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X-cube model code[1]

Description

A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include several generalized X-cube models [2]. A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [3], is equivalent to the X-cube model [4] (see also Refs. [5,6]).

Decoding

Parallelized matching decoder [7].

Code Capacity Threshold

Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [8].

Cousins

  • Quantum-inspired classical block code— According to Ref. [9], a classical analogue of the X-cube model is the eight-vertex model [10–12].
  • Newman-Moore code— Generalized X-cube models [2] are constructed from a balanced product of the quantum repetition (1D Ising) code and the Newman-Moore code.
  • Quantum repetition code— Generalized X-cube models [2] are constructed from a balanced product of the quantum repetition (1D Ising) code and the Newman-Moore code.
  • Balanced product (BP) code— Generalized X-cube models [2] are constructed from a balanced product of the quantum repetition (1D Ising) code and the Newman-Moore code.
  • Double-semion stabilizer code— A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [3], is equivalent to the X-cube model [4] (see also Refs. [5,6]).
  • String-net code— A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [3], is equivalent to the X-cube model [4] (see also Refs. [5,6]).
  • Kitaev surface code— The X-cube model can be constructed by coupling layers of the surface code [3,13].
  • 3D surface code— The X-cube model admits a topological defect network construction out of 3D surface codes [14].
  • Plaquette Ising code— The 3D plaquette Ising model can be used to obtain the X-cube model by gauging [15–17,17] its subsystem symmetry [1].
  • Fracton Floquet code— The ISG of the X-cube Floquet code can be that of the X-cube model code or the checkerboard model code.
  • X-cube Floquet code— The ISG of the X-cube Floquet code can be that of the X-cube model code or that of several decoupled surface codes. A rewinding of the original measurement sequence yields a period-six sequence whose ISGs are those of the X-cube model, some 3D surface codes, and some decoupled surface codes [18].
  • Majorana checkerboard code— The Majorana checkerboard code is equivalent via a constant-depth unitary to a semionic version of the X-cube model and some decoupled fermionic modes [5].
  • Checkerboard model code— The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [19].
  • Four Color Cube (FCC) fracton model code— The FCC fracton model code is obtained from four coupled X-cube models using p-membrane condensation. [3].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum
Parents
Qubit CSS code
Qudit X-cube model codeFracton stabilizer Lattice stabilizer QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
The qudit X-cube model code reduces to the X-cube model code for \(q=2\). The X-cube model is a foliated type-I fracton code [20,21].
X-cube model code

References

[1]
S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[2]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
[3]
H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
[4]
W. Shirley, K. Slagle, and X. Chen, “Fractional excitations in foliated fracton phases”, Annals of Physics 410, 167922 (2019) arXiv:1806.08625 DOI
[5]
T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
[6]
S. Pai and M. Hermele, “Fracton fusion and statistics”, Physical Review B 100, (2019) arXiv:1903.11625 DOI
[7]
B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
[8]
H. Song, J. Schönmeier-Kromer, K. Liu, O. Viyuela, L. Pollet, and M. A. Martin-Delgado, “Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry”, Physical Review Letters 129, (2022) arXiv:2112.05122 DOI
[9]
G. M. Nixon and B. J. Brown, “Correcting Spanning Errors With a Fractal Code”, IEEE Transactions on Information Theory 67, 4504 (2021) arXiv:2002.11738 DOI
[10]
B. Sutherland, “Two-Dimensional Hydrogen Bonded Crystals without the Ice Rule”, Journal of Mathematical Physics 11, 3183 (1970) DOI
[11]
R. J. Baxter, “Eight-Vertex Model in Lattice Statistics”, Physical Review Letters 26, 832 (1971) DOI
[12]
R. J. Baxter, “Partition function of the Eight-Vertex lattice model”, Annals of Physics 70, 193 (1972) DOI
[13]
Z. Song, A. Dua, W. Shirley, and D. J. Williamson, “Topological Defect Network Representations of Fracton Stabilizer Codes”, PRX Quantum 4, (2023) arXiv:2112.14717 DOI
[14]
D. Aasen, D. Bulmash, A. Prem, K. Slagle, and D. J. Williamson, “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020) arXiv:2002.05166 DOI
[15]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[16]
L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[17]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[18]
A. Dua, N. Tantivasadakarn, J. Sullivan, and T. D. Ellison, “Engineering 3D Floquet Codes by Rewinding”, PRX Quantum 5, (2024) arXiv:2307.13668 DOI
[19]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order in the checkerboard model”, Physical Review B 99, (2019) arXiv:1806.08633 DOI
[20]
W. Shirley, K. Slagle, and X. Chen, “Universal entanglement signatures of foliated fracton phases”, SciPost Physics 6, (2019) arXiv:1803.10426 DOI
[21]
A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
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Zoo Code ID: xcube

Cite as:
“X-cube model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/xcube
BibTeX:
@incollection{eczoo_xcube, title={X-cube model code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/xcube} }
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“X-cube model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/xcube

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/fracton/xcube.yml.