dicke state mapping

codes/classical/bits/easy/hadamard.yml

@@ -1,3 +1,8 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
 code_id: hadamard
 physical: bits
 logical: bits
@@ -6,33 +11,36 @@ name: 'Hadamard code'
 
 description: |
   Also known as a \textit{Walsh code} or \textit{Walsh-Hadamard code}.
-  An \([2^m,m,2^{m-1}]\) balanced binary code dual to an extended Hamming Code.
+  An \([2^m,m,2^{m-1}]\) balanced binary code dual to an extended Hamming code.
 
-  The \([2^m,m+1,2^{m-1}]\) \textit{augmented Hadamard code} can be constructed by adding the all-zero bit.
-  Its codewords are the rows of the \(2^m\)-dimensional Hadamard matrix \(H\) and its negation \(-H\) with the mapping \(+1\to 0\) and \(-1\to 1\). Its codewords form a \(2^m\)-dimensional biorthogonal spherical code under the antipodal mapping.
+  The \([2^m,m+1,2^{m-1}]\) \textit{augmented Hadamard code} (a.k.a. the RM\((1,m)\)) can be constructed by adding the all-ones vector.
+  Its codewords are the rows of the \(2^m\)-dimensional Hadamard matrix \(H\) and its negation \(-H\) with the mapping \(+1\to 0\) and \(-1\to 1\).
+  Its codewords form a \(2^m\)-dimensional biorthogonal spherical code under the antipodal mapping.
 
   The \([2^m-1,m,2^{m-1}]\) \textit{shortened Hadamard code} is a binary simplex code. Its codewords form a \(2^m\)-simplex spherical code under the antipodal mapping.
 
 relations:
   parents:
-    - code_id: long
-      detail: 'The Hadamard code is a subcode of the long code and can be obtained by restricting the long-code construction to only linear functions.'
+    - code_id: binary_ltc
+      detail: 'The Hadamard code is the first code to be identified as a (three-query) LTC \cite{doi:10.1145/100216.100225,doi:10.1016/0022-0000(93)90044-W}.'
     - code_id: balanced
       detail: 'Each Hadamard codeword has length \(2^m\) and Hamming weight of \(2^{m-1}\), making this code balanced.'
     - code_id: q-ary_lcc
       detail: 'Hadamard codes are two-query LDCs and LCCs \cite{doi:10.1561/0400000030,manual:{Gopi, Sivakanth. Locality in coding theory. Diss. Princeton University, 2018.}}.'
-    - code_id: binary_ltc
-      detail: 'The Hadamard code is the first code to be identified as a (three-query) LDC \cite{doi:10.1145/100216.100225,doi:10.1016/0022-0000(93)90044-W}.'
   cousins:
+    - code_id: long
+      detail: 'The Hadamard code is a subcode of the long code and can be obtained by restricting the long-code construction to only linear functions.'
     - code_id: dual
       detail: 'The Hadamard code is the dual of the extended Hamming Code.
       Conversely, the shortened Hadamard code is the dual of the Hamming Code.'
     - code_id: hamming
-      detail: 'The shortened Hadamard code is the dual of the Hamming Code.'
+      detail: 'The Hadamard code is the dual of the extended Hamming Code.
+      Conversely, the shortened Hadamard code is the dual of the Hamming Code.'
     - code_id: extended_hamming
-      detail: 'The Hadamard code is the dual of the extended Hamming Code.'
+      detail: 'The Hadamard code is the dual of the extended Hamming Code.
+      Conversely, the shortened Hadamard code is the dual of the Hamming Code.'
     - code_id: reed_muller
-      detail: 'The augmented Hadamard code is the RM\((1,m)\) code.'
+      detail: 'The shortened Hadamard code is the RM\(^*(1,m)\) code, while the augmented Hadamard code is the RM\((1,m)\) code.'
     - code_id: simplex_spherical
       detail: 'The shortened Hadamard code maps to a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping \cite[Sec. 6.5.2]{manual:{Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.}}\cite[pg. 18]{preset:EricZin}.'
     - code_id: biorthogonal

