## Description

A geometrically local 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]).

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.

## 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]. Some non-locality is necessary to circumvent these bounds [12].

## 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 [13]. A refinement can be made that expresses the bound in terms of higher-group symmetries of the topological phases underlying the codes [14; 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 [15]. The code capacity threshold of such a code family is upper bounded by \(1/\nu\) [15].

## Decoding

## Parent

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

## Children

- 2D 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 [19]. Some notions may be extendable to non-stabilizer codes [20; 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.
- \((1,3)\) 4D toric code
- Loop toric code
- 3D lattice stabilizer code
- Modular-qudit color code — Modular-qudit color codes are defined analogous to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizers commute [21; Sec. III].

## Cousins

- Symmetry-protected topological (SPT) code — Lattice CSS codes in \(D\) dimensions can be converted in SPT Hamiltonians in one less dimension [22].
- Quantum LDPC (QLDPC) code — Chain complexes describing some QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [23]. In addition, chain complexes describing QLDPC codes can be converted to 2D lattice stabilizer codes [24].
- Good QLDPC code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [23,25]. 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 \(n\) [26].
- 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.
- 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

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- 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
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## 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