# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021072
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## An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

 1 Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark 2 IMUVA-Mathematics Research Institute, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain 3 Department of Electrical Engineering, Federal University of Campina Grande, Rua Aprígio Veloso 882, 58190-970, Campina Grande, Paraíba, Brazil 4 School of Science and Technology, University of Camerino, I-62032 Camerino, Italy 5 INFN, Sezione di Perugia, Via A. Pascoli, 06123 Perugia, Italy

*Corresponding author: René B. Christensen

Received  May 2021 Revised  November 2021 Early access January 2022

Fund Project: This work was supported in part by Grant PGC2018-096446-B-C21 funded by MCIN/AEI/10.13039/501100011033 and by \ERDF A way of making Europe", by Grant RYC- 2016-20208 funded by MCIN/AEI/10.13039/501100011033 and by \ESF Investing in your future", and by the European Union's Horizon 2020 research and innovation programme, under grant agreement QUARTET No 862644

We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $c$, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $c$ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.

Citation: René B. Christensen, Carlos Munuera, Francisco R. F. Pereira, Diego Ruano. An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve. Advances in Mathematics of Communications, doi: 10.3934/amc.2021072
##### References:
 [1] A. Allahmadi, A. Alkenani, R. Hijazi, N. Muthana, F. Özbudak and P. Solé, New constructions of entanglement-assisted quantum codes, Cryptography and Communications. [2] A. Ashikhmin, S. Litsyn and M. Tsfasman, Asymptotically good quantum codes, Physical Review A, 63 (2001), 032311.  doi: 10.1103/PhysRevA.63.032311. [3] D. Bartoli, M. Montanucci and G. Zini, On certain self-orthogonal AG codes with applications to quantum error- correcting codes, Des. Codes Cryptogr., 89 (2021), 1221-1239.  doi: 10.1007/s10623-021-00870-y. [4] T. Brun, I. Devetak and M.-H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563. [5] A. Calderbank, E. Rains, P. Shor and N. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315. [6] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly, 54 (1947), 589-592.  doi: 10.2307/2304500. [7] C. Galindo, F. Hernando, R. Matsumoto and D. Ruano, Entanglement-assisted quantum error-correcting codes over arbitrary finite fields, Quantum Inf. Process., 18 (2019), Paper No. 116, 18 pp. doi: 10.1007/s11128-019-2234-5. [8] D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A, 54 (1996), 1862-1868.  doi: 10.1103/PhysRevA.54.1862. [9] G. G. L. Guardia, Quantum Error Correction, Quantum Science and Technology. Springer, Cham, 2020. doi: 10.1007/978-3-030-48551-1. [10] A. Guo, S. Kopparty and M. Sudan, New affine-invariant codes from lifting, Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science, ACM Press, (2013), 529–540. [11] F. Hernando, G. McGuire, F. Monserrat and J. J. Moyano-Fernández, Quantum codes from a new construction of self-orthogonal algebraic geometry codes, Quantum Inf. Process., 19 (2020), Paper No. 117, 25 pp. doi: 10.1007/s11128-020-2616-8. [12] T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes, Handbook of Coding Theory, North-Holland, Amsterdam, (1998), 871–961. [13] A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612. [14] J.-L. Kim and G. L. Matthews, Quantum error-correcting codes from algebraic curves, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, 5 (2008), 419–444. doi: 10.1142/9789812794017_0012. [15] R. Lidl and H. Niederreiter, Finite Fields, 2$^{nd}$ edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. [16] E. Lucas, Théorie des Nombres, Gauthier-Villars et fils, Paris, 1891. [17] R. Matsumoto, Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inform. Theory, 48 (2002), 2122-2124.  doi: 10.1109/TIT.2002.1013156. [18] C. Munuera, W. Tenório and F. Torres, Quantum error-correcting codes from algebraic geometry codes of Castle type, Quantum Inf. Process., 15 (2016), 4071-4088.  doi: 10.1007/s11128-016-1378-9. [19] F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Application of complementary dual AG codes to entanglement-assisted quantum codes, 2019 IEEE International Symposium on Information Theory (ISIT), (2019), 2559–2563. [20] F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Entanglement-assisted quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 67 (2021), 7110-7120.  doi: 10.1109/TIT.2021.3113367. [21] P. K. Sarvepalli and A. Klappenecker, Nonbinary quantum codes from Hermitian curves, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer Berlin Heidelberg, 3857 (2006), 136–143. doi: 10.1007/11617983_13. [22] H. Stichtenoth, Algebraic Function Fields and Codes, 2$^{nd}$ edition, Graduate Texts in Mathematics, Springer, 2009. [23] H. Tiersma, Remarks on codes from Hermitian curves, IEEE Trans. Inform. Theory, 33 (1987), 605-609.  doi: 10.1109/TIT.1987.1057327. [24] K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, Coding Theory and Algebraic Geometry Lecture Notes in Mathematics, 1518 (1992), 99-107.  doi: 10.1007/BFb0087995.

