Lattice stabilizer code[14] 

Also known as Topological stabilizer code.

Description

A geometrically local stabilizer code with sites organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). On an infinite lattice, its stabilizer group is generated by few-site Pauli-type operators and their translations, in which case the code is called translationally invariant stabilizer code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions. Lattice defects and boundaries between different codes can also be introduced.

Modular- and Galois-qudit lattice stabilizer codes

Translationally-invariant prime-qudit (\(q=p\)) stabilizer codes with \(m\) qudits per unit cell have been classified in dimensions \(D\in\{1,2\}\) in the thermodynamic limit, up to equivalence under local constant-depth Clifford circuits. Any 1D (2D) code can be converted to several copies of the 1D repetition code (prime-qudit 2D surface code) along with some trivial codes [5] ([6]). See 3D lattice stabilizer code entry for the 3D classification.

Pauli-to-polynomial mapping: A single modular- or Galois-qudit Pauli operator can be specified by the lattice coordinate of the site and the symplectic vector representation of the Pauli operator within the site. In an extension of the sympletic representation, each lattice coordinate can be represented by a Laurent monomial of \(D\) formal variables. For example, when \(D=2\) and \(m=1\), the product of an \(X\) acting on the qubit at lattice coordinate \((-1,2)\) and a \(Z\) acting on the qubit at \((1,0)\) can be represented by the vector \( (x^{-1} y^2 | x) \). The multiplicative group of finitely supported Pauli operators modulo phase factors on the lattice of dimension \(D\) with \(m\) prime-dimensional qubits per site is isomorphic to the additive group of Laurent polynomial column vectors of length \(2m\) in \(D\) formal variables (see Ref. [5] and Sec. IV of Ref. [7]).

For periodic boundary conditions, this mapping can be thought of as a quantum extension of the cyclic-to-polynomial correspondence. For open boundary conditions, this mapping extends the mapping used in quantum convolutional codes to multiple spatial dimensions.

Bosonic lattice stabilizer codes

Bosonic lattice stabilizer codes can contain discrete or continuous subgroups and can admit logical qudit and/or oscillator logical subspaces. Such codes can realize topological phases of matter that are expected not to be realizable with qudit stabilizer codes [4].

Rate

BPT bound: Lattice qubit codes are limited by the Bravyi-Poulin-Terhal (BPT) bound [8] (see also [911]), which states that \(d \leq O(n^{1-1/D})\) and \(k d^{2/D-1} = O(n)\) for \(D\)-dimensional lattice geometries. The Bravyi-Terhal (BT) bound states that \(d = O(L^{D-1})\) [9]. Codes on a \(D\)-dimensional homogeneous Riemannian manifold with diameter \(L\) satisfy \(k = O(L^{D-2})\) [12]. Some non-locality is necessary to circumvent these bounds [13].

Gates

Bravyi-Koenig bound: Logical gates implemented via constant-depth quantum circuits on a \(D\)-dimensional lattice stabilizer code whose distance increases at least logarithmically with \(n\) lie in the \(D\)th level of the Clifford hierarchy [14]. A refinement can be made that expresses the bound in terms of higher-group symmetries of the topological phases underlying the codes [15; Sec. 5.4.2]. Conversely, the distance of a code on an \(L^{D}\) lattice is upper bounded by order \(O(L^{D+1-\nu})\) if the code implements an \(\nu\)th-level Clifford hierarchy gate [16]. The code capacity threshold of such a code family is upper bounded by \(1/\nu\) [16].

Decoding

A local decoder (a.k.a. automaton decoder) applies local rules to each small region of sites in a lattice geometry. Such decoders do not require any potentially non-local classical post-processing of error syndromes.Clustering decoder [17,18].Quantum neural-network (QNN) decoder [19].Almost linear-time decoder [20].

Parent

  • Quasi-cyclic QLDPC code — Lattice stabilizer codes are QLDPC codes that are invariant under translations by a lattice unit cell.

