Bacon-Shor code[1][2]

Description

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~, \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} \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 \([[9,1,3]]\) and has 6 gauge operators, symmetric in both \(X\) and \(Z\), reducing to the Shor code for a particular gauge configuration. The error-detecting \([[4,1,2]]\) Bacon-Shor code, which reduces to a subcode of the \([[4,2,2]]\) code for a particular gauge configuration, has gauge operators \(\{XIXI,IIXX,ZIZI,IZIZ\}\).

Protection

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 \([[L^3,1,L]]\).

Rate

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].

Gates

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

Decoding

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 [7][8].

Fault Tolerance

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

Threshold

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 [2].

Realizations

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 [9].

Parent

Cousins

Zoo code information

Internal code ID: bacon_shor

Your contribution is welcome!

on github.com (edit & pull request)

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Zoo Code ID: bacon_shor

Cite as:
“Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bacon_shor
BibTeX:
@incollection{eczoo_bacon_shor, title={Bacon-Shor code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/bacon_shor} }
Permanent link:
https://errorcorrectionzoo.org/c/bacon_shor

References

[1]
P. W. Shor, “Scheme for reducing decoherence in quantum computer memory”, Physical Review A 52, R2493 (1995). DOI
[2]
D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006). DOI; quant-ph/0506023
[3]
P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007). DOI; quant-ph/0610063
[4]
X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000). DOI; quant-ph/0002039
[5]
Theodore J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”. 1705.01686
[6]
Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
[7]
Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell, “Fiber Bundle Codes: Breaking the $N^{1/2} \operatorname{polylog}(N)$ Barrier for Quantum LDPC Codes”. 2009.03921
[8]
M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021). DOI; 2107.02194
[9]
Laird Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”. 2009.11482

Cite as:

“Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bacon_shor

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/subsystem/bacon_shor.yml.