show all references

##### References:
 [1] A. Allahmadi, A. Alkenani, R. Hijazi, N. Muthana, F. Özbudak and P. Solé, New constructions of entanglement-assisted quantum codes, Cryptography and Communications. [2] A. Ashikhmin, S. Litsyn and M. Tsfasman, Asymptotically good quantum codes, Physical Review A, 63 (2001), 032311.  doi: 10.1103/PhysRevA.63.032311. [3] D. Bartoli, M. Montanucci and G. Zini, On certain self-orthogonal AG codes with applications to quantum error- correcting codes, Des. Codes Cryptogr., 89 (2021), 1221-1239.  doi: 10.1007/s10623-021-00870-y. [4] T. Brun, I. Devetak and M.-H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563. [5] A. Calderbank, E. Rains, P. Shor and N. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315. [6] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly, 54 (1947), 589-592.  doi: 10.2307/2304500. [7] C. Galindo, F. Hernando, R. Matsumoto and D. Ruano, Entanglement-assisted quantum error-correcting codes over arbitrary finite fields, Quantum Inf. Process., 18 (2019), Paper No. 116, 18 pp. doi: 10.1007/s11128-019-2234-5. [8] D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A, 54 (1996), 1862-1868.  doi: 10.1103/PhysRevA.54.1862. [9] G. G. L. Guardia, Quantum Error Correction, Quantum Science and Technology. Springer, Cham, 2020. doi: 10.1007/978-3-030-48551-1. [10] A. Guo, S. Kopparty and M. Sudan, New affine-invariant codes from lifting, Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science, ACM Press, (2013), 529–540. [11] F. Hernando, G. McGuire, F. Monserrat and J. J. Moyano-Fernández, Quantum codes from a new construction of self-orthogonal algebraic geometry codes, Quantum Inf. Process., 19 (2020), Paper No. 117, 25 pp. doi: 10.1007/s11128-020-2616-8. [12] T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes, Handbook of Coding Theory, North-Holland, Amsterdam, (1998), 871–961. [13] A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612. [14] J.-L. Kim and G. L. Matthews, Quantum error-correcting codes from algebraic curves, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, 5 (2008), 419–444. doi: 10.1142/9789812794017_0012. [15] R. Lidl and H. Niederreiter, Finite Fields, 2$^{nd}$ edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. [16] E. Lucas, Théorie des Nombres, Gauthier-Villars et fils, Paris, 1891. [17] R. Matsumoto, Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inform. Theory, 48 (2002), 2122-2124.  doi: 10.1109/TIT.2002.1013156. [18] C. Munuera, W. Tenório and F. Torres, Quantum error-correcting codes from algebraic geometry codes of Castle type, Quantum Inf. Process., 15 (2016), 4071-4088.  doi: 10.1007/s11128-016-1378-9. [19] F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Application of complementary dual AG codes to entanglement-assisted quantum codes, 2019 IEEE International Symposium on Information Theory (ISIT), (2019), 2559–2563. [20] F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Entanglement-assisted quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 67 (2021), 7110-7120.  doi: 10.1109/TIT.2021.3113367. [21] P. K. Sarvepalli and A. Klappenecker, Nonbinary quantum codes from Hermitian curves, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer Berlin Heidelberg, 3857 (2006), 136–143. doi: 10.1007/11617983_13. [22] H. Stichtenoth, Algebraic Function Fields and Codes, 2$^{nd}$ edition, Graduate Texts in Mathematics, Springer, 2009. [23] H. Tiersma, Remarks on codes from Hermitian curves, IEEE Trans. Inform. Theory, 33 (1987), 605-609.  doi: 10.1109/TIT.1987.1057327. [24] K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, Coding Theory and Algebraic Geometry Lecture Notes in Mathematics, 1518 (1992), 99-107.  doi: 10.1007/BFb0087995.
