Symplectic halving, pair_partition_css, mitten (arXiv:2609.30069)

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/qcga.yml

@@ -40,7 +40,8 @@ features:
     - 'Certain bivariate bicycle codes admit a cup product structure and can thus have logical gates in the \term{Clifford hierarchy} implemented by constant-depth Clifford circuits \cite{arxiv:2410.16250}.'
 
   rate: 'When ancilla qubit overhead is included, the encoding rate surpasses that of the surface code.
-  A general \([[n,k,d]]\) bivariate bicycle code requires \(n\) ancilla qubits for encoding, meaning that its \textit{ancilla-added encoding rate} is \(k/2n\).'
+  A general \([[n,k,d]]\) bivariate bicycle code requires \(n\) ancilla qubits for encoding, meaning that its \textit{ancilla-added encoding rate} is \(k/2n\).
+  BB codes have zero design rate, so all of their logical qubits come from dependent stabilizer generators \cite{arxiv:2609.30069}.'
 
   fault_tolerance:
     - 'Fault-tolerant state initialization using lattice surgery techniques \cite{arxiv:2110.10794,arxiv:2308.08648} and an ancillary surface code \cite{arxiv:2308.07915}.'
@@ -87,6 +88,8 @@ relations:
 _meta:
   # Change log - most recent first
   changelog:
+    - user_id: VictorVAlbert
+      date: '2026-09-27'
     - user_id: VictorVAlbert
       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/mitten.yml

@@ -27,6 +27,11 @@ description: |
   Instances include a \([[540,108,18]]\) code over \(C_9 \rtimes C_{12}\) \cite{arxiv:2607.28795}.
   See Ref. \cite[Table I]{arxiv:2607.28795} for eight instances from \([[150,30,10]]\) to \([[975,195,\leq 24]]\) and their lift groups.
 
+  Choosing the entries of \(B\) as the images of the entries of \(A\) under inversion followed by an involutive automorphism \(\varphi\) of \(G\) yields an \(XZ\)-duality \cite{arxiv:2607.27644}.
+  This duality has fixed points in three of the five qubit blocks \cite[App. C]{arxiv:2609.30069}.
+  Twisting it by a central element \(z\) with \(\varphi(z)=z\) removes the fixed points when \(z\) lies outside the image of \(g\mapsto g\varphi(g)\), which requires \(|Z(G)|\) to be even \cite[App. C]{arxiv:2609.30069}.
+  \hyperref[topic:symplectic-doubling]{Symplectic halving} along the twisted duality yields non-CSS codes, e.g., a \([[160,32,\leq 11]]\) code from a \([[320,64,\leq 12]]\) code over \(\mathbb{Z}_4\rtimes_3\mathbb{Z}_{16}\) \cite[Table V]{arxiv:2609.30069}.
+
 protection: |
   For an Abelian lift group, the minimum-weight codeword of the first base matrix produces a logical operator of that weight \cite{arxiv:2607.28795}.
   This caps the distance at the weight of the base matrices, hence at six for this shape.
@@ -80,6 +85,8 @@ relations:
 _meta:
   # Change log - most recent first
   changelog:
+    - user_id: VictorVAlbert
+      date: '2026-09-27'
     - user_id: VictorVAlbert
       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/tensor/singlesector/hypergraph/hypergraph_product.yml

@@ -43,6 +43,7 @@ protection: |
   An algebraic reformulation of HGP codes, together with several rate-improved variants, was given in Ref. \cite{arxiv:1202.0928}.
   Seed parity-check matrices with as many rows as columns yield \([[2n_1 n_2,2k_1 k_2,\min(d_1,d_2)]]\) codes.
   Symmetric seed matrices, \(H_i=H_i^T\), yield \([[n_1 n_2,k_1 k_2,\min(d_1,d_2)]]\) codes, and two-tile cyclic constructions yield \([[n_1^2,2k_1^2,d_1]]\) codes.
+  The former are \hyperref[topic:symplectic-doubling]{symplectic halves} of the HGP codes along the qubit involution exchanging the two blocks of qubits and are not CSS in general \cite{arxiv:1202.0928}\cite[Sec. IV.A]{arxiv:2609.30069}.
   These variants improve the rate of the original construction from full-rank seed matrices by factors up to four at small block length \cite{arxiv:1202.0928}.
 
