qubit_8_4_2

Author: valbert4

codes/classical/bits/reed_muller/reed_muller.yml

@@ -23,6 +23,7 @@ description: |
   Another way to generate RM codewords is to list all outcomes of all polynomials of \(m\) binary variables of degree at most \(r\) \cite{arxiv:2002.03317} (see also \cite[Ch. 13]{preset:MacSlo}).
 
   The automorphism group of the RM\((r,m)\) (RM\(^*(r,m)\)) code is \(GA(m,\mathbb{F}_2)\) (\(GL(m,\mathbb{F}_2)\)) for \(1 \leq r \leq m-2\) \cite{preset:MacSlo}.
+  For \(m\geq 5\) and \(m=3\), the only binary linear codes of length \(2^m-1\) whose automorphism group contains \(GL(m,\mathbb{F}_2)\) are the punctured and shortened Reed-Muller codes \cite{doi:10.1112/S002461079900839X,arxiv:2604.15111}.
 
 protection: |
   The \textit{Schwartz-Zippel lemma} provides a distance lower bound on RM codes, extending the degree-based distance argument familiar from RS codes.

codes/quantum/properties/block/block_quantum.yml

@@ -61,6 +61,7 @@ features:
       For subsets of size one, gates are sometimes called \textit{strongly transversal} if the single-subsystem unitaries are identical, and \textit{weakly transversal} otherwise.
       A universal gate set for a finite-dimensional block quantum code cannot be transversal for any code that detects single-block errors due to the Eastin-Knill theorem \cite{arxiv:0811.4262}.
       \end{defterm}'
+    - 'A qudit code of length \(n\) with permutation automorphism subgroups \(N\triangleleft G\leq \mathrm{PAut}(Q)\) and simple non-Abelian quotient \(G/N\) must satisfy \(n\geq \mu(G/N)\) \cite[Thm. S5]{arxiv:2604.15111}.'
 
 
 notes:

codes/quantum/qubits/qubits_into_qubits.yml

@@ -119,6 +119,7 @@ features:
   rate: 'Exact two-way assisted capacities have been obtained for the erasure and dephasing channels \cite{arxiv:1510.08863}. There are many bounds on the quantum capacity of the depolarizing channel (e.g., \cite{arxiv:quant-ph/0607039}); see review \cite{arxiv:1801.02019}. The optimal asymptotic error exponent of entanglement distillation is given by the reverse relative entropy of entanglement, a single-letter quantity \cite{arxiv:2408.07067}.'
   transversal_gates:
     - 'A qubit code is \(U\)-\textit{quasi-transversal} if it can realize the logical gate \(U\) in the third level of the \term{Clifford hierarchy} using the physical gate \(C T^{\otimes n}\), where \(C\) is some Clifford gate \cite[Def. 4]{arxiv:1606.01904}.'
+    - 'If a qubit code \(Q\) of length \(n\) has compact subgroups \(N\triangleleft G\leq \mathrm{Aut}(Q)\) such that \(G/N\) is finite, non-Abelian, simple, and not \(A_5\), then \(n\) is at least the minimal permutation degree \(\mu(G/N)\) \cite[Thm. 1]{arxiv:2604.15111}.'
   general_gates:
     - 'Computing using \hyperref[topic:clifford]{Clifford gates} only can be efficiently simulated on a classical computer, according to the \textit{Gottesman-Knill theorem} \cite{arxiv:quant-ph/9807006,manual:{E. Knill, private communication, ca. 1998}}.
     There is a canonical form for \hyperref[topic:clifford]{Clifford circuits} \cite{arxiv:2003.09412,arxiv:2408.15202} and many algorithms for simulating them \cite{arxiv:quant-ph/0406196,arxiv:1712.03554,arxiv:2301.02356}.

codes/quantum/qubits/small_distance/hypercube_quantum.yml

@@ -19,6 +19,7 @@ description: |
   Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices.
   These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy.
   The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples \cite{arxiv:2404.19005}.
+  Puncturing the \([[2^D,D,2]]\) hypercube quantum code yields the \([[2^D-1,D,2]]\) punctured-hypercube family.
 
