Description
Lattice stabilizer code in one Euclidean dimension.
Any modular-qudit code can be converted to several copies of the 1D repetition code along with some trivial codes via a local constant-depth Clifford circuit [1]. There is no 1D bosonic topological order at nonzero temperature [2,3].
Fault Tolerance
A fault-tolerant quantum computation scheme exists for a 1D qubit lattice with nearest-neighbor connectivity [4].Cousins
- Analog cluster-state code— Analog cluster states defined on a 1D array of modes are called quantum wires [5,6], not to be confused with the Kitaev quantum wire, a fermion code. Analog cluster states defined on a 1D ladder are sometimes called dual-rail, not to be confused with the dual-rail code.
- Local Haar-random circuit qubit code— In a 1D geometry, the local Haar-random circuit qubit code code approximately detects any error with support on a segment of length \(\leq n/4\), with deviations exponentially suppressed in \(n\). There is a phase transition in error correction power vs error rate \(p\), with a critical depth of order \(O(1/p)\) [7].
- Modular-qudit cluster-state code— Qudit cluster states defined on 1D lattices are representatives of various SPT phases [8].
Primary Hierarchy
Parents
1D lattice stabilizer code
Children
Log-depth Clifford circuits on a 1D geometry yield approximate codes whose encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors [9].
The codespace of the quantum repetition code is the ground-state space of a frustration-free 1D classical Ising model with nearest-neighbor interactions.
Quantum convolutional codes are lattice 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]. 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 [1].
References
- [1]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [2]
- K. Kato and F. G. S. L. Brandão, “Locality of edge states and entanglement spectrum from strong subadditivity”, Physical Review B 99, (2019) arXiv:1804.05457 DOI
- [3]
- W. De Roeck, M. Fraas, and B. de O. Carvalho, “Pure gapped ground states of spin chains are short-range entangled”, (2025) arXiv:2511.14699
- [4]
- C. Gidney and T. Bergamaschi, “A Constant Rate Quantum Computer on a Line”, (2025) arXiv:2502.16132
- [5]
- N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-Way Quantum Computing in the Optical Frequency Comb”, Physical Review Letters 101, (2008) arXiv:0804.4468 DOI
- [6]
- S. T. Flammia, N. C. Menicucci, and O. Pfister, “The optical frequency comb as a one-way quantum computer”, Journal of Physics B: Atomic, Molecular and Optical Physics 42, 114009 (2009) arXiv:0811.2799 DOI
- [7]
- J. Nelson, J. Rajakumar, and M. J. Gullans, “Error correction phase transition in noisy random quantum circuits”, (2025) arXiv:2510.07512
- [8]
- L. Tsui, H.-C. Jiang, Y.-M. Lu, and D.-H. Lee, “Quantum phase transitions between a class of symmetry protected topological states”, Nuclear Physics B 896, 330 (2015) arXiv:1503.06794 DOI
- [9]
- G. Liu, Z. Du, Z.-W. Liu, and X. Ma, “Approximate Quantum Error Correction with 1D Log-Depth Circuits”, (2025) arXiv:2503.17759
- [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
Page edit log
- Victor V. Albert (2026-02-23) — most recent
Cite as:
“1D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/1d_stabilizer