Bacon-Shor code

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~, \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.

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].Subsystem lattice surgery [7].

Decoding

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].

Threshold

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].

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 [16].

Notes

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

Parents

Cousins

Hamiltonian-based code — The 2D Bacon-Shor code Hamiltonian is the compass model [18,19].
Self-correcting quantum code — 3D Bacon-Shor codes were conjectured to be self-correcting [2], but there remain issues to be resolved in order to validate this conjecture (see [20; Sec. IX.B]).
\([[9,1,3]]\) Shor code — The \([[9,1,3,3]]\) Bacon-Shor code reduces to the Shor code for a particular gauge configuration.
\([[4,2,2]]\) CSS code — The error-detecting \([[4,1,1,2]]\) Bacon-Shor code, which reduces to a subcode of the \([[4,2,2]]\) code for a particular gauge configuration, has gauge operators \(\{XXII,IIXX,ZIZI,IZIZ\}\).
Quantum parity code (QPC) — Bacon-Shor codes reduce to QPCs for particular gauge configurations.
Heavy-hexagon code — Bacon-Shor stabilizers are used to measure the X-type stabilizers of the code.
GNU permutation-invariant code — Symmetrized versions of the Bacon-Shor codes are GNU codes

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) arXiv:quant-ph/0506023 DOI
[3]: P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610063 DOI
[4]: X. Zhou, D. W. Leung, and I. L. Chuang, “Methodology for quantum logic gate construction”, Physical Review A 62, (2000) arXiv:quant-ph/0002039 DOI
[5]: T. J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”, (2017) arXiv:1705.01686
[6]: Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
[7]: H. Poulsen Nautrup, N. Friis, and H. J. Briegel, “Fault-tolerant interface between quantum memories and quantum processors”, Nature Communications 8, (2017) arXiv:1609.08062 DOI
[8]: E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[9]: A. T. Schmitz, “Thermal Stability of Dynamical Phase Transitions in Higher Dimensional Stabilizer Codes”, (2020) arXiv:2002.11733
[10]: H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
[11]: M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber Bundle Codes: Breaking the \(N^{1/2} \operatorname{polylog}(N)\) Barrier for Quantum LDPC Codes”, (2020) arXiv:2009.03921
[12]: M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
[13]: N. C. Brown, M. Newman, and K. R. Brown, “Handling leakage with subsystem codes”, New Journal of Physics 21, 073055 (2019) arXiv:1903.03937 DOI
[14]: C. Gidney and D. Bacon, “Less Bacon More Threshold”, (2023) arXiv:2305.12046
[15]: D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories”, Physical Review A 73, (2006) DOI
[16]: L. Egan et al., “Fault-Tolerant Operation of a Quantum Error-Correction Code”, (2021) arXiv:2009.11482
[17]: B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
[18]: K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
[19]: J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
[20]: B. J. Brown et al., “Quantum memories at finite temperature”, Reviews of Modern Physics 88, (2016) arXiv:1411.6643 DOI

Page edit log

Victor V. Albert (2022-07-31) — most recent
Mazin Karjikar (2022-06-28)
Victor V. Albert (2022-03-15)
Srilekha Gandhari (2022-01-20)
Victor V. Albert (2021-12-03)

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.