su3 and su4 tverberg_spin

Author: valbert4

codes/quantum/groups/stabilizer/css.yml

@@ -14,6 +14,8 @@ description: |
   A stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type operators.
   The two sets of stabilizer generators can often be related to parts of a chain complex over the appropriate ring or field.
 
+  CSS codes can also be viewed as an instance of a two-step convex-geometric construction: one first chooses an intermediate subspace stabilized by one type of generator, and then imposes the other type of generator so that the remaining diagonal error slopes vanish via an application of the Tverberg theorem \cite[Sec. 4.3]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
 relations:
   parents:
     - code_id: stabilizer

codes/quantum/spins/single_spin/su3_spin.yml

@@ -13,9 +13,6 @@ introduced: '\cite{arxiv:2312.00162}'
 description: |
   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 \(SU(3)\).
 
-  A distinct family of \(SU(3)\) single-spin codes uses the Tverberg theorem construction \cite{arxiv:quant-ph/9908066} in the totally symmetric \(n\)-particle irrep of \(\mathfrak{su}(3)\) (dimension \(\binom{n+2}{2}\)) with the \(\mathfrak{su}(3)\) Lie algebra as the error set.
-  Taking the subspace \(\mathcal{B} = \mathrm{span}\{|a_1 a_2 a_3\rangle : a_1 - a_2 \equiv 0 \pmod{3}\}\) (the \(T(n,3)\) sublattice of the \(\Delta(n,3)\) simplex), the Tverberg construction yields a distance-1 error-detecting code of dimension \(\approx \binom{n+2}{2}/3\) \cite{arxiv:1205.4517}\cite[Ex. 6.4]{manual:{I. Shors, “Quantum Error Detection and Lie Theory”, undergraduate thesis, UC Davis Mathematics REU, 2022, \href{https://reu.math.ucdavis.edu/application/files/8316/7872/4883/shors-final.pdf}{URL}}}.
-
 
 relations:
   parents:

codes/quantum/spins/single_spin/su3_tverberg_spin.yml

@@ -0,0 +1,43 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: su3_tverberg_spin
+physical: spins
+logical: spins
+
+name: '\(SU(3)\) Tverberg spin code'
+introduced: '\cite{arxiv:1205.4517}\cite[Ex. 6.4]{manual:{I. Shors, “Quantum Error Detection and Lie Theory”, undergraduate thesis, UC Davis Mathematics REU, 2022, \href{https://reu.math.ucdavis.edu/application/files/8316/7872/4883/shors-final.pdf}{URL}}}'
+
+description: |
+  \(SU(3)\) single-spin code family obtained from the two-step Tverberg construction \cite{arxiv:quant-ph/9908066} in the totally symmetric \(N\)-particle irrep \(\mathcal{H}=\mathrm{Sym}^N(\mathbb{C}^3)\) of \(\mathfrak{su}(3)\), which has dimension \(\binom{N+2}{2}\).
+  Its weight basis is indexed by the \hyperref[topic:discrete-simplex]{discrete simplex} \(\Delta_{3,N}\), whose centered form is the triangular \(A_2\) lattice.
+
+  A distance-two intermediate subspace can be chosen as
+  \begin{align}
+    \mathcal{B}=\mathrm{span}\{|a_1a_2a_3\rangle: a_1-a_2\equiv0\pmod 3\}~,
+  \end{align}
+  giving \(\dim\mathcal{B}\approx\binom{N+2}{2}/3\) \cite{arxiv:1205.4517}\cite[Ex. 6.4]{manual:{I. Shors, “Quantum Error Detection and Lie Theory”, undergraduate thesis, UC Davis Mathematics REU, 2022, \href{https://reu.math.ucdavis.edu/application/files/8316/7872/4883/shors-final.pdf}{URL}}}.
+  A symmetric partition of this sublattice into pairs in the central hexagon and triples near the corners yields an error-detecting code for single Lie-algebra errors of dimension \(\frac{4}{27}\binom{N+2}{2}+O(N)\) \cite[Sec. 6.2.1]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
+protection: |
+  Detects errors in the \(\mathfrak{su}(3)\) Lie-algebra error set.
+  More generally, for the Lie-type graph metric \(V_t=\mathrm{span}(\mathfrak{su}(3)\oplus \mathbb{C}I)^t\), a distance-\(d\) construction can be obtained by first choosing a graph-distance-\(d\) subset of the discrete simplex \(\Delta_{3,N}\) \cite[Secs. 5.1,6.2]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
+features:
+  rate: |
+    For general distance \(d\), the two-step construction uses an asymptotically optimal distance-\(d\) sublattice of the centered \(\Delta_{3,N}\) discrete simplex for the intermediate space.
+    If \(d=2t\), then \(\dim\mathcal{B}=\dim\mathcal{H}/(3t^2)+O(N)\); if \(d=2t+1\), then \(\dim\mathcal{B}=\dim\mathcal{H}/(3t^2+3t+1)+O(N)\) \cite[Sec. 6.2.2]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
+relations:
+  parents:
+    - code_id: single_spin
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'

