Quantum Reed-Solomon code[1] 

Description

Also called prime-qudit polynomial code (QPyC). Prime-qudit CSS code constructed using two Reed-Solomon codes.

The original construction [1] was for a qubit code (\(p=2\)) by using a basis for a larger Galois field over \(GF(2)\), yielding an \([[kN,k(N-2K),K+1]]\) qubit code from a \([N,K,\delta]_{GF(2^k)}\) RS code with \(N=2^k-1\) and \(K=N-\delta+1\).

An alternative construction [2] yields an \([[n,k,d]]_{p>n}\) prime-qudit CSS code with \(d=\min(n-g,g+2-k)\) that is constructed using two Reed-Solomon codes over \(GF(p)=\mathbb{Z}_p\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(\mathbb{Z}_p\), and let \(g\) be a number satisfying \(0\leq k \leq g < n\). Then, define degree-\(g\) polynomials \begin{align} f_{\mu\cup c}\left(x\right)=\mu_{0}+\mu_{1}x+\cdots+\mu_{k-1}x^{k-1}+c_{k}x^{k}+\cdots+c_{g}x^{g}\,, \tag*{(1)}\end{align} where the first \(k\) coefficients are indexed by the coefficient vector \(\mu\in\mathbb{Z}_p^{ k}\), and the remaining coefficients are indexed by the vector \(c\in\mathbb{Z}_p^{ (g+1-k)}\). Logical states, labeled by \(\mu\), are superpositions of canonical basis states whose \(i\)th bit is \(f_{\mu\cup c}\), evaluated at \(\alpha_i\) and summed over all possible vectors \(c\), \begin{align} |\overline{\mu}\rangle=\sum_{c\in\mathbb{Z}_{p}^{(g+1-k)}}|f_{\mu\cup c}(\alpha_{1}),|f_{\mu\cup c}(\alpha_{2}),\cdots,|f_{\mu\cup c}(\alpha_{n})\rangle. \tag*{(2)}\end{align}

Magic

Triorthogonal quantum RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero [3].

Parents

Child

  • Three-qutrit 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 \(m=2\) [4].

Cousin

  • Triorthogonal code — Triorthogonality can be generalized to qudit codes. Quantum RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero [3].

References

[1]
M. Grassl, W. Geiselmann, and T. Beth, “Quantum Reed—Solomon Codes”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes 231 (1999) arXiv:quant-ph/9910059 DOI
[2]
D. Gottesman. Surviving as a quantum computer in a classical world
[3]
A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
[4]
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
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Zoo Code ID: polynomial

Cite as:
“Quantum Reed-Solomon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polynomial
BibTeX:
@incollection{eczoo_polynomial, title={Quantum Reed-Solomon code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/polynomial} }
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Cite as:

“Quantum Reed-Solomon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/polynomial

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