Normalized reductions for $q = 3$ and $m = 22$
 $i$ $f_i$ $\mathfrak{r}'(f_i^q)$ $\mathfrak{r}(f_i^q)=\mathfrak{n}(\mathfrak{r}(f_i^q))$ $1$ $1$ $1$ $1$ $2$ $x$ $x^3$ $x^3$ $3$ $y$ $x^4$ $+$ $2y$ $x^4$ $+$ $2y$ $4$ $x^2$ $x^6$ $x^6$ $5$ $xy$ $x^7$ $+$ $2x^3y$ $x^7$ $+$ $2x^3y$ $6$ $y^2$ $x^8$ $+$ $x^4y$ $+$ $y^2$ $x^8$ $+$ $x^4y$ $+$ $y^2$ $7$ $x^3$ $x$ $x$ $8$ $x^2y$ $2x^6y$ $+$ $x^2$ $x^6y$ $+$ $2x^2$ $9$ $xy^2$ $x^7y$ $+$ $x^3y^2$ $+$ $x^3$ $x^7y$ $+$ $x^3y^2$ $+$ $x^3$ $10$ $x^4$ $x^4$ $x^4$ $11$ $x^3y$ $x^5$ $+$ $2xy$ $x^5$ $+$ $2xy$ $12$ $x^2y^2$ $x^6y^2$ $+$ $x^6$ $+$ $x^2y$ $x^6y^2$ $+$ $x^6$ $+$ $x^2y$ $13$ $x^5$ $x^7$ $x^7$ $14$ $x^4y$ $x^8$ $+$ $2x^4y$ $x^8$ $+$ $2x^4y$ $15$ $x^3y^2$ $x^5y$ $+$ $xy^2$ $+$ $x$ $x^5y$ $+$ $xy^2$ $+$ $x$ $16$ $x^6$ $x^2$ $x^2$ $17$ $x^5y$ $2x^7y$ $+$ $x^3$ $x^7y$ $+$ $2x^3$ $18$ $x^4y^2$ $x^8y$ $+$ $x^4y^2$ $+$ $x^4$ $x^8y$ $+$ $x^4y^2$ $+$ $x^4$ $19$ $x^7$ $x^5$ $x^5$ $20$ $x^6y$ $x^6$ $+$ $2x^2y$ $x^6$ $+$ $2x^2y$
 $i$ $f_i$ $\mathfrak{r}'(f_i^q)$ $\mathfrak{r}(f_i^q)=\mathfrak{n}(\mathfrak{r}(f_i^q))$ $1$ $1$ $1$ $1$ $2$ $x$ $x^3$ $x^3$ $3$ $y$ $x^4$ $+$ $2y$ $x^4$ $+$ $2y$ $4$ $x^2$ $x^6$ $x^6$ $5$ $xy$ $x^7$ $+$ $2x^3y$ $x^7$ $+$ $2x^3y$ $6$ $y^2$ $x^8$ $+$ $x^4y$ $+$ $y^2$ $x^8$ $+$ $x^4y$ $+$ $y^2$ $7$ $x^3$ $x$ $x$ $8$ $x^2y$ $2x^6y$ $+$ $x^2$ $x^6y$ $+$ $2x^2$ $9$ $xy^2$ $x^7y$ $+$ $x^3y^2$ $+$ $x^3$ $x^7y$ $+$ $x^3y^2$ $+$ $x^3$ $10$ $x^4$ $x^4$ $x^4$ $11$ $x^3y$ $x^5$ $+$ $2xy$ $x^5$ $+$ $2xy$ $12$ $x^2y^2$ $x^6y^2$ $+$ $x^6$ $+$ $x^2y$ $x^6y^2$ $+$ $x^6$ $+$ $x^2y$ $13$ $x^5$ $x^7$ $x^7$ $14$ $x^4y$ $x^8$ $+$ $2x^4y$ $x^8$ $+$ $2x^4y$ $15$ $x^3y^2$ $x^5y$ $+$ $xy^2$ $+$ $x$ $x^5y$ $+$ $xy^2$ $+$ $x$ $16$ $x^6$ $x^2$ $x^2$ $17$ $x^5y$ $2x^7y$ $+$ $x^3$ $x^7y$ $+$ $2x^3$ $18$ $x^4y^2$ $x^8y$ $+$ $x^4y^2$ $+$ $x^4$ $x^8y$ $+$ $x^4y^2$ $+$ $x^4$ $19$ $x^7$ $x^5$ $x^5$ $20$ $x^6y$ $x^6$ $+$ $2x^2y$ $x^6$ $+$ $2x^2y$
