Three-qutrit code

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

In addition to thinking about the encoding of states, it is also interesting to look at the trasformation of operators from the physical space into the logical space. Due to the unique structure and recovery protocol of the three qutrit code, the representation of a logical operator \( \overline{O} \) is not unique. Instead, \( \overline{O} \) can be constructed from unitary matricies with support on only two out of the three qutrits. Therefore, the logical operator has valid representations constructed from support on different sets of two qutrits. This operator construction is directly analogous to the construction of operators in the bulk (at the center) of the AdS\(_3\)-Rindler reconstruction. The three-qutrit code can then be used to describe how these local bulk operators are protected against localized boundary errors [2].

Decoding

The quantum information (the secret) can be recovered from a unitary transformation acting on only two qutrits, \( U_{ij} \otimes I \), where \(U_{ij}\) acts on qutrits \(i,j\) and \(I\) is the identity on the remaining qutrit. By the cyclic structure of the codewords, this unitary transformation performs a permutation that recovers the information and stores it in one of the two qutrits involved in recovery.

Notes

Connections to Ads/CFT from the perspetive of how arbitrary operators are encoded into the logical space. This encoding is analagous and helps explain why operators acting on the bulk are protected against localized boundary errors [2].

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 Reed-Solomon 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.