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Non-binary entanglement-assisted stabilizer codes

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Abstract

Using pre-shared entangled states between the encoder and the decoder, we provide a previously unreported coding-theoretic framework for constructing entanglement-assisted stabilizer codes over qudits of dimension \(p^k\) from first principles, where p is prime and \(k \in {\mathbb {Z}}^+\). We introduce the concept of mathematically decomposing a qudit of dimension \(p^k\) into k subqudits, each of dimension p. Our contributions toward the entanglement-assisted stabilizer coding framework over qudits are multi-fold as follows: (a) We study the properties of the code and derive an analytical expression for the minimum number of pre-shared entangled subqudits required to construct the code. (b) We provide a code construction procedure that involves obtaining the explicit form of the stabilizers of the code. (c) We show that the proposed entanglement-assisted qudit stabilizer codes are analogous to classical additive codes over \({\mathbb {F}}_{p^k}\). (d) We provide the quantum coding bounds, such as the quantum Hamming bound, the quantum Singleton bound, and the quantum Gilbert–Varshamov bound for non-degenerate entanglement-assisted stabilizer codes over qudits. (e) We finally demonstrate that the error correction capability of the code can be increased with entanglement assistance. The proposed framework is useful for realizing coded quantum computing and communication systems over \(p^k\)-dimensional qudits.

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Notes

  1. We note that stabilizer codes are analogous to classical additive codes. Classical additive codes are more generalized compared to classical linear codes.

  2. We note that the quantum Hamming bound \(\sum \nolimits _{ i=0}^{\lfloor \frac{d-1}{2}\rfloor }\left( {\begin{array}{c}n\\ i\end{array}}\right) 3^i \le 2^{n-m+c_e}\) for an \([[n,m,d';c_e]]\) qubit stabilizer code [30], i.e., \(((n,2^m,d';c_e))\) code, is obtained by substituting \(q=p=2\), \(k=1\), \(n_e'=c_e\), and \(K=2^m\) (as K is the code dimension) in the bound provided in Theorem 9. We note that the rest of the bounds for other special cases can be obtained similarly.

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Acknowledgements

The PhD work of P. J. Nadkarni is funded by a fellowship from the Ministry of Electronics & Information Technology (MeitY), Government of India. S. S. Garani acknowledges the Ministry of Human Resource and Development (MHRD), Government of India, for support.

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  1. Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, India, 560012

    Priya J. Nadkarni & Shayan Srinivasa Garani

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  1. Priya J. Nadkarni
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Correspondence to Shayan Srinivasa Garani.

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Nadkarni, P.J., Garani, S.S. Non-binary entanglement-assisted stabilizer codes. Quantum Inf Process 20, 256 (2021). https://doi.org/10.1007/s11128-021-03174-1

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