## Description

One-dimensional translationally invariant qubit stabilizer code whose whose stabilizer group can be partitioned into constant-size subsets of constant support and of constant overlap between neighboring sets. Initially formulated as a quantum analogue of convolutional codes, which were designed to protect a continuous and never-ending stream of information. Precise formulations sometimes begin with a finite-dimensional lattice, with the intent to take the thermodynamic limit; logical dimension can be infinite as well.

Quantum convolutional codes, like their classical counterparts, can also be understood in terms of frames. Let each encoding frame take in \(n\) qubits, carry \(m\) qubits of information between frames, and act on them with \(n-k\) Pauli generators. Each generator, countably infinite in length, must commute with each \(n\) register shift of itself, but need not commute with the other generators [3]. The \(m\) qubits of information carried between each frame are also stabilized by additional memory Pauli operators. It is known that the minimal value for \(m\) is given by \(\text{dim}(M)-\frac{1}{2}\text{rank}(M)\), with \(M\) being the matrix containing the required commutation relations of the memory qubits [4–6]. These operators can be efficiently determined [7].

## Encoding

## Decoding

## Notes

## Parents

- Qubit stabilizer code
- Lattice stabilizer code — Quantum convolutional codes are lattice stabilizer codes on an semi-infinite or infinite lattice in one dimension [16]. Some notions may be extendable to non-stabilizer codes [2; 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 [17].
- Tensor-network code — Quantum convolutional encoding circuits can be viewed as matrix-product-state tensor networks [11].

## Children

## Cousins

- Convolutional code
- EA quantum convolutional code
- Hybrid convolutional code
- Quantum data-syndrome (QDS) code — The QDS code framework has been extended to quantum convolutional codes [18].
- Quantum Reed-Muller code — Quantum convolutional codes can be derived from quantum Reed-Muller codes [19].

## References

- [1]
- H. Ollivier and J.-P. Tillich, “Description of a Quantum Convolutional Code”, Physical Review Letters 91, (2003) arXiv:quant-ph/0304189 DOI
- [2]
- H. Ollivier and J.-P. Tillich, “Quantum convolutional codes: fundamentals”, (2004) arXiv:quant-ph/0401134
- [3]
- M. Grassl and M. Rotteler, “Constructions of Quantum Convolutional Codes”, 2007 IEEE International Symposium on Information Theory (2007) arXiv:quant-ph/0703182 DOI
- [4]
- M. Houshmand, S. Hosseini-Khayat, and M. M. Wilde, “Minimal-Memory, Noncatastrophic, Polynomial-Depth Quantum Convolutional Encoders”, IEEE Transactions on Information Theory 59, 1198 (2013) arXiv:1105.0649 DOI
- [5]
- M. M. Wilde, M. Houshmand, and S. Hosseini-Khayat, “Examples of minimal-memory, non-catastrophic quantum convolutional encoders”, 2011 IEEE International Symposium on Information Theory Proceedings (2011) arXiv:1011.5535 DOI
- [6]
- M. M. Wilde and T. A. Brun, “Optimal entanglement formulas for entanglement-assisted quantum coding”, Physical Review A 77, (2008) arXiv:0804.1404 DOI
- [7]
- L. G. Gunderman, “Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers”, Physical Review A 107, (2023) arXiv:2206.13040 DOI
- [8]
- M. Grassl and M. Rotteler, “Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes”, 2006 IEEE International Symposium on Information Theory (2006) arXiv:quant-ph/0602129 DOI
- [9]
- M. Houshmand, S. Hosseini-Khayat, and M. M. Wilde, “Minimal-Memory Requirements for Pearl-Necklace Encoders of Quantum Convolutional Codes”, IEEE Transactions on Computers 61, 299 (2012) arXiv:1004.5179 DOI
- [10]
- M. M. Wilde, “Quantum-shift-register circuits”, Physical Review A 79, (2009) arXiv:0903.3894 DOI
- [11]
- A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014) arXiv:1312.4578 DOI
- [12]
- G. D. Forney, M. Grassl, and S. Guha, “Convolutional and Tail-Biting Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 53, 865 (2007) arXiv:quant-ph/0511016 DOI
- [13]
- M. Grassl and M. Rotteler, “Quantum block and convolutional codes from self-orthogonal product codes”, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. (2005) arXiv:quant-ph/0703181 DOI
- [14]
- M. Wilde, “Quantum convolutional codes”, Quantum Error Correction 231 (2013) DOI
- [15]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [16]
- J. Haah, “Lattice quantum codes and exotic topological phases of matter”, (2013) arXiv:1305.6973
- [17]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [18]
- W. Zeng et al., “Quantum convolutional data-syndrome codes”, 2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (2019) arXiv:1902.07395 DOI
- [19]
- S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes”, (2007) arXiv:quant-ph/0701037

## Page edit log

- Victor V. Albert (2023-01-04) — most recent
- Lane G. Gunderman (2023-01-04)

## Cite as:

“Quantum convolutional code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_convolutional