# Loop toric code[1]

Also known as Kitaev tesseract code, 4D surface code, All-loop toric code, \((2,2)\) 4D toric code.

## Description

A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((2,2)\) toric code because it admits 2D membrane \(Z\)-type and \(X\)-type logical operators. Both types of operators create 1D (i.e., loop) excitations at their edges. The code serves as a self-correcting quantum memory [1,2].

Loop toric code often either refers to the construction on the four-dimensional torus or is an alternative name for the general construction.

The construction has been extended to modular qudits [3].

## Encoding

Lindbladian-based dissipative encoding, for which codespace is steady-state space of a Lindbladian [4].

## Transversal Gates

Only logical Clifford gates can be implemented transversally when defined on a hypercubic lattice [5].

## Gates

Logical \(CZ\), \(S\), and Hadamard gates when defined on a hypercubic lattice [6].

## Decoding

Measurement-free local error correction circuit (LEC) using reinforcement learning [7].

## Code Capacity Threshold

Independent \(X,Z\) noise: \(2.117\%\) with Hastings decoder [8] and \(7.3\%\) with RG decoder for 4D surface code [9]. It is conjectured via a statistical-mechanical mapping that the optimal ML decoder yields a threshold of \(11.003\%\) [10].

## Threshold

Phenomenological noise model for the 4D loop toric code: \(4.35\%\) with RG decoder [9], and \(4.3\%\) under improved BP-OSD decoder [11].Gate-based depolarizing noise: \(0.31\%\) with RG decoder for 4D loop toric code [9].\(1.59\%\) for independent \(X,Z\) noise and faulty syndrome measurements using the Hastings decoder [8].

## Realizations

Trapped ions: single-shot QEC realized using the \([[33,1,4]]\) loop toric code on the Quantinuum H2 device [12].

## Parents

- Homological code — The 4D loop toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with only loop excitations [3].
- Lattice stabilizer code
- Self-correcting quantum code — The 4D loop toric code is a self-correcting quantum memory [1,2].
- Single-shot code — Single-shot QEC has been realized using the \([[33,1,4]]\) loop toric code on the Quantinuum H2 device [12].
- Abelian topological code — The 4D loop toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with only loop excitations [3].

## Cousin

- Haah cubic code (CC) — The energy of any partial implementation of CC1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.

## References

- [1]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [2]
- R. Alicki et al., “On thermal stability of topological qubit in Kitaev’s 4D model”, (2008) arXiv:0811.0033
- [3]
- X. Chen et al., “Loops in 4+1d topological phases”, SciPost Physics 15, (2023) arXiv:2112.02137 DOI
- [4]
- F. Pastawski, L. Clemente, and J. I. Cirac, “Quantum memories based on engineered dissipation”, Physical Review A 83, (2011) arXiv:1010.2901 DOI
- [5]
- T. Jochym-O’Connor and T. J. Yoder, “Four-dimensional toric code with non-Clifford transversal gates”, Physical Review Research 3, (2021) arXiv:2010.02238 DOI
- [6]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
- [7]
- M. Park et al., “Enhancing Quantum Memory Lifetime with Measurement-Free Local Error Correction and Reinforcement Learning”, (2024) arXiv:2408.09524
- [8]
- N. P. Breuckmann et al., “Local Decoders for the 2D and 4D Toric Code”, (2016) arXiv:1609.00510
- [9]
- K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
- [10]
- K. Takeda and H. Nishimori, “Self-dual random-plaquette gauge model and the quantum toric code”, Nuclear Physics B 686, 377 (2004) arXiv:hep-th/0310279 DOI
- [11]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
- [12]
- N. Berthusen et al., “Experiments with the 4D Surface Code on a QCCD Quantum Computer”, (2024) arXiv:2408.08865

## Page edit log

- Victor V. Albert (2024-04-26) — most recent

## Cite as:

“Loop toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/4d_surface