   Higher-distance generalizations include a \([[2^{2D},D,4]]\) hyperoctahedron family and a \([[2^D(2^D+1),D,4]]\) family built from distance-two \(D\)-dimensional toric/surface-code blocks \cite{arxiv:2404.19005}.
   Various other concatenations give families with increasing distance (see cousins).
@@ -45,7 +46,7 @@ relations:
     - code_id: quantum_reed_muller
       detail: '\([[2^D,D,2]]\) hypercube quantum codes are special cases of the \([[2^m,{m \choose r}, 2^r]]\) quantum RM codes for \(m=D\) and \(r=1\) \cite[Exam. 8]{arxiv:1910.09333}\cite{arxiv:1606.01904,arxiv:1606.01906,arxiv:1709.02832,arxiv:2410.23263}.'
     - code_id: phantom
-      detail: 'The \([[2^D,D,2]]\) hypercube quantum codes are phantom codes: all ordered-pair in-block logical CNOT gates can be implemented by physical-qubit permutations \cite{arxiv:2601.20927}.'
+      detail: 'The \([[2^D,D,2]]\) hypercube quantum codes are phantom codes: all ordered-pair in-block logical CNOT gates can be implemented by physical-qubit permutations \cite{arxiv:2601.20927}. The punctured hypercube family is unique among binary CSS phantom codes saturating \(n\geq 2^k-1\) for \(k=3\) and \(k\geq 5\) \cite[Thm. 4]{arxiv:2604.15111}.'
     - code_id: self_complementary
       detail: 'A basis of hypercube quantum codewords of the form \(|c\rangle+|\overline{c}\rangle\) can be obtained via the \hyperref[code:qubit_css]{qubit CSS codeword construction} since their sole \(X\)-type stabilizer generator acts on all qubits.'
   cousins:

codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml

@@ -26,6 +26,16 @@ description: |
     Z & X & I & X & Z
   \end{array}~.
   \end{align}
+  A basis of codewords for the above stabilizer is \cite[Ch. 3]{preset:GottesmanBook}
+  \begin{align}
+  \begin{split}
+  |\overline{0}\rangle &= \tfrac{1}{4}(|00000\rangle + |10010\rangle + |01001\rangle - |11011\rangle + |10100\rangle - |00110\rangle - |11101\rangle - |01111\rangle \\
+  &\quad + |01010\rangle - |11000\rangle - |00011\rangle - |10001\rangle - |11110\rangle - |01100\rangle - |10111\rangle + |00101\rangle)\\
+  |\overline{1}\rangle &= \tfrac{1}{4}(|11111\rangle + |01101\rangle + |10110\rangle - |00100\rangle + |01011\rangle - |11001\rangle - |00010\rangle - |10000\rangle \\
+  &\quad + |10101\rangle - |00111\rangle - |11100\rangle - |01110\rangle - |00001\rangle - |10011\rangle - |01000\rangle + |11010\rangle)~.
+  \end{split}
+  \end{align}
+  Logical Pauli operators are \(\bar{X} = XXXXX\) and \(\bar{Z} = ZZZZZ\) \cite[Table 3.4]{preset:GottesmanBook}.
   The code's automorphism group is the dihedral group of order 10 \cite{arxiv:2109.12735}.
   A graph-code realization of the code uses a pentagon graph with an additional central input node \cite{arxiv:quant-ph/0012111}.
   The \hyperref[topic:encoder-respecting]{encoder-respecting form} can be taken to have this shape \cite{arxiv:2411.14448}.