 #notes:
@@ -113,6 +114,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-09-27'
     - user_id: VictorVAlbert
       date: '2026-09-23'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/pair_partition_css.yml

@@ -9,32 +9,50 @@ physical: qubits
 logical: qubits
 
 name: 'Pair-partition CPM code'
-introduced: '\cite{arxiv:2607.14091}'
+introduced: '\cite{arxiv:2607.14091,arxiv:2609.30069}'
 
 description: |
-  Qubit CSS code whose \(X\)- and \(Z\)-type stabilizer generator matrices are \(J\)-by-\(L\) arrays of \(P\)-by-\(P\) circulant permutation matrices (CPMs), with the CSS condition enforced by pairing up block columns \cite{arxiv:2607.14091}.
+  Qubit CSS code whose \(X\)- and \(Z\)-type stabilizer generator matrices are \(J\)-by-\(L\) arrays of \(P\)-by-\(P\) circulant permutation matrices (CPMs), with the CSS condition enforced by pairing up block columns \cite{arxiv:2607.14091,arxiv:2609.30069}.
   Each block of \(H_XH_Z^T\) is a sum of \(L\) CPMs, one per block column.
   A \textit{pair partition}, one per block, splits the block columns into \(L/2\) pairs, and the CPM exponents are constrained so that the two CPMs in each pair coincide and cancel over \(\mathbb{F}_2\).
   CSS orthogonality thereby becomes a homogeneous linear condition on the exponents \cite{arxiv:2607.14091}.
 
-  Fix a column weight \(J\), an even row weight \(L>J\), and a prime lift size \(P\).
-  Let \(C(s)\) be the \(P\)-by-\(P\) CPM with shift \(s\in\mathbb{F}_P\), and let \((e_{i\ell})\) and \((d_{j\ell})\) be \(J\)-by-\(L\) exponent arrays over \(\mathbb{F}_P\).
+  Fix a column weight \(J\), an even row weight \(L>J\), and a lift size \(P\).
+  Let \(C(s)\) be the \(P\)-by-\(P\) CPM with shift \(s\in\mathbb{Z}_P\), and let \((e_{i\ell})\) and \((d_{j\ell})\) be \(J\)-by-\(L\) exponent arrays over \(\mathbb{Z}_P\).
   The generator matrices \(H_X=(C(e_{i\ell}))\) and \(H_Z=(C(d_{j\ell}))\) are \(G\)-\hyperref[topic:lifting]{lifts} of the complete \(J\)-by-\(L\) protograph over the cyclic group of order \(P\), with length \(n=LP\), row weight \(L\), and column weight \(J\).
   A \(J\)-by-\(J\) array \((M_{ij})\) of pair partitions, one for each \(X\)-type block row \(i\) and \(Z\)-type block row \(j\), is fixed before the exponents.
   For each pair \(\{u,v\}\in M_{ij}\), the mixed-difference equation \(e_{iu}-d_{ju}=e_{iv}-d_{jv}\) is imposed.
   It makes the contributions of block columns \(u\) and \(v\) to block \((i,j)\) of \(H_XH_Z^T\) identical.
-  The exponents are solutions of the resulting homogeneous system over \(\mathbb{F}_P\) \cite{arxiv:2607.14091}.
+  The exponents are solutions of the resulting homogeneous system over \(\mathbb{Z}_P\) \cite{arxiv:2607.14091}.
+  The rate is at least the design rate \(1-2J/L\) \cite{arxiv:2609.30069}.
+
+  Suppose the exponent arrays satisfy \(d_{j\ell}=\eta\,e_{\rho(j)\sigma(\ell)}+\alpha_j+\beta_\ell\) for a block-row permutation \(\rho\), a fixed-point-free involution \(\sigma\) of the block columns, \(\eta^2\equiv 1 \pmod P\), and offsets with \(\beta_{\sigma(\ell)}=-\eta\beta_\ell\).
+  Then the qubit involution \((\ell,t)\mapsto(\sigma(\ell),\eta(t+\beta_\ell))\) exchanges the two check spaces, and the code can be \hyperref[topic:symplectic-doubling]{symplectically halved} into a non-CSS stabilizer code on \(LP/2\) qubits \cite{arxiv:2609.30069}.
+  Halved codes reach a given distance at shorter length than pair-partition CPM codes with the same \((J,L)\) \cite[Sec. IV.B]{arxiv:2609.30069}.
+  Examples include the \([[90,21,11]]\), \([[140,31,15]]\), and \([[200,43,20]]\) codes with check weight ten \cite[Table II]{arxiv:2609.30069}.
+
+  For odd \(P\), suppose both check matrices have rank \(J\) over each nontrivial field factor of \(R=\mathbb{F}_2[\mathbb{Z}_P]\).
+  Then each logical space is the \(R\)-module \(R^{L-2J}\oplus\mathbf{1}^{2J-2}\), i.e., \(L-2J\) regular orbits of the cyclic shift plus \(2J-2\) shift-invariant logical classes \cite{arxiv:2609.30069}.
 
   Reported instances include the \((3,10)\)-regular \([[2230,896,24]]\) code with \(P=223\) and Tanner girth eight \cite[Table I]{arxiv:2607.14091}.
-  See Ref. \cite[Table I]{arxiv:2607.14091} for the full list of reported codes.
+  See Refs. \cite[Table I]{arxiv:2607.14091}\cite[Table VIII]{arxiv:2609.30069} for lists of instances.
 
 protection: |
   Every CPM lift of the complete \(J\)-by-\(L\) protograph with \(J\geq 2\) and \(L\geq 3\) has Tanner girth at most twelve \cite{arxiv:2607.14091}.
-  For odd \(P\), \(J\geq 2\), \(L\geq 2J+1\), and maximal binary ranks \(\operatorname{rank}H_X=\operatorname{rank}H_Z=J(P-1)+1\), permanent-based logical witnesses give the lift-size-independent bound \(d_X,d_Z,d\leq (J+1)!\) \cite{arxiv:2607.14091}.
+  For odd \(P\), \(J\geq 2\), \(L\geq 2J+1\), and maximal binary ranks \(\operatorname{rank}H_X=\operatorname{rank}H_Z=J(P-1)+1\), permanent-based logical witnesses give the lift-size-independent bound \(d_X,d_Z,d\leq (J+1)!\) \cite{arxiv:2607.14091}\cite[App. A]{arxiv:2609.30069}.
 
   The thirty-six reported codes have exact distances reaching \(24\) and rates reaching \(0.574\), attained by the \([[1414,812,12]]\) code \cite{arxiv:2607.14091}.
   Twenty-nine of them have Tanner girth six and seven have girth eight \cite[Table I]{arxiv:2607.14091}.
 
+features:
+  decoders:
+    - 'Sum-product or normalized min-sum BP followed by \textit{CPM-aware residual-syndrome decoding}, a bounded-weight correction search that exploits invariance under simultaneous cyclic shifts \cite[App. F]{arxiv:2609.30069}.'
+    - 'Hierarchical decoder with BP, relay-BP, and integer-programming stages \cite{arxiv:2604.16209,arxiv:2609.30069}.'
+
+  fault_tolerance:
+    - 'Block-wise scheduling of the protograph gives syndrome-extraction circuits of depth equal to the check weight for the \([[530,216,12]]\) and \([[518,228,16]]\) codes \cite[Sec. VI.B]{arxiv:2609.30069}. Their circuit-level distances are at most \(10\) and \(14\), respectively.'
+    - 'Idling-free circuit-level noise of Ref. \cite{arxiv:2604.16209} at \(0.1\%\): per-round logical error rates of \(2.1\times 10^{-9}\) for the \([[530,216,12]]\) code and \(1.5\times 10^{-10}\) for the \([[518,228,16]]\) code \cite[Sec. VI.B]{arxiv:2609.30069}.'
+
 relations:
   parents:
     - code_id: qubit_css
@@ -44,7 +62,7 @@ relations:
       detail: |
         Both stabilizer generator matrices of a pair-partition CPM code are arrays of circulant permutation matrices.
         This structure makes the code invariant under simultaneous cyclic shifts of the lift coordinates.
-        This automorphism has order equal to the prime lift size \cite{arxiv:2607.14091}.
+        This automorphism has order equal to the lift size \(P\) \cite{arxiv:2607.14091}.
   cousins:
     - code_id: two_branch_css
       detail: |
@@ -56,12 +74,19 @@ relations:
         Replacing the circulant permutation matrices by affine permutation matrices extends the pair-partition construction.
         A structured subfamily of that extension has the same check matrices as the active rows of Kasai codes.
         A separate condition on the pair partitions reproduces the complementary latent rows \cite{arxiv:2607.14091,arxiv:2601.08824}.
+    - code_id: perm_self_dual_css
+      detail: |
+        A pair-partition CPM code is a PSD code when its exponent arrays satisfy \(d_{j\ell}=\eta\,e_{\rho(j)\sigma(\ell)}+\alpha_j+\beta_\ell\) \cite{arxiv:2609.30069}.
+        Here \(\rho\) permutes block rows, \(\sigma\) is a fixed-point-free involution of block columns, \(\eta^2\equiv 1 \pmod P\), and \(\beta_{\sigma(\ell)}=-\eta\beta_\ell\).
+        The duality is the fixed-point-free qubit involution \((\ell,t)\mapsto(\sigma(\ell),\eta(t+\beta_\ell))\), so the code is the \hyperref[topic:symplectic-doubling]{symplectic double} of a non-CSS code on \(LP/2\) qubits.
 
