I. Shors thesis

Author: valbert4

codes/quantum/spins/single_spin/okada.yml

@@ -10,17 +10,22 @@ name: 'Okada spin code'
 introduced: '\cite{arxiv:2502.14165}'
 
 description: |
-  An \(SU(2)\) single-spin code encoding a logical qutrit for angular momentum \(J = 3\).
+  Non-diagonal \(SU(2)\) single-spin code in the spin-\(J = 3m\) irrep for integer \(m \geq 1\), encoding a logical \((2m+1)\)-dimensional space.
+  The construction uses a \textit{non-diagonal} subspace (one for which the projected error space \(P_{\mathcal{B}}\mathcal{E}P_{\mathcal{B}}\) is block-diagonal rather than diagonal) to exceed the dimension bound achievable by the Tverberg theorem construction \cite{arxiv:quant-ph/9908066}.
 
-  Admits the following basis of unnormalized codewords,
+  The code is defined in terms of a subspace \(\mathcal{B} = \mathrm{span}\{|k, n-k\rangle : k \equiv 0 \text{ or } 1 \pmod{3}\}\) of the spin-\(n/2\) Hilbert space \(\mathcal{H}_n\) (with \(n = 2J = 6m\)), where \(|k, n-k\rangle\) denotes the state with \(k\) particles in the first mode and \(n-k\) in the second \cite[Ex. 7.1]{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}}}.
+  The codespace has dimension \(\dim \mathcal{C} = 2m+1\), which is approximately \((n+1)/3\).
+
+  The \(m = 1\) (\(J = 3\)) instance encodes a logical qutrit and admits the unnormalized codewords
   \begin{align}
   \begin{split}
     |\overline{0}\rangle&=|_{0}^{3}\rangle\\|\overline{1}\rangle&\propto\sqrt{2}|_{-2}^{3}\rangle-|_{4}^{3}\rangle\\|\overline{2}\rangle&\propto|_{-4}^{3}\rangle+\sqrt{2}|_{2}^{3}\rangle~.
   \end{split}
   \end{align}
 
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-  The code has distance two, detecting any linear combination of the angular momentum operators \cite{arxiv:2502.14165}.
+  Detects distance-1 errors from the \(\mathfrak{su}(2)\) Lie algebra, i.e., any linear combination of the angular momentum operators \(\{E, F, H\}\) \cite{arxiv:2502.14165}.
+  The codespace dimension \(2m+1 \approx (n+1)/3\) improves on the Tverberg theorem construction \cite{arxiv:quant-ph/9908066}, which gives \(\lceil (n+1)/4 \rceil\) \cite{arxiv:1205.4517}.
 
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codes/quantum/spins/single_spin/single_spin.yml

@@ -25,7 +25,10 @@ protection: |
 
 
 features:
-  rate: 'For every \(K,t \geq 2\), there are explicitly constructible \(K\)-dimensional single-spin codes for \(SU(q=N)\) with total spin \(N=(K-1)t(t+1)\) and distance \(t+1\); there also exist families with logical dimension \(K = o(2^N)\) and distance of \hyperref[topic:asymptotics]{order} \(o(N/\log N)\) \cite{arxiv:2509.20545}.'
+  rate: |
+    For every \(K,t \geq 2\), there are explicitly constructible \(K\)-dimensional single-spin codes for \(SU(q=N)\) with total spin \(N=(K-1)t(t+1)\) and distance \(t+1\); there also exist families with logical dimension \(K = o(2^N)\) and distance of \hyperref[topic:asymptotics]{order} \(o(N/\log N)\) \cite{arxiv:2509.20545}.
+
+    For the spin-\(n/2\) irrep of \(\mathfrak{su}(2)\), the Tverberg theorem construction \cite{arxiv:quant-ph/9908066} yields a distance-1 error-detecting code of dimension \(\lceil (n+1)/4 \rceil\) against the \(\mathfrak{su}(2)\) Lie algebra error set \(\{E, F, H\}\) \cite{arxiv:1205.4517}\cite[Ex. 6.3]{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}}}.
 
 
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codes/quantum/spins/single_spin/su3_spin.yml

@@ -13,6 +13,9 @@ 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}}}.
+
 
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