codes/quantum/qubits/small_distance/small/7/xz_7_3_2.yml

@@ -24,11 +24,15 @@ description: |
   \end{array}~.
   \end{align}
 
+protection: 'Distance two. As the \(k=3\) punctured hypercube code, it saturates the general qubit phantom-code bound \(n\geq 2^k-1\) \cite{arxiv:2604.15111}.'
+
 
 relations:
   parents:
     - code_id: phantom
       detail: 'This code is the punctured version of the \([[8,3,2]]\) hypercube quantum code and is a phantom code \cite{arxiv:2601.20927}.'
+    - code_id: quantum_reed_muller
+      detail: 'The punctured hypercube family is a binary quantum Reed-Muller family built from shortened and punctured classical Reed-Muller codes \cite{arxiv:2604.15111}.'
     - code_id: small_distance_qubit_stabilizer
   cousins:
     - code_id: stab_8_3_2

codes/quantum/qubits/small_distance/small/8/qubit_8_4_2.yml

@@ -0,0 +1,38 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: qubit_8_4_2
+physical: qubits
+logical: qubits
+
+name: '\(((8,16,2))\) \(PG(3,2)\) code'
+introduced: '\cite{arxiv:2604.15111}'
+
+description: |
+  Eight-qubit code encoding four logical qubits whose logical basis consists of a GHZ state together with fifteen states built from the incidence geometry of the projective space \(PG(3,2)\).
+  
+protection: |
+  Distance two, and this is optimal because no \(((8,16,3))\) code exists \cite{arxiv:2604.15111}.
+  No Pauli stabilizer subsystem phantom code of type \([[8,4,r,d\geq2]]\) exists, so this exceptional \(k=4\) phantom code is necessarily nonstabilizer \cite{arxiv:2604.15111}.
+
+features:
+  transversal_gates:
+    - 'Even physical-qubit permutations act as \(GL(4,\mathbb{F}_2)\) on the logical basis, and odd permutations extend the permutation automorphism group to the full symmetric group \(S_8\) \cite{arxiv:2604.15111}.'
+    - '\(T^{\otimes 8}\) is a transversal non-Clifford gate implementing \(2\ketbra{\overline{0}}-\mathbbm{1}\) on the logical subspace \cite{arxiv:2604.15111}.'
+    - 'A specific odd permutation implements a non-Clifford logical involution, and the full permutation automorphism group is \(S_8\) \cite{arxiv:2604.15111}.'
+
+relations:
+  parents:
+    - code_id: small_distance_quantum
+  cousins:
+    - code_id: phantom
+      detail: 'This is the exceptional nonstabilizer \(k=4\) qubit phantom code of minimal length eight that violates the generic bound \(n\geq 2^k-1\) \cite{arxiv:2604.15111}.'
+    - code_id: self_complementary
+      detail: 'The logical basis of the \(((8,16,2))\) \(PG(3,2)\) code contains a GHZ state and linear combinations of self-complementary states \cite{arxiv:2604.15111}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  changelog:

codes/quantum/qubits/small_distance/small/9/shor_nine.yml

@@ -37,7 +37,7 @@ description: |
   \end{align}
   The \hyperref[topic:encoder-respecting]{encoder-respecting form} of the Shor code is a star-shaped tree graph \cite{arxiv:2411.14448}.
   The code works by \hyperref[code:qubit_concatenated]{concatenating} each qubit of a phase-flip \hyperref[code:quantum_repetition]{repetition code} with a bit-flip \hyperref[code:quantum_repetition]{repetition code}. Therefore, the code can correct both types of errors simultaneously.
-  The code is degenerate: for example, two \(Z\) errors in the same three-qubit block act identically on all codewords \cite[Ch. 2]{preset:GottesmanBook}.
+  The code is degenerate: for example, two \(Z\) errors in the same three-qubit block act identically on all codewords \cite[Ch. 3]{preset:GottesmanBook}.
 