Algorithm 1 for $q = 3$
 $f_i$ $\nu(f_i)$ $\mathfrak{r}(f_i^q)$ $\nu(\mathfrak{r}(f_i^q))$ $\phi_i$ $\nu(\phi_i)$ $1$ 0 $1$ 0 $1$ 0 $x$ 3 $x^3$ 9 $x^3$ 9 $y$ 4 $x^4+2y$ 12 $x^4+2y$ 12 $x^2$ 6 $x^6$ 18 $x^6$ 18 $xy$ 7 $x^7+2x^3y$ 21 $x^7+2x^3y$ 21 $y^2$ 8 $x^8+x^4y+y^2$ 24 $x^8+x^4y+y^2$ 24 $x^3$ 9 $x$ 3 $x$ 3 $x^2y$ 10 $x^6y+2x^2$ 22 $x^6y+2x^2$ 22 $xy^2$ 11 $x^7y+x^3y^2+x^3$ 25 $x^7y+x^3y^2+x^3$ 25 $x^4$ 12 $x^4$ 12 $y$ 4 $x^3y$ 13 $x^5+2xy$ 15 $x^5+2xy$ 15 $x^2y^2$ 14 $x^6y^2+x^6+x^2y$ 26 $x^6y^2+x^6+x^2y$ 26 $x^5$ 15 $x^7$ 21 $x^3y$ 13
 $f_i$ $\nu(f_i)$ $\mathfrak{r}(f_i^q)$ $\nu(\mathfrak{r}(f_i^q))$ $\phi_i$ $\nu(\phi_i)$ $1$ 0 $1$ 0 $1$ 0 $x$ 3 $x^3$ 9 $x^3$ 9 $y$ 4 $x^4+2y$ 12 $x^4+2y$ 12 $x^2$ 6 $x^6$ 18 $x^6$ 18 $xy$ 7 $x^7+2x^3y$ 21 $x^7+2x^3y$ 21 $y^2$ 8 $x^8+x^4y+y^2$ 24 $x^8+x^4y+y^2$ 24 $x^3$ 9 $x$ 3 $x$ 3 $x^2y$ 10 $x^6y+2x^2$ 22 $x^6y+2x^2$ 22 $xy^2$ 11 $x^7y+x^3y^2+x^3$ 25 $x^7y+x^3y^2+x^3$ 25 $x^4$ 12 $x^4$ 12 $y$ 4 $x^3y$ 13 $x^5+2xy$ 15 $x^5+2xy$ 15 $x^2y^2$ 14 $x^6y^2+x^6+x^2y$ 26 $x^6y^2+x^6+x^2y$ 26 $x^5$ 15 $x^7$ 21 $x^3y$ 13
Examples of code's parameters and comparative analysis by means of coding bounds
 Parameters Singleton defect Exceeding GV $[[27, 1, 19; 16]]_3$ 6 ✔ $[[27, 4, 16; 13]]_3$ 6 ✔ $[[27, 13, 7; 4]]_3$ 6 ✔ $[[27, 16, 4; 1]]_3$ 6 ✔ $[[64, 5, 42; 35]]_4$ 12 ✔ $[[64, 16, 30; 22]]_4$ 12 ✔ $[[64, 35, 12; 3]]_4$ 10 ✔ $[[64, 39, 8; 1]]_4$ 12 ✔ $[[125, 1,101; 96]]_5$ 20 ✔ $[[125, 9, 91; 84]]_5$ 20 ✔ $[[125, 36, 56; 41]]_5$ 20 ✔ $[[125, 70, 26; 15]]_5$ 20 ✔ $[[125, 90, 10; 1]]_5$ 18 ✔