 
 # Begin Entry Meta Information
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       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/qldpc.yml

@@ -35,6 +35,9 @@ protection: |
 
 features:
   rate: |
+    Suppose each qubit takes part in \(\bar{J}\) stabilizer generators on average and each generator has average weight \(\bar{L}\).
+    Counting qubit-generator incidences gives \(n\bar{J}/\bar{L}\) generators, so the rate is the \textit{design rate} \(1-\bar{J}/\bar{L}\) plus the number of generator dependencies divided by \(n\) \cite{arxiv:2609.30069}.
+
     Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction.
     Bounds generalizing the \hyperref[topic:bpt-bound]{BPT bound} to qubit QLDPC codes depend on the separation profile of the code's underlying connectivity graph \cite{arxiv:2106.00765,arxiv:2307.03283}.
     A constant relative minimum distance can be achieved only for graphs that contain expanders \cite{arxiv:2106.00765}.
@@ -146,6 +149,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-09-27'
     - user_id: VictorVAlbert
       date: '2026-08-24'
     - user_id: VictorVAlbert

codes/quantum/qudits/subsystem/qudit_subsystem_css.yml

@@ -44,7 +44,13 @@ description: |
   In the case of a stabilizer code, the stabilizer generator matrix is mapped instead to yield a \hyperref[code:two_block_quantum]{two-block CSS code} (see \cite[Thm. 1]{arxiv:1212.6703} for the case of qubit stabilizer codes).
   For geometrically local 2D stabilizer codes with twist defects, this mapping yields a twisted double cover of the underlying qudit geometry \cite{arxiv:2406.09951}.
 
-  For qubit stabilizer codes, this map is also known as the \textit{BLT \(n\to 2n\) mapping}, and its output can be packaged as an \([[n,k,d]]_4\) \hyperref[code:galois_css]{Galois-qudit CSS code} \cite[Lemma 1]{arxiv:2605.15344}\cite{arxiv:1004.3791}.
+  \textit{Symplectic halving} is the inverse map, defined for qubit CSS codes on \(2n\) qubits that admit a fixed-point-free involutory \hyperref[code:perm_self_dual_css]{\(ZX\) duality} \cite[Thm. 3.5]{arxiv:2406.09951}\cite{arxiv:2609.30069}.
+  Order the qubits so that the duality exchanges the two halves.
+  For a suitable choice of generators, the check matrices then take the form \(H_X=(A|B)\) and \(H_Z=(B|A)\), and the halved code has symplectic check matrix \((A|B)\) \cite{arxiv:2609.30069}.
+  Halving an \([[2n,2k,d^{\prime}]]\) code yields an \([[n,k,d]]\) code with \(\lceil d^{\prime}/2\rceil\leq d\leq d^{\prime}\) and does not increase check weights \cite{arxiv:2609.30069}.
+  Swapping the two members of a pair acts as a Hadamard gate on the halved qubit, so all orientations of a given duality yield locally Clifford-equivalent codes \cite{arxiv:2609.30069}.
+
+  For qubit stabilizer codes, symplectic doubling is also known as the \textit{BLT \(n\to 2n\) mapping}, and its output can be packaged as an \([[n,k,d]]_4\) \hyperref[code:galois_css]{Galois-qudit CSS code} \cite[Lemma 1]{arxiv:2605.15344}\cite{arxiv:1004.3791}.
   The Galois-qudit distance is exactly \(d\) because the Galois-qudit support of the image coincides with the symplectic support of the input, while splitting each Galois qudit into two qubits can double the weight of a logical operator.
   Conversely, a Galois-qudit CSS code arises as a symplectic double of a qubit stabilizer code if and only if it is \hyperref[code:galois_css]{Hermitian self-dual} \cite{arxiv:2601.20927,arxiv:2605.15344}.
 
@@ -82,6 +88,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-09-27'
     - user_id: VictorVAlbert
       date: '2026-08-25'
     - user_id: VictorVAlbert

codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/scalar/generalized_bicycle.yml

@@ -28,6 +28,7 @@ description: |
   \end{align}
   where \(A=a(P)\) and \(B=b(P)\) are  \(\ell\times\ell\) circulant matrices, and \(P\) is the permutation matrix of a one-step length-\(\ell\) cyclic shift.
   This refinement is a special case of \hyperref[topic:symplectic-doubling]{symplectic doubling}.
+  The corresponding symplectic halving of a qubit GB code yields a stabilizer code on \(\ell\) qubits that is not CSS in general \cite{arxiv:1212.6703}\cite[Sec. IV.A]{arxiv:2609.30069}.
 
   With any GB\((a,b)\) code, there is an associated \(q\)-ary cyclic classical code \(C_{h(x)}^{\perp}=C_{g(x)}\) of length \(\ell\), with the check and generating polynomials
   \begin{align}
@@ -124,6 +125,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-09-27'
     - user_id: VictorVAlbert
       date: '2026-08-28'
     - user_id: VictorVAlbert