 
 # Specifically, a state is phase-flip error-corrected by a three-qubit phase-flip \hyperref[code:quantum_repetition]{repetition code}, with stabilizer generators \(X_0 X_1I_2\) and \(X_0I_1X_2\) in \(X\) basis, where the subscript represents the qubit index. Each logical qubit is encoded using

codes/quantum/qubits/stabilizer/css/phantom.yml

@@ -12,14 +12,15 @@ introduced: '\cite{arxiv:2601.20927}'
 
 description: |
   Qubit CSS code for which, in some logical basis, every ordered-pair logical \(\overline{\mathrm{CNOT}}_{ab}\) gate between logical qubits in the same code block can be implemented by a physical-qubit permutation \cite{arxiv:2601.20927}.
+  The definition has been extended to non-CSS and non-qubit codes \cite{arxiv:2604.15111}.
 
 protection: |
   CSS phantom codes obey a Hamming-type constraint: if \(d=d_{\mu}\) for \(\mu\in\{X,Z\}\), then \(\eta(2^k-1)\leq B(n,d)\), where \(\eta\) counts weight-\(d\) logical operators in a fixed \(\mu\)-type logical equivalence class and \(B(n,d)\leq {n \choose d}\) is the maximum size of a binary length-\(n\) code whose pairwise sums have weight at least \(d\) \cite{arxiv:2601.20927}.
-  All phantom codes identified in Ref. \cite{arxiv:2601.20927} are non-LDPC and encode \(k=O(\log n)\) logical qubits.
+  Any qubit phantom code of distance \(d\geq 2\) encoding \(k\geq 2\) logical qubits with \(k\neq 4\) obeys \(n\geq 2^k-1\), equivalently \(k\leq \log_2(n+1)\); this parameter bound also holds for non-CSS phantom codes and for qubit subspace or subsystem phantom-LU codes \cite{arxiv:2604.15111}.
 
 features:
   transversal_gates:
-    - 'Interblock \(\overline{\mathrm{CNOT}}\) gates are transversal because phantom codes are CSS codes. Combining transversal interblock CNOTs with in-block permutation CNOTs implements any logical CNOT circuit on \(2^a\) phantom-code blocks in physical depth at most \(4(2^a-1)\), up to a residual logical-qubit permutation; for unidirectional CNOT circuits, the bound is \(2(2^a-1)\) while preserving logical-qubit order \cite{arxiv:2601.20927}.'
+    - 'For CSS phantom codes, interblock \(\overline{\mathrm{CNOT}}\) gates are transversal. Combining transversal interblock CNOTs with in-block permutation CNOTs implements any logical CNOT circuit on \(2^a\) phantom-code blocks in physical depth at most \(4(2^a-1)\), up to a residual logical-qubit permutation; for unidirectional CNOT circuits, the bound is \(2(2^a-1)\) while preserving logical-qubit order \cite{arxiv:2601.20927}.'
     - 'A stabilizer code supporting a logical gate by qubit permutations cannot admit any strictly transversal logical gate that does not commute with that permutation-implemented logical gate, ruling out strictly transversal implementations of several gates on phantom codes \cite{arxiv:2601.20927}.'
     - 'Additional logical Clifford and non-Clifford gates can arise from code automorphisms combining local Cliffords and qubit permutations, from fold-diagonal gates using patterned one- and two-qubit diagonal interactions, and from non-uniform diagonal single-qubit rotations \cite{arxiv:2601.20927}.'
   general_gates:

codes/quantum/qubits/subsystem/subsystem_qubits_into_qubits.yml

@@ -15,6 +15,9 @@ alternative_names:
 description: |
   Subsystem QECC encoding into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space.
 
+features:
+  transversal_gates:
+    - 'If a subsystem qubit code \(Q\) of length \(n\) has compact subgroups \(N\triangleleft G\leq \mathrm{Aut}(Q)\) such that \(G/N\) is finite, non-Abelian, simple, and not \(A_5\), then \(n\) is at least the minimal permutation degree \(\mu(G/N)\) \cite[Thm. 1]{arxiv:2604.15111}.'
 
 relations:
   parents: