## 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 \(D\) such that each lattice point, referred to as a site, contains \(m\) qudits of dimension \(q\). The stabilizer group of the translationally invariant code is generated by site-local Pauli operators and their translations. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional codes. Infinite dimensional formulations are also possible, with the 1D lattice version reducing to quantum convolutional codes.

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. [5]).

Translationally-invariant prime-qudit (\(q=p\)) stabilizer codes have been classified in dimensions \(D\in\{1,2\}\), 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] ([6]). Three-dimensional qubit codes can be characterized by four coarse classes [7]:

1. Abelian topological phase: Excitations are mobile in all 3 dimensions, as is typical in a topological code. Such codes are conjectured to be equivalent to a \(\mathbb{Z}_2\) gauge theory, i.e., multiple copies of the 3D surface code or its variant where the charge excitation is a fermion.

2. Foliated type-I fracton phase: Excitations are mobile in less than 3 dimensions, but codes can be grown by foliation, i.e., stacking copies of the 2D surface code.

3. Fractal type-I fracton phase: Excitations are mobile in less than 3 dimensions, and codes are not foliated.

4. Type-II fracton phase: Excitations are not mobile in any dimension and there are no string operators.

## Decoding

## Parents

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

## Children

- Crystalline-circuit qubit code
- Transverse-field Ising model (TFIM) code
- Quantum spatially coupled (SC-QLDPC) code — Stabilizer generator matrices of SC-QLDPC codes on infinite-length chains or grids define a class of translationally-invariant stabilizer codes.
- Quantum parity code (QPC)
- Quantum convolutional code — Quantum convolutional codes are translationally-invariant stabilizer codes on an semi-infinite or infinite lattice in one dimension [10]. Some notions may be extendable to non-stabilizer codes [11; Sec. 4].

## Cousins

- Modular-qudit stabilizer code — Modular-qudit stabilizer codes can be thought of as translationally-invariant stabilizer codes for dimension \(D = 0\), with the lattice consisting of a single site.
- Kitaev surface code — Translation-invariant 2D qubit topological stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [4,12,13].
- Abelian quantum double stabilizer code — Translation-invariant 2D prime-qudit topological stabilizer codes are equivalent to several copies of the prime-qudit surface code via a local constant-depth Clifford circuit [6].
- Abelian TQD stabilizer code — Translationally-invariant stabilizer codes can realize 2D modular gapped abelian topological orders [14]. Conversely, abelian TQD codes need not be translationally invariant, and can realize multiple topological phases on one lattice.
- Fracton code — Translationally-invariant stabilizer codes can realize fracton orders. Conversely, fracton codes need not be translationally invariant, and can realize multiple phases on one lattice.
- Self-correcting quantum code — 3D translationally-invariant qubit stabilizer code families with constant \(k\) support logical string operators and thus cannot be self-correcting [15]. For non-constant \(k\), such families can support at most a logarithmic energy barrier [1].

## 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, L. Fidkowski, and M. B. Hastings, “Nontrivial Quantum Cellular Automata in Higher Dimensions”, Communications in Mathematical Physics (2022) arXiv:1812.01625 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, 012201 (2021) arXiv:1812.11193 DOI
- [7]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [8]
- J. W. Harrington, Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes, California Institute of Technology, 2004 DOI
- [9]
- S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI
- [10]
- J. Haah, “Lattice quantum codes and exotic topological phases of matter”, (2013) arXiv:1305.6973
- [11]
- H. Ollivier and J.-P. Tillich, “Quantum convolutional codes: fundamentals”, (2004) arXiv:quant-ph/0401134
- [12]
- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
- [13]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
- [14]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [15]
- B. Yoshida, “Feasibility of self-correcting quantum memory and thermal stability of topological order”, Annals of Physics 326, 2566 (2011) arXiv:1103.1885 DOI

## Page edit log

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

## Cite as:

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