Children

Cousins

  • Symmetry-protected topological (SPT) code — Lattice CSS codes in \(D\) dimensions can be converted in SPT Hamiltonians in one less dimension [24].
  • Quantum LDPC (QLDPC) code — Chain complexes describing some QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [25]. In addition, chain complexes describing QLDPC codes can be converted to 2D lattice stabilizer codes [26].
  • Good QLDPC code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [25,27]. Applying this procedure to good QLDPC codes yiels \([[n,n^{1-2/D},n^{1-1/D}]]\) lattice stabilizer codes in \(D\) spatial dimensions that saturate the BPT bound, up to corrections poly-logarithmic in \(n\) [28].
  • Lattice subsystem code — Lattice subsystem codes reduce to lattice stabilizer codes when there are no gauge qudits. The former (latter) is required to admit few-site gauge-group (stabilizer-group) generators on a lattice with boundary conditions.
  • Cluster-state code — Cluster states defined on various lattices are representatives of SPT phases, and states realizing these phases can be resources for MBQC. In 1D, cluster states are examples of SPT phases with global symmetries [2933] and enable MBQC on a single qubit [34,35]. The square-lattice cluster state, which is the prototypical resource for universal MBQC [34,35], and other 2D cluster states [3638] have SPT order protected by subsystem symmetries [36,39,40]. States like AKLT states and SPT fixed-point states can be efficiently converted into cluster states using local measurements and subsequently used as resources for MBQC [30,4145]. In 3D, cluster states belong to SPT phases protected by higher-form symmetries [46] and enable universal fault-tolerant MBQC [47]. A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [48]. Cluster states can be created on various lattices [49].
  • Modular-qudit DA code — DA codes are typically defined on 2D and 3D lattices, but they are not conventional stabilizer codes in that they use code switching for error correction and gates.

