Merge pull request #367 from Fe-r-oz/fa/double-homological-product

Double Homological Product Code

Author: valbert4

codes/quantum/qubits/small_distance/small/8/stab_8_3_3.yml

@@ -39,7 +39,7 @@ relations:
 _meta:
   # Change log - most recent first
   changelog:
-    - user_id: FerozAhmad
+    - user_id: FerozAhmedMian
       date: '2024-03-14'
     - user_id: VictorVAlbert
       date: '2023-11-28'

codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/double_homological_product.yml

@@ -0,0 +1,57 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: double_homological_product
+physical: qubits
+logical: qubits
+
+name: 'Double homological product code'
+introduced: '\cite{arxiv:1805.09271}'
+
+description: |
+  The double homological product code is derived from two applications of the homological product to a classical code, resulting in a length-\(4\) chain complex:  
+  \[
+  \breve{C}_{-2} \xrightarrow{\breve{\delta}_{-2}} \breve{C}_{-1} \xrightarrow{\breve{\delta}_{-1}} \breve{C}_0 \xrightarrow{\breve{\delta}_0} \breve{C}_1 \xrightarrow{\breve{\delta}_1} \breve{C}_2.
+  \]  
+ 
+  The boundary maps \(\breve{\delta}_j\) are constructed using tensor products of the original boundary maps, ensuring the chain condition \(\breve{\delta}_{j+1} \breve{\delta}_j = 0\).  
+  
+  The construction method allows for the use of two different classical codes as inputs. However, \cite{arxiv:1805.09271} assumes identical input codes for simplicity.
+  
+  The code supports metachecks to detect measurement errors, enabling single-shot error correction.  
+
+protection: |
+  - Given a classical \([n, k, d]\) code, the double homological product yields a quantum code with parameters:  
+  \[
+  [[n_Q = n^4 + 4n^2(n-k)^2 + (n-k)^4, k_Q = k^4, d_Q \geq d, d_{ss} = \infty]].
+  \]  
+  - The code is \((d, f)\)-sound with \(f(x) = x^3/4\), meaning small syndromes can be corrected by small errors.
+  - The check redundancy is bounded (\(\breve{\upsilon} < 2\)), and the construction preserves LDPC properties if the original code is LDPC.  
+
+relations:
+  parents:
+    - code_id: multisector_hypergraph
+  cousins:
+    - code_id: higher_dimensional_toric
+      detail: While topological codes like the \(2D\) or \(2D\) toric codes offer intuitive geometric structures, they face fundamental trade-offs in code parameters (e.g., rate vs. distance). The double homological product code, though more abstract, shares key advantages with high-dimensional topological codes—such as single-shot error correction—while circumventing these limitation.
+    - code_id: hypergraph_product
+      detail: Notational conventions for chain complexes permit some flexibility. \cite{arxiv:1805.09271} select conventions ensuring that the multi-sector homological product construction directly corresponds to the hypergraph product of \cite{arxiv:0903.0566}.
+
+features:
+  fault_tolerance:
+    - Corrects errors using only one round of noisy measurements, with residual errors bounded by the soundness function.  
+    - Metachecks enable detection of measurement errors, improving fault tolerance.  
+
+  decoders:
+    - 'The meta-check-based decoder operates through a two-stage process: first, it identifies a minimal correction \( s_{\text{rec}} \) to the syndrome \( s \) such that the repaired syndrome \( s + s_{\text{rec}} \) satisfies all metachecks (\( H(s + s_{\text{rec}}) = 0 \)). Second, it computes a minimal-weight physical error \( E_{\text{rec}} \) consistent with the repaired syndrome. This approach uniquely tolerates up to \( \lfloor (d_{ss} - 1)/2 \rfloor \) measurement errors in a single round, eliminating the need for repeated syndrome measurements—a hallmark of single-shot error correction.'
+    - 'The minimum-weight decoder optimizes the recovery operation \( E_{\text{rec}} \) to minimize the residual error \( E_{\text{rec}} \cdot E \) given a noisy syndrome \( s = \sigma(E) + u \). The decoder’s performance is intrinsically tied to the code’s soundness: when the code is \((t, f)\)-sound, the minimum-weight decoder guarantees that the residual error’s min-weight scales as \( f(2|u|) \) for measurement errors \( |u| < t/2 \) \cite{arxiv:1805.09271}. This property is particularly robust in double homological codes, where soundness follows a cubic scaling (\( f(x) \sim x^3 \)).'
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: FerozAhmedMian
+      date: '2025-07-28'
+

users/users_db.yml

@@ -562,8 +562,8 @@
   name: 'Purva Thakre'
   githubusername: purva-thakre
 
-- user_id: FerozAhmad
-  name: 'Feroz Ahmad'
+- user_id: FerozAhmedMian
+  name: 'Feroz Ahmed Mian'
   githubusername: Fe-r-oz
 
 - user_id: vtomole