Bacon-Shor code[1,2] 


CSS subsystem stabilizer code defined on an \(m_1 \times m_2\) lattice of qubits. It is said to be symmetric when \(m_1=m_2\). The \(X\)-type and \(Z\)-type stabilizers defined as \(X\) and \(Z\) operators acting on all qubits on adjacent columns and rows, respectively. Let \(O_{i,j}\) denote an operator acting on the qubit at a position \((i,j)\) on the lattice, with \(i\in\{0,1,\ldots ,m_1-1\}\) and \(j\in\{0,1,\ldots,m_2-1\}\). The code's stabilizer group is \begin{align} \mathsf{S}=\langle X_{i,*}X_{i+1,*},Z_{*,j}Z_{*,j+1}\rangle~, \tag*{(1)}\end{align} with generators expressed as products of nearest-neightbour 2-qubit gauge operators, \begin{align} \begin{split} X_{i,*}X_{i+1,*}= \bigotimes_{k=0}^{m_2-1} X_{i,k}X_{i+1,k} \\ Z_{*,j}Z_{*,j+1}=\bigotimes_{k=0}^{m_1-1} Z_{k,j}Z_{k,j+1}~. \end{split} \tag*{(2)}\end{align} Syndrome extraction can be done by measuring these gauge operators, which are on fewer qubits and local.

The shortest error-correcting Bacon-Shor code is \([[9,1,3,3]]\), with four stabilizer generators \begin{align} \begin{array}{ccccccccc} X & X & X & X & X & X & I & I & I\\ I & I & I & X & X & X & X & X & X\\ Z & Z & I & Z & Z & I & Z & Z & I\\ I & Z & Z & I & Z & Z & I & Z & Z \end{array}~, \tag*{(3)}\end{align} which generate the gauge group with the help of eight additional generators \begin{align} \begin{array}{ccccccccc} X & I & I & X & I & I & I & I & I\\ I & X & I & I & X & I & I & I & I\\ I & I & I & X & I & I & X & I & I\\ I & I & I & I & X & I & I & X & I\\ Z & Z & I & I & I & I & I & I & I\\ I & I & I & Z & Z & I & I & I & I\\ I & Z & Z & I & I & I & I & I & I\\ I & I & I & I & Z & Z & I & I & I \end{array}~. \tag*{(4)}\end{align} If the physical qubits are arranged in a three-by-three square, the \(Z\)-type (\(X\)-type) gauge operators are supported on qubits in the same row (column). The code reduces to the Shor code for a particular gauge configuration.


The \([[m_1 m_2,1,min(m_1,m_2)]]\) variant has distance \(d=min(m_1,m_2)\). In a symmetric 3-dimensional case (defined on a cubic lattice) with \(L^3\) qubits, the code has the parameters \([[L^3,1,L]]\).


A non-LDPC family of Bacon-Shor codes achieves a distance of \(\Omega(n^{1-\epsilon})\) with sparse gauge operators.

Transversal Gates

Logical Hadamard is transversal in symmetric Bacon-Shor codes up to a qubit permutation [3] and can be implemented with teleportation [4]. Bacon-Shor codes on an \(m \times mk\) lattice admit transversal \(k\)-qubit-controlled \(Z\) gates [5].


Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively [6].Subsystem lattice surgery [7].


Utilizing the mapping of the effect of the noise to a statistical mechanical model [8,9] yields several copies of the 1D Ising model [10; Sec. V.B].While check operators are few-body, stabilizer weights scale with the number of qubits, and stabilizer expectation values are obtained by taking products of gauge-operator expectation values. It is thus not clear how to extract stabilizer values in a fault-tolerant manner [11,12].

Fault Tolerance

Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively [6].


The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a threshold [13]. However, a threshold can be obtained from concatenated Bacon-Shor codes restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code [14].A lower bound of \(1.94 \times 10^{-4}\) for the accuracy threshold was proved for Bacon-Shor code with 5 levels of concatenation, using Steane method of FTEC [3].The three dimensional version offers the possibility of being a self-correcting quantum memory [15].


Trapped-ion qubits: state preparation, logical measurement, and stabilizer measurement for nine-qubit Bacon-Shor code demonstrated on a 13-qubit device by M. Cetina and C. Monroe groups [16].


See [17; Sec. III.C1] for an exposition.




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“Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_bacon_shor, title={Bacon-Shor code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.