 Parameters Singleton defect Exceeding GV $[[27, 1, 19; 16]]_3$ 6 ✔ $[[27, 4, 16; 13]]_3$ 6 ✔ $[[27, 13, 7; 4]]_3$ 6 ✔ $[[27, 16, 4; 1]]_3$ 6 ✔ $[[64, 5, 42; 35]]_4$ 12 ✔ $[[64, 16, 30; 22]]_4$ 12 ✔ $[[64, 35, 12; 3]]_4$ 10 ✔ $[[64, 39, 8; 1]]_4$ 12 ✔ $[[125, 1,101; 96]]_5$ 20 ✔ $[[125, 9, 91; 84]]_5$ 20 ✔ $[[125, 36, 56; 41]]_5$ 20 ✔ $[[125, 70, 26; 15]]_5$ 20 ✔ $[[125, 90, 10; 1]]_5$ 18 ✔
Monomials in the support of $\mathfrak{r}(f_{24}^q)$ for $q = 5$
 $j$ $f_j$ $\nu(f_j)$ $\nu(f_j)\bmod{(q^2-1)}$ $108$ $x^{21}y^2$ $117$ $21$ $84$ $x^{15}y^3$ $93$ $21$ $36$ $x^9$ $45$ $21$ $12$ $x^3y$ $21$ $21$
 $j$ $f_j$ $\nu(f_j)$ $\nu(f_j)\bmod{(q^2-1)}$ $108$ $x^{21}y^2$ $117$ $21$ $84$ $x^{15}y^3$ $93$ $21$ $36$ $x^9$ $45$ $21$ $12$ $x^3y$ $21$ $21$
Results of the modified algorithm for $q = 3$
 $f_i$ $\nu(f_i)$ $\mathfrak{r}(f_i^q)$ $\nu(\mathfrak{r}(f_i^q))$ $\phi_i$ $\nu(\phi_i)$ $1$ 0 0 0 $x$ 3 9 9 $y$ 4 $x^4+2y$ 12 $x^4+2y$ 12 $x^2$ 6 18 18 $xy$ 7 $x^7+2x^3y$ 21 $x^7+2x^3y$ 21 $y^2$ 8 24 24 $x^3$ 9 3 3 $x^2y$ 10 22 22 $xy^2$ 11 25 25 $x^4$ 12 $x^4$ 12 $y$ 4 $x^3y$ 13 15 15 $x^2y^2$ 14 26 26 $x^5$ 15 $x^7$ 21 $x^3y$ 13
 $f_i$ $\nu(f_i)$ $\mathfrak{r}(f_i^q)$ $\nu(\mathfrak{r}(f_i^q))$ $\phi_i$ $\nu(\phi_i)$ $1$ 0 0 0 $x$ 3 9 9 $y$ 4 $x^4+2y$ 12 $x^4+2y$ 12 $x^2$ 6 18 18 $xy$ 7 $x^7+2x^3y$ 21 $x^7+2x^3y$ 21 $y^2$ 8 24 24 $x^3$ 9 3 3 $x^2y$ 10 22 22 $xy^2$ 11 25 25 $x^4$ 12 $x^4$ 12 $y$ 4 $x^3y$ 13 15 15 $x^2y^2$ 14 26 26 $x^5$ 15 $x^7$ 21 $x^3y$ 13
Hermitian EAQECCs exceding the GV bound
 $q$ $n$ $c$ 2 8 0–3 3 27 1–16 4 64 3–45 5 125 4–96 7 343 10–288 8 512 9–441 9 729 14–640 11 1331 38–1200 13 2197 51–2016 16 4096 45–3825
 $q$ $n$ $c$ 2 8 0–3 3 27 1–16 4 64 3–45 5 125 4–96 7 343 10–288 8 512 9–441 9 729 14–640 11 1331 38–1200 13 2197 51–2016 16 4096 45–3825
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