codes/quantum/groups/rotors/rotor.yml

@@ -41,5 +41,7 @@ relations:
 _meta:
   # Change log - most recent first
   changelog:
+    - user_id: AustinHe
+      date: '2024-04-19'
     - user_id: VictorVAlbert
       date: '2022-07-27'

codes/quantum/qubits/permutation_invariant/gnu/gnu_permutation_invariant.yml

@@ -20,6 +20,7 @@ description: |
   \end{align}
   Here, \(n\) is the number of particles used for encoding \(1\) qubit, and \(g, m \leq n\) are arbitrary positive integers.
   Codes with higher logical dimension are developed in Ref. \cite{arxiv:1512.02469}.
+  Each Dicke state in the code can be \textit{shifted} by adding a shift \(s\) to both \(n\) and \(g\).
 
 
 protection: 'Depends on the family. One family which is completely symmetrized versions of Bacon-Shor codes (parameterized by \(t\)) protects against arbitrary weight-\(t\) spin errors. Additionally, codes with large enough length \((t+1)(3t+1)+t\) can approximately correct \(t\) spontaneous decay errors.'

codes/quantum/qubits/permutation_invariant/qubit_permutation_invariant.yml

@@ -14,12 +14,13 @@ description: |
 
   \begin{defterm}{Dicke states}
   \label{topic:dicke}
-  For \(n\)-qubit block codes, an often used basis for the \(n/2\)-dimensional permutation-invariant subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized permutation-invariant states of \(w\) excitations, i.e., a normalized sum over all binary-string basis elements with \(w\) ones and \(n - w\) zeroes.
+  For \(n\)-qubit block codes, an often used basis for the \(n+1\)-dimensional permutation-invariant subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized permutation-invariant states of \(w\) excitations, i.e., a normalized sum over all binary-string basis elements with \(w\) ones and \(n - w\) zeroes.
   For example, the single-excitation Dicke state on three qubits is
   \begin{align}
     |D_{1}^{3}\rangle=\frac{1}{\sqrt{3}}\left(|001\rangle+|010\rangle+|100\rangle\right)~.
   \end{align}
-  Each Dicke state in the code can be \textit{shifted} by adding a shift \(s\) to both \(n\) and \(w\).
+  The \(n+1\)-dimensional permutation-invariant space can be thought of as a standalone spin-\(n/2\) quantum system, yielding a way to convert between permutation-invatiant qubit codes and \(SU(2)\) spin codes.
+  A single-spin code for the \(SU(2)\) group correcting spherical tensors can be mapped into a permutation-invariant qubit code with an analogous distance \cite{arxiv:2304.08611}\cite[Thm. 1]{arxiv:2310.17652}.
   \end{defterm}
 
 protection: |

codes/quantum/spins/amo/ae.yml

@@ -38,7 +38,7 @@ relations:
     - code_id: binomial
       detail: 'Many well-performing Æ codes can be mapped into shifted versions of binomial codes via the Holstein-Primakoff mapping.'
     - code_id: gnu_permutation_invariant
-      detail: 'Many well-performing Æ codes can be mapped into GNU codes via the Dicke state mapping.'
+      detail: 'Many well-performing Æ codes can be mapped into GNU codes via the \hyperref[topic:dicke]{Dicke state mapping}.'
     - code_id: single_spin
       detail: 'Since Æ codes are defined in a subspace of fixed total angular momentum and protect against errors linear in the momentum generators, they can also be thought of a single-spin codes.'
 

codes/quantum/spins/single_spin/j_gross.yml

@@ -11,7 +11,7 @@ name: 'Clifford spin code'
 introduced: '\cite{arXiv:2005.10910,arxiv:2304.08611}'
 
 description: |
-  A spin code designed to realize a discrete group of gates using \(SU(2)\) rotations, which are realized transversally if the single spin is treated as a collective spin of several spin-half subsystems.
+  A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations.
   Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\).
 
