase-space values \(\boldsymbol{\alpha}\) lie on a torus embedded in the complex sphere of fixed-energy coherent coherent states, satisfying \(|\alpha_j|^2 = 1\).
+
+protection: |
+  Tiger codes protect against losses and gains of occupation numbers along with rotation noise stemming from modal dephasing.
+  Protection against the latter type of noise is characterized by the minimum Euclidean distance between coherent states in different (continuous) logical constellations.
+
+relations:
+  parents:
+    - code_id: oscillators
+  cousins:
+    - code_id: fock_state
+      detail: 'Tiger codes encoding logical qudits are Fock-state codes.'
+    - code_id: coherent_constellation
+      detail: 'Tiger codewords consist of continuous and compact coherent-state constellations \cite{arxiv:2411.09668}.'
+    - code_id: qsc
+      detail: 'Tiger (quantum spherical) codewords consist of continuous and compact (discrete and finite) coherent-state constellations. Both codes protect against losses and gains of occupation numbers along with rotation noise stemming from modal dephasing. Protection against the latter type of noise is characterized by the minimum Euclidean distance between coherent states in different logical constellations.'
+    - code_id: generalized_homological_product_css
+      detail: 'Tiger codes are CSS-like multi-mode bosonic non-stabilizer codes constructed from chain complexes over the integers \cite{arxiv:2411.09668}. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension.'
+    - code_id: homological_number-phase
+      detail: 'Tiger codes of infinite Fock-state support can be thought of as appropriately regularized homological number-phase codes \cite{arxiv:2411.09668}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-12-06'

codes/quantum/oscillators/uncategorized/homological_number-phase.yml

@@ -32,8 +32,7 @@ relations:
       detail: 'Homological number-phase codes are bosonic codes encoding logical qudits and/or logical rotors.'
   cousins:
     - code_id: homological_rotor
-      detail: 'Homological number-phase codes are mappings of \hyperref[code:homological_rotor]{homological rotor codes} into harmonic oscillators.
-      Codewords of both codes are right eigenstates of powers of Susskind–Glogower phase operators and bosonic rotation operators, with the semigroup of the former (latter) continuous (discrete).'
+      detail: 'Homological number-phase codes can be thought of as homological rotor codes but whose underlying rotors consist of the number and phase degrees of freedom of physical modes.'
     - code_id: oscillator_stabilizer
       detail: 'Homological number-phase codewords span the joint right eigenspace of powers of the non-unitary Susskind–Glogower phase operators and unitary bosonic rotation operators.'
     - code_id: gkp-stabilizer

codes/quantum/qubits/stabilizer/mbqc/cluster_state.yml

@@ -123,6 +123,8 @@ relations:
       detail: 'Cluster states can be created on various lattices \cite{arxiv:1909.11817}.'
 
 
+# In 1D, cluster states are examples of SPT phases with global symmetries and allow MBQC on a single qubit. In 2D, cluster states with subsystem symmetries are universal resources for MBQC. In 3D, cluster states with higher-form symmetries enable universal fault-tolerant MBQC.
+
 # Begin Entry Meta Information
 _meta:
   # Change log - most recent first