References

[1]
J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
[2]
J. Haah, “Commuting Pauli Hamiltonians as Maps between Free Modules”, Communications in Mathematical Physics 324, 351 (2013) arXiv:1204.1063 DOI
[3]
J. Haah, Lattice Quantum Codes and Exotic Topological Phases of Matter, California Institute of Technology, 2013 DOI
[4]
J. C. M. de la Fuente, T. D. Ellison, M. Cheng, and D. J. Williamson, “Topological stabilizer models on continuous variables”, (2024) arXiv:2411.04993
[5]
J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
[6]
J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices”, Journal of Mathematical Physics 62, (2021) arXiv:1812.11193 DOI
[7]
J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial Quantum Cellular Automata in Higher Dimensions”, Communications in Mathematical Physics 398, 469 (2022) arXiv:1812.01625 DOI
[8]
S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010) arXiv:0909.5200 DOI
[9]
S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009) arXiv:0810.1983 DOI
[10]
S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
[11]
S. T. Flammia, J. Haah, M. J. Kastoryano, and I. H. Kim, “Limits on the storage of quantum information in a volume of space”, Quantum 1, 4 (2017) arXiv:1610.06169 DOI
[12]
J. Haah, “A degeneracy bound for homogeneous topological order”, SciPost Physics 10, (2021) arXiv:2009.13551 DOI
[13]
S. Dai and R. Li, “Locality vs Quantum Codes”, (2024) arXiv:2409.15203
[14]
S. Bravyi and R. König, “Classification of Topologically Protected Gates for Local Stabilizer Codes”, Physical Review Letters 110, (2013) arXiv:1206.1609 DOI
[15]
M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry in finite gauge theory and stabilizer codes”, SciPost Physics 16, (2024) arXiv:2211.11764 DOI
[16]
F. Pastawski and B. Yoshida, “Fault-tolerant logical gates in quantum error-correcting codes”, Physical Review A 91, (2015) arXiv:1408.1720 DOI
[17]
J. W. Harrington, Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004 DOI
[18]
S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI
[19]
W. Zhong, O. Shtanko, and R. Movassagh, “Advantage of Quantum Neural Networks as Quantum Information Decoders”, (2024) arXiv:2401.06300
[20]
Q. Eggerickx, A. Wills, T.-C. Lin, K. De Greve, and M.-H. Hsieh, “Almost Linear Decoder for Optimal Geometrically Local Quantum Codes”, (2024) arXiv:2411.02928
[21]
J. Haah, “Lattice quantum codes and exotic topological phases of matter”, (2013) arXiv:1305.6973
[22]
H. Ollivier and J.-P. Tillich, “Quantum convolutional codes: fundamentals”, (2004) arXiv:quant-ph/0401134
[23]
F. H. E. Watson, E. T. Campbell, H. Anwar, and D. E. Browne, “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
[24]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[25]
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
[26]
X. Li, T.-C. Lin, and M.-H. Hsieh, “Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension”, (2024) arXiv:2408.01769
[27]
T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2024) arXiv:2309.16104
[28]
E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
[29]
R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states”, Physical Review A 68, (2003) arXiv:quant-ph/0301052 DOI
[30]
A. Miyake, “Quantum computational capability of a 2D valence bond solid phase”, Annals of Physics 326, 1656 (2011) arXiv:1009.3491 DOI
[31]
W. Son, L. Amico, and V. Vedral, “Topological order in 1D Cluster state protected by symmetry”, Quantum Information Processing 11, 1961 (2011) arXiv:1111.7173 DOI
[32]
D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, “Symmetry-Protected Phases for Measurement-Based Quantum Computation”, Physical Review Letters 108, (2012) arXiv:1201.4877 DOI
[33]
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
[34]
R. Raussendorf, D. Browne, and H. Briegel, “The one-way quantum computer--a non-network model of quantum computation”, Journal of Modern Optics 49, 1299 (2002) arXiv:quant-ph/0108118 DOI
[35]
R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer”, Physical Review Letters 86, 5188 (2001) DOI
[36]
D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, “Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter”, Quantum 3, 142 (2019) arXiv:1806.08780 DOI
[37]
T. Devakul and D. J. Williamson, “Universal quantum computation using fractal symmetry-protected cluster phases”, Physical Review A 98, (2018) arXiv:1806.04663 DOI
[38]
A. K. Daniel, R. N. Alexander, and A. Miyake, “Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices”, Quantum 4, 228 (2020) arXiv:1907.13279 DOI
[39]
Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, “Subsystem symmetry protected topological order”, Physical Review B 98, (2018) arXiv:1803.02369 DOI
[40]
R. Raussendorf, C. Okay, D.-S. Wang, D. T. Stephen, and H. P. Nautrup, “Computationally Universal Phase of Quantum Matter”, Physical Review Letters 122, (2019) arXiv:1803.00095 DOI
[41]
X. Chen, R. Duan, Z. Ji, and B. Zeng, “Quantum State Reduction for Universal Measurement Based Computation”, Physical Review Letters 105, (2010) arXiv:1002.1567 DOI
[42]
T.-C. Wei, I. Affleck, and R. Raussendorf, “Two-dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation”, Physical Review A 86, (2012) arXiv:1009.2840 DOI
[43]
T.-C. Wei, I. Affleck, and R. Raussendorf, “Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource”, Physical Review Letters 106, (2011) arXiv:1102.5064 DOI
[44]
T.-C. Wei, P. Haghnegahdar, and R. Raussendorf, “Hybrid valence-bond states for universal quantum computation”, Physical Review A 90, (2014) arXiv:1310.5100 DOI
[45]
H. P. Nautrup and T.-C. Wei, “Symmetry-protected topologically ordered states for universal quantum computation”, Physical Review A 92, (2015) arXiv:1509.02947 DOI
[46]
S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, “Symmetry-protected topological order at nonzero temperature”, Physical Review A 96, (2017) arXiv:1611.05450 DOI
[47]
R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation”, New Journal of Physics 9, 199 (2007) arXiv:quant-ph/0703143 DOI
[48]
N. Tantivasadakarn and A. Vishwanath, “Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps”, Physical Review Letters 129, (2022) arXiv:2107.04019 DOI
[49]
M. Newman, L. A. de Castro, and K. R. Brown, “Generating Fault-Tolerant Cluster States from Crystal Structures”, Quantum 4, 295 (2020) arXiv:1909.11817 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: translationally_invariant_stabilizer

Cite as:
“Lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer
BibTeX:
@incollection{eczoo_translationally_invariant_stabilizer, title={Lattice stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer

Cite as:

“Lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/translationally_invariant_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/lattice/translationally_invariant_stabilizer.yml.