# Three-qutrit code[1]

## Description

A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. with stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound.

The codewords are \begin{align} \begin{split} | \overline{0} \rangle &= \frac{1}{\sqrt{3}} (| 000 \rangle + | 111 \rangle + | 222 \rangle) \\ | \overline{1} \rangle &= \frac{1}{\sqrt{3}} (| 012 \rangle + | 120 \rangle + | 201 \rangle) \\ | \overline{2} \rangle &= \frac{1}{\sqrt{3}} (| 021 \rangle + | 102 \rangle + | 210 \rangle)~. \end{split} \tag*{(1)}\end{align} The elements in the superposition of each logical codeword are related to each other via cyclic permutations.

## Protection

Detects single qutrit errors and protects against a single-qutrit erasure. It is the smallest single-erasure correcting qudit code for \(q>2\), and there does not exist a three-qubit code with analogous properties.

The code is an example of a \( ((n = 3, k = 2)) \) threshold scheme where a secret (the quantum information) is split into \( n \) shares and can be reconstructed by \( k \) pieces.

They key property of this code is that the reduced density matrix of any single qutrit is maximally mixed, meaning no information can be extracted from that qutrit. Therefore, a single qutrit tells you nothing about the encoded message, but access to any two pairs of qutrits will reveal the secret.

## Encoding

## Decoding

## Notes

## Parents

- Prime-qudit RS code — The three-qutrit code is the smallest member of a family of \([[2m-1,1,m]]_{p}\) prime-qudit quantum RS codes for \(p=3\) and \(m=2\) [1].
- Holographic tensor-network code — The three-qutrit code is a radius-one holographic tensor-network code and serves as a minimal model for holography [2,3].
- Perfect-tensor code — The three-qutrit code stems from the \([[4,0,3]]_3\) AME state [4–6].
- Quantum maximum-distance-separable (MDS) code — The three-qutrit code is the smallest nontrivial quantum MDS code.
- Small-distance block quantum code

## Cousins

- Approximate secret-sharing code — The three-qutrit code defines a minimal secret-sharing scheme [1] that is substantially generalized by approximate secret-sharing codes.
- Three-rotor code

## References

- [1]
- R. Cleve, D. Gottesman, and H.-K. Lo, “How to Share a Quantum Secret”, Physical Review Letters 83, 648 (1999) arXiv:quant-ph/9901025 DOI
- [2]
- A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015) arXiv:1411.7041 DOI
- [3]
- D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction”, Communications in Mathematical Physics 354, 865 (2017) arXiv:1607.03901 DOI
- [4]
- W. Helwig, “Absolutely Maximally Entangled Qudit Graph States”, (2013) arXiv:1306.2879
- [5]
- D. Goyeneche et al., “Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices”, Physical Review A 92, (2015) arXiv:1506.08857 DOI
- [6]
- Z. Raissi, “Modifying Method of Constructing Quantum Codes From Highly Entangled States”, IEEE Access 8, 222439 (2020) arXiv:2005.01426 DOI

## Page edit log

- Victor V. Albert (2022-08-12) — most recent
- Felix Huber (2022-08-12)
- Victor V. Albert (2021-12-29)
- Elizabeth R. Bennewitz (2021-12-03)

## Cite as:

“Three-qutrit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_3_1_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/small/stab_3_1_2.yml.