kb/data/en.wikipedia.org/wiki/Bacon–Shor_code-0.md

---
title: "Bacon–Shor code"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Bacon–Shor_code"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:06:36.713955+00:00"
instance: "kb-cron"
---

The Bacon–Shor code is a subsystem error correcting code. In a subsystem code, information is encoded in a subsystem of a Hilbert space. Subsystem codes lend to simplified error correcting procedures unlike codes which encode information in the subspace of a Hilbert space.  This simplicity led to the first claim of fault tolerant circuit demonstration on a quantum computer. It is named after Dave Bacon and Peter Shor.
Given the stabilizer generators of Shor's code: 
  
    
      
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    {\displaystyle \langle X_{0}X_{1}X_{2}X_{3}X_{4}X_{5},X_{0}X_{1}X_{2}X_{6}X_{7}X_{8},Z_{0}Z_{1},Z_{1}Z_{2},Z_{3}Z_{4},Z_{4}Z_{5},Z_{6}Z_{7},Z_{7}Z_{8}\rangle }
  
, 4 stabilizers can be removed from this generator by recognizing gauge symmetries in the code to get: 
  
    
      
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    {\displaystyle \langle X_{0}X_{1}X_{2}X_{3}X_{4}X_{5},X_{0}X_{1}X_{2}X_{6}X_{7}X_{8},Z_{0}Z_{1}Z_{3}Z_{4}Z_{6}Z_{7},Z_{1}Z_{2}Z_{4}Z_{5}Z_{7}Z_{8}\rangle }
  
. Error correction is now simplified because 4 stabilizers are needed to measure errors instead of 8. A gauge group can be created from the stabilizer generators:
  
    
      
        ⟨
        
          Z
          
            1
          
        
        
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        ⟩
      
    
    {\displaystyle \langle Z_{1}Z_{2},X_{2}X_{8},Z_{4}Z_{5},X_{5}X_{8},Z_{0}Z_{1},X_{0}X_{6},Z_{3}Z_{4},X_{3}X_{6},X_{1}X_{7},X_{4}X_{7},Z_{6}Z_{7},Z_{7}Z_{8}\rangle }
  
. Given that the Bacon–Shor code is defined on a square lattice where the qubits are placed on the vertices; laying the qubits on a grid in a way that corresponds to the gauge group shows how only 2 qubit nearest-neighbor measurements are needed to infer the error syndromes. The simplicity of  deducing the syndromes reduces the overhead for fault tolerant error correction.


== See also ==
Five-qubit error correcting code


== References ==