   The first realization \cite{arXiv:2005.10910} used the single-qubit Clifford group (i.e., the binary octahedral (\(2O\)) subgroup of \(SU(2)\)).
@@ -32,8 +32,6 @@ description: |
   Finally, \(|\overline{0} \rangle\) is defined as the \(+1\) eigenvalue of \(\overline{\sigma}_z\) and \(|\overline{1} \rangle = \overline{\sigma}_x |\overline{0} \rangle \).
 
 features:
-  transversal_gates:
-    - 'Discrete subgroups of \(SU(2)\) can be realized transversally.'
   general_gates:
     - 'Universal computation results from being able to prepare a single logical state, perform one measurement, and the following logical gates: the phase gate (\( \overline{S} \)), the Hadamard gate (\(\overline{H}\)), the conditional phase gate (\(\overline{CZ}\)), and the square root of the phase gate (\(\overline{T}\)). Single-qubit Cliffords can be generated using \(\overline{S}\) and \(\overline{H}\), the extension to multiple-qubit Cliffords is done using \(\overline{CZ}\), and \(\overline{T}\) is to transform to non-Clifford states. Together these gates can be used to create all logical unitaries, while preparation and measurement complete universal quantum computation.'
 
@@ -43,7 +41,8 @@ relations:
     - code_id: group_representation
       detail: 'Clifford spin codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\) \cite{arxiv:2306.11621}.'
     - code_id: qubit_permutation_invariant
-      detail: 'Clifford codes yield permutation-invariant qubit codes with non-trivial distance when the single spin is treated as a collective spin of several qubits via the Dicke-state mapping \cite{arxiv:2304.08611}\cite[Thm. 1]{arxiv:2310.17652}.'
+      detail: 'Clifford codes for spins housing representations of \(SU(2)\) yield permutation-invariant qubit codes with non-trivial distance when the single spin-\(n/2\) is treated as the permutationally invariant subspace of \(n\) qubits via the Dicke-state mapping.
+      The subgroup of gates of a Clifford spin code is implemented transversally via this mapping.'
 
 
 # Begin Entry Meta Information

codes/quantum/spins/single_spin/single_spin.yml

@@ -12,9 +12,6 @@ description: |
   An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group.
   In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space.
 
-  For the simplest case of \(SU(2)\), a single-spin code can be thought of as a permutation invariant qubit code encoding a \(K\)-dimensional Hilbert space into the maximally symmetric subspace or \textit{collective spin} of \(2\ell\) spin-half systems.
-  This \(2\ell+1\)-dimensional Hilbert space can be thought of as a standalone spin-\(\ell\) quantum system.
-
 protection: |
   Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not.
 
@@ -38,8 +35,7 @@ relations:
     - code_id: qecc_finite
   cousins:
     - code_id: qubit_permutation_invariant
-      detail: 'Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the permutation-invariant subspace of a set of spins.
-      A single-spin code for the \(SU(2)\) group correcting spherical tensors can be mapped into a permutation-invariant qubit code with an analogous distance via the Dicke-state mapping \cite[Thm. 1]{arxiv:2310.17652}.'
+      detail: 'Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the permutation-invariant subspace of a set of spins via the \hyperref[topic:dicke]{Dicke state mapping}.'
 
 
 # Begin Entry Meta Information

codes/quantum/spins/single_spin/su3_spin.yml

@@ -11,7 +11,7 @@ name: '\(SU(3)\) spin code'
 introduced: '\cite{arXiv:2312.00162}'
 
 description: |
-  An extension of Clifford codes to the group \(SU(3)\), whose codespace is a projection onto a particular irrep of a subgroup of \(SU(3)\) of an underlying spin that houses some particular irrep of \(SU(3)\).
+  An extension of Clifford single-spin codes to the group \(SU(3)\), whose codespace is a projection onto a particular irrep of a subgroup of \(SU(3)\) of an underlying spin that houses some particular irrep of \(SU(3)\).
 
 
 relations: