Description
Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise.
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.
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 \([[L^3,1,L]]\). Bacon-Shor code parameteres can be optimized by changing the block geometry, yielding good performance against biased noise [3].
Rate
A non-LDPC family of Bacon-Shor codes achieves a distance of order \(\Omega(n^{1-\epsilon})\) with sparse gauge operators.
Transversal Gates
Logical Hadamard is transversal in symmetric Bacon-Shor codes up to a qubit permutation [4] and can be implemented with teleportation [5].Bacon-Shor codes on an \(m \times m^k\) lattice admit transversal \(k\)-qubit-controlled \(Z\) gates [6].
Gates
Piecably fault-tolerant circuits can be employed to construct non-transversal gates effectively [7].Subsystem lattice surgery [8].Measurement-free deformation protocol realizing the \(CCZ\) gate [9].
Decoding
Both Steane error correction and Shor error correction can be used for syndrome extraction, with the former outperforming the latter [10].Utilizing the mapping of the effect of the noise to a statistical mechanical model [11,12] yields several copies of the 1D Ising model [13; 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 [14,15].Continuous-time QEC [16].
Fault Tolerance
Fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes that is effective against highly biased noise [17].Pieceably fault-tolerant circuits can be employed to construct non-transversal gates effectively [7].
Code Capacity Threshold
The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a topological threshold in the \(m_1,m_2 \to \infty\) limit [18]. However, a threshold can be obtained from concatenated Bacon-Shor codes that are further restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code [19]. This threshold differs from a concatenated threshold in that there are no long-range connectivity requirements.Lower bounds for the concatenated threshold of various small Bacon-Shor codes are tabulated in [4; Table I].
Threshold
Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [20].The Bacon-Shor code has a measurement threshold of zero [21].
Notes
See [22; Sec. III.C1] for an exposition.
Parents
- Bravyi-Bacon-Shor (BBS) code
- Subsystem hypergraph product (SHP) code
- Compass code — A compass code on a fully non-colored lattice reduces to the Bacon-Shor code.
Children
- \([[4,1,1,2]]\) Four-qubit subsystem code — The four-qubit subsystem code is the shortest error-detecting Bacon-Shor code.
- \([[9,1,3,3]]\) Nine-qubit Bacon-Shor code — The nine-qubit Bacon-Shor code is the shortest error-correcting Bacon-Shor code.
Cousins
- Hamiltonian-based code — The 2D Bacon-Shor gauge-group Hamiltonian is the compass model [23–25].
- Hastings-Haah Floquet code — The Bacon-Shor code admits a Floquet version with a particular stabilizer measurement schedule [26].
- Hybrid stabilizer code — There are several ways to convert Bacon-Shor codes to hybrid qubit stabilizer codes [27,28]
- Asymmetric quantum code — Bacon-Shor code parameters against bit- and phase-noise can be optimized by changing the block geometry, yielding good performance against biased noise [3]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [17].
- 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 [29; Sec. IX.B]).
- Majorana subsystem stabilizer code — Bacon-Shor codes can be fermionized into fermionic subsystem codes with two-body terms [30].
- GNU PI code — GNU codes of length \((2t+1)^2\) result from projecting Bacon-Shor codes into the PI qubit subspace [31].
- Quantum parity code (QPC) — Bacon-Shor codes reduce to QPCs when all \(X\)-type gauge generators are fixed [32; pg. 6].
- Heavy-hexagon code — Bacon-Shor stabilizers are used to measure the X-type stabilizers of the code.
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]
- J. Napp and J. Preskill, “Optimal Bacon-Shor codes”, (2012) arXiv:1209.0794
- [4]
- P. Aliferis and A. W. Cross, “Subsystem Fault Tolerance with the Bacon-Shor Code”, Physical Review Letters 98, (2007) arXiv:quant-ph/0610063 DOI
- [5]
- 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
- [6]
- T. J. Yoder, “Universal fault-tolerant quantum computation with Bacon-Shor codes”, (2017) arXiv:1705.01686
- [7]
- Yoder, Theodore., DSpace@MIT Practical Fault-Tolerant Quantum Computation (2018)
- [8]
- 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
- [9]
- S. Veroni, A. Paler, and G. Giudice, “Universal quantum computation via scalable measurement-free error correction”, (2024) arXiv:2412.15187
- [10]
- G. Escobar-Arrieta and M. Gutiérrez, “Improved performance of the Bacon-Shor code with Steane’s syndrome extraction method”, (2024) arXiv:2403.01659
- [11]
- E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [12]
- A. T. Schmitz, “Thermal Stability of Dynamical Phase Transitions in Higher Dimensional Stabilizer Codes”, (2020) arXiv:2002.11733
- [13]
- H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
- [14]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n \({}^{\text{1/2}}\) polylog( n ) barrier for Quantum LDPC codes”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing 1276 (2021) arXiv:2009.03921 DOI
- [15]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [16]
- G. Sarma and H. Mabuchi, “Gauge subsystems, separability and robustness in autonomous quantum memories”, New Journal of Physics 15, 035014 (2013) arXiv:1212.3564 DOI
- [17]
- P. Brooks and J. Preskill, “Fault-tolerant quantum computation with asymmetric Bacon-Shor codes”, Physical Review A 87, (2013) arXiv:1211.1400 DOI
- [18]
- 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
- [19]
- C. Gidney and D. Bacon, “Less Bacon More Threshold”, (2023) arXiv:2305.12046
- [20]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
- [21]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [22]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI
- [23]
- K. I. Kugel’ and D. I. Khomskiĭ, “The Jahn-Teller effect and magnetism: transition metal compounds”, Soviet Physics Uspekhi 25, 231 (1982) DOI
- [24]
- J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
- [25]
- Z. Nussinov and J. van den Brink, “Compass and Kitaev models -- Theory and Physical Motivations”, (2013) arXiv:1303.5922
- [26]
- M. S. Alam and E. Rieffel, “Dynamical Logical Qubits in the Bacon-Shor Code”, (2024) arXiv:2403.03291
- [27]
- A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) arXiv:2012.05896 DOI
- [28]
- G. Dauphinais, D. W. Kribs, and M. Vasmer, “Stabilizer Formalism for Operator Algebra Quantum Error Correction”, Quantum 8, 1261 (2024) arXiv:2304.11442 DOI
- [29]
- B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, “Quantum memories at finite temperature”, Reviews of Modern Physics 88, (2016) arXiv:1411.6643 DOI
- [30]
- A. Chapman, S. T. Flammia, and A. J. Kollár, “Free-Fermion Subsystem Codes”, (2022) arXiv:2201.07254
- [31]
- Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
- [32]
- M. Li, D. Miller, M. Newman, Y. Wu, and K. R. Brown, “2D Compass Codes”, Physical Review X 9, (2019) arXiv:1809.01193 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