## Description

Block quantum code constructed using a tensor-network-based graphical framework from atomic tensors a.k.a. quantum Lego blocks [3], which can be encoding isometries for smaller quantum codes. The class of codes constructed using the framework depends on the choice of atomic Lego blocks.

The individual Lego blocks and resulting quantum Lego codes can be stabilizer [2,4] or non-stabilizer [3,5]. They need not be isometries, meaning that this class of codes generalizes planar-perfect tensor-network codes. However, both the logical and physical degrees of freedom must have the same local dimension.

For example, any stabilizer code can be built out of atomic blocks like the 2-site repetition code, single-site trivial stabilizer codes, and tensor products of the \(|0\rangle\) state. Specifically, the HaPPY holographic code is a quantum Lego code whose atomic Lego block is the five-qubit perfect qubit code.

Many known codes can be created using this code's methods in order to further their understanding, including a 6 qubit implementaion of the generalized Bacon-Shor code, the Toric code, and the \([[7,1,3]]\) Steane code. For example, a simple \( [[4,2,2]] \) stabilizer code can be written as a rank 6 tensor. Attaching two of these via gluing together one logical leg from each can produce a \([[6,4,2]]\) stabilizer code. Code optimization in this framework can be done using reinforcement learning [6,7].

To construct a Lego code, the encoding map \(V\) for each code that is to be used in the construction is converted to a tensor by decomposing it using the formula \begin{align} V = \sum_{i_j} V_{i_1 \ldots i_{n+k}} | i_{k+1} \ldots i_{k+n} \rangle \langle i_1 \ldots i_k |~. \tag*{(1)}\end{align} We then look at the codes graphically, treating each \(i_j\) as an edge dangling out of the tensor vertex \(V_{i_1 \ldots i_{n+k}}\). These edges are either connected to another tensor vertex's edges or left dangling. If the block codes are stabilizer, then each local tensor has unitary product stabilizers (UPS). The goal is to push each UPS through the tensor network until each dangling edge has only trivial support. Otherwise, a matching value is pushed through the edge and the process is repeated on the next tensor. If a UPS can be pushed through the whole network, then a UPS for the larger network has been found. The dangling legs (edges) and UPS of the whole network can then be converted to physical/logical elements and stabilizers/logical operators for a new quantum code.

## Protection

## Encoding

## Transversal Gates

## Decoding

## Notes

## Parent

## Children

- Holographic tensor-network code — Quantum Lego codes whose encoders are tensor networks discretizing hyperbolic space can be thought of as holographic codes. More generally, holograhpic tensor-network codes are types of quantum LEGO codes made from stabilizer codes where logical and physical legs are pre-assigned and logical legs are not contracted. In other words, logical legs resulting from the conversion of codes to tensors must remain logical in the final tensor network, and the same for physical. Contracting logical legs is another word for gluing two logical legs together.
- Planar-perfect-tensor code
- Concatenated quantum code — Encoders for a concatenated codes correspond to tree tensor networks.
- Quantum convolutional code — Quantum convolutional encoding circuits can be viewed as matrix-product-state tensor networks [1].
- Quantum parity code (QPC) — Encoders for a recursively concatenated QPCs are related to quantum trees [12,13] and tree tensor networks [1].
- Branching MERA code — Encoders for branching MERA codes are related to branching MERA tensor networks [1,14].
- Modular-qudit stabilizer code — Modular-qudit stabilizer codes are quantum Lego codes built out of atomic blocks such as the 2-qudit repetition code, single-qudit trivial stabilizer codes, and tensor-products of the \(|0\rangle\) state [4].

## Cousins

- \([[7,1,3]]\) Steane code — The Steane code can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [3].
- \([[6,2,2]]\) \(C_6\) code — The \([[6,4,2]]\) error-detecting code can be constructed out of two \([[4,2,2]]\) codes in the quantum Lego code framework.
- \([[4,2,2]]\) Four-qubit code — The Steane and \([[6,4,2]]\) error-detecting codes can be built from two \([[4,2,2]]\) codes in the quantum Lego code framework [3].
- Rotated surface code — A tensor-network based modification of the rotated surface code improves performance against depolarizing noise by \(\approx 2\%\) [4].
- Honeycomb (6.6.6) color code — Larger triangular color codes can be constructed by contracting legs of tensors of smaller codes [4; Fig. 5].
- Asymmetric quantum code — Quantum Lego and more general tensor-network code optimization for biased noise can be done using reinforcement learning [6,7].
- Tensor-network code — Quantum Lego and more general tensor-network code optimization for biased noise can be done using reinforcement learning [6,7].
- Tensor-network code — Quantum Lego and more general tensor-network code optimization for biased noise can be done using reinforcement learning [6,7].
- Quantum polar code — Quantum polar encoding circuits can be viewed as branching-tree tensor networks [1].
- XP stabilizer code — XP stabilizer codes can be understood through the Quantum Lego formalism [10].
- \([[15,1,3]]\) quantum Reed-Muller code — The quantum Lego framework yields an \([[8,1,2]]\) stabilizer code admits a transversal logical \(T\) gate that originates from that of a trivial (distance-one) \([[7,1]]\) code. This code, in turn, is obtained from the \([[15,1,3]]\) code [10].

## References

- [1]
- A. J. Ferris and D. Poulin, “Tensor Networks and Quantum Error Correction”, Physical Review Letters 113, (2014) arXiv:1312.4578 DOI
- [2]
- T. Farrelly et al., “Tensor-Network Codes”, Physical Review Letters 127, (2021) arXiv:2009.10329 DOI
- [3]
- C. Cao and B. Lackey, “Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks”, PRX Quantum 3, (2022) arXiv:2109.08158 DOI
- [4]
- T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
- [5]
- C. Cao et al., “Quantum Lego Expansion Pack: Enumerators from Tensor Networks”, (2024) arXiv:2308.05152
- [6]
- V. P. Su et al., “Discovery of Optimal Quantum Error Correcting Codes via Reinforcement Learning”, (2023) arXiv:2305.06378
- [7]
- C. Mauron, T. Farrelly, and T. M. Stace, “Optimization of Tensor Network Codes with Reinforcement Learning”, (2023) arXiv:2305.11470
- [8]
- J. I. de Vicente and M. Huber, “Multipartite entanglement detection from correlation tensors”, Physical Review A 84, (2011) arXiv:1106.5756 DOI
- [9]
- W. Laskowski et al., “Correlation-tensor criteria for genuine multiqubit entanglement”, Physical Review A 84, (2011) arXiv:1110.4108 DOI
- [10]
- R. Shen, Y. Wang, and C. Cao, “Quantum Lego and XP Stabilizer Codes”, (2023) arXiv:2310.19538
- [11]
- T. Farrelly et al., “Parallel decoding of multiple logical qubits in tensor-network codes”, (2020) arXiv:2012.07317
- [12]
- B. Ferté and X. Cao, “Solvable Model of Quantum-Darwinism-Encoding Transitions”, Physical Review Letters 132, (2024) arXiv:2305.03694 DOI
- [13]
- S. A. Yadavalli and I. Marvian, “Noisy Quantum Trees: Infinite Protection Without Correction”, (2024) arXiv:2306.14294
- [14]
- A. J. Ferris and D. Poulin, “Branching MERA codes: a natural extension of polar codes”, (2013) arXiv:1312.4575

## Page edit log

- Victor V. Albert (2022-05-25) — most recent
- Thomas Wrona (2022-05-18)

## Cite as:

“Tensor-network code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_lego