## Description

A geometrically local qubit, modular-qudit, or Galois-qudit stabilizer code with qudits 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 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.

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 [4] ([5]). See 3D lattice stabilizer code entry for the 3D classification.

Pauli-to-polynomial mapping: A single-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. [4] and Sec. IV of Ref. [6]).

## Rate

BPT bound: Lattice qubit codes are limited by the Bravyi-Poulin-Terhal (BPT) bound [7] (see also [8–10]), 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 \(d = O(L^{D-1})\) [8]. Codes on a \(D\)-dimensional homogeneous Riemannian manifold with diameter \(L\) satisfy \(k = O(L^{D-2})\) [11].

## Decoding

## Parents

- Quantum low-density parity-check (QLDPC) code — Lattice stabilizer codes are geometrically local.
- Quasi-cyclic quantum code — Lattice stabilizer codes are invariant under translations by a lattice unit cell.

## Children

- 2D lattice stabilizer code
- 3D lattice stabilizer code
- Crystalline-circuit qubit code
- Transverse-field Ising model (TFIM) code
- Quantum convolutional code — Quantum convolutional codes are lattice stabilizer codes on an semi-infinite or infinite lattice in one dimension [15]. Some notions may be extendable to non-stabilizer codes [16; Sec. 4]. Any prime-qudit code can be converted using a constant-depth Clifford circuit to several copies of the 1D repetition code along with some trivial codes [4].
- Quantum spatially coupled (SC-QLDPC) code — Stabilizer generator matrices of SC-QLDPC codes on infinite-length chains or grids define a class of lattice stabilizer codes.

## Cousins

- Hamiltonian-based code — Lattice stabilizer code Hamiltonians are stable with respect to small perturbations [17,18], meaning that the notion of a phase can be defined.
- Good QLDPC code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [19,20]. 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 \(n\) [21].
- 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.
- Dynamical automorphism (DA) code — DA codes are 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. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [5]
- 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
- [6]
- 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
- [7]
- 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
- [8]
- 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
- [9]
- S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
- [10]
- S. T. Flammia et al., “Limits on the storage of quantum information in a volume of space”, Quantum 1, 4 (2017) arXiv:1610.06169 DOI
- [11]
- J. Haah, “A degeneracy bound for homogeneous topological order”, SciPost Physics 10, (2021) arXiv:2009.13551 DOI
- [12]
- J. W. Harrington, Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004 DOI
- [13]
- S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI
- [14]
- W. Zhong, O. Shtanko, and R. Movassagh, “Advantage of Quantum Neural Networks as Quantum Information Decoders”, (2024) arXiv:2401.06300
- [15]
- J. Haah, “Lattice quantum codes and exotic topological phases of matter”, (2013) arXiv:1305.6973
- [16]
- H. Ollivier and J.-P. Tillich, “Quantum convolutional codes: fundamentals”, (2004) arXiv:quant-ph/0401134
- [17]
- S. Bravyi and M. B. Hastings, “A Short Proof of Stability of Topological Order under Local Perturbations”, Communications in Mathematical Physics 307, 609 (2011) arXiv:1001.4363 DOI
- [18]
- S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations”, Journal of Mathematical Physics 51, (2010) arXiv:1001.0344 DOI
- [19]
- M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
- [20]
- T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2023) arXiv:2309.16104
- [21]
- E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755

## Page edit log

- Victor V. Albert (2022-05-15) — most recent
- Tony Lau (2022-04-02)

## Cite as:

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