codes/quantum/spins/single_spin/su4_tverberg_spin.yml

@@ -0,0 +1,50 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: su4_tverberg_spin
+physical: spins
+logical: spins
+
+name: '\(SU(4)\) Tverberg spin code'
+introduced: '\cite[Sec. 6.3]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}'
+
+description: |
+  Single-spin code family in the totally symmetric \(N\)-particle irrep \(\mathcal{H}=\mathrm{Sym}^N(\mathbb{C}^4)\) of \(\mathfrak{su}(4)\), whose weight diagram is the \hyperref[topic:discrete-simplex]{discrete simplex} \(\Delta_{4,N}\), equivalently its centered tetrahedral realization.
+  The construction uses the two-step Tverberg-theorem method \cite{arxiv:quant-ph/9908066}: first choose an intermediate subspace from a distance-two subset of \(\Delta_{4,N}\), then combine basis states whose convex hulls contain the origin.
+
+  For \(N\) divisible by eight, an optimal distance-two intermediate lattice is obtained as the kernel of
+  \begin{align}
+    a_1L_1+a_2L_2+a_3L_3+a_4L_4 \mapsto a_1+2a_2+3a_3 \pmod 4~.
+  \end{align}
+  The second step pairs opposite points in the central octahedral region and can form additional triples and quadruples near the tetrahedron's corners; for \(N=8\), this partition is optimal within the two-step framework \cite[Sec. 6.3 and Ex. 6.10]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
+  More explicitly, write the normalized monomial basis of \(\mathrm{Sym}^N(\mathbb{C}^4)\) as \(|a_1a_2a_3a_4\rangle\), where \((a_1,a_2,a_3,a_4)\in\Delta_{4,N}\), and let \(\Lambda_B^N\) be the kernel above.
+  For each block \(Y\subset\Lambda_B^N\) whose convex hull contains the origin, choose barycentric weights \(\{\beta_{\mathbf{a}}\}_{\mathbf{a}\in Y}\) satisfying \(\beta_{\mathbf{a}}\geq0\), \(\sum_{\mathbf{a}\in Y}\beta_{\mathbf{a}}=1\), and \(\sum_{\mathbf{a}\in Y}\beta_{\mathbf{a}}\mathbf{a}=0\) in centered \(\Delta_{4,N}\) coordinates.
+  The corresponding codeword is
+  \begin{align}
+    |\psi_Y\rangle=\sum_{\mathbf{a}\in Y}\sqrt{\beta_{\mathbf{a}}}\,|\mathbf{a}\rangle~.
+  \end{align}
+  Thus opposite pairs give \((|p\rangle+|-p\rangle)/\sqrt{2}\); triples of the form \(\{p,p',-(p+p')\}\) give \((|p\rangle+|p'\rangle+|-(p+p')\rangle)/\sqrt{3}\); and zero-sum quadruples give equal four-term superpositions.
+
+protection: |
+  Detects single Lie-algebra errors from the \(\mathfrak{su}(4)\) error set, equivalently errors in \(V_1=\mathfrak{su}(4)\oplus \mathbb{C}I\) in the Lie-type graph metric.
+  The first step suppresses off-diagonal root-space errors by choosing weight vectors separated by graph distance at least two in the centered \(\Delta_{4,N}\) discrete simplex, leaving only a commuting diagonal error algebra for the convex-geometric second step \cite[Secs. 5.1,5.2,6.3]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
+features:
+  rate: |
+    The distance-two intermediate subset has asymptotic density \(1/4\) in \(\Delta_{4,N}\).
+    The generic Tverberg step gives an asymptotic code dimension at least \(\dim\mathcal{H}/16\), while the extra triangle constructions improve finite-size instances but do not change the leading asymptotic rate in the construction described in Ref. \cite[Sec. 6.3]{manual:{Ruochuan Xu, “Classical Construction of Quantum Codes”, undergraduate thesis, January 2023}}.
+
+relations:
+  parents:
+    - code_id: single_spin
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-05-26'