SCIENCE CHINA Information Sciences, Volume 60, Issue 4: 042501(2017) https://doi.org/10.1007/s11432-015-0932-y

## Non-binary entanglement-assisted quantum stabilizer codes

• AcceptedApr 29, 2016
• PublishedSep 13, 2016
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### Abstract

In this paper, we present the $p^m$-ary entanglement-assisted (EA) stabilizer formalism, where $p$ is a prime and $m$ is a positive integer. Given an arbitrary non-abelian stabilizer", the problem of code construction and encoding is settled perfectly in the case of $m=1$. The optimal number of required maximally entangled pairs is discussed and an algorithm to determine the encoding and decoding circuits is proposed. We also generalize several bounds on $p$-ary EA stabilizer codes, such as the BCH bound, the G-V bound and the linear programming bound. However, the issue becomes tricky when it comes to $m>1$, in which case, the former construction method applies only when the non-commuting stabilizer" satisfies a sophisticated limitation.

### References

[1] Calderbank A R, Shor P W. Good quantum error-correcting codes exist. Phys Rev A, 1995, 54: 1098-1105 Google Scholar

[2] Gottesman D. Stabilizer codes and quantum error correction. Dissertation for Ph.D. Degree. Psadena: California Institute of Technology, 1997. 17--35. Google Scholar

[3] Steane A M. Error correcting codes in quantum theory. Phys Rev Lett, 1997, 77: 793-797 Google Scholar

[4] Gottesman D. An introduction to quantum error correction. In: Lomonaco S J, ed. Quantum Computation: a Grand Mathematical Challenge for the Twenty-First Century and the Millennium. Providence: American Mathematical Society, 2002. 221--235. Google Scholar

[5] Nielsen M A, Chuang I L. Quantum computation and quantum information. Am J Phys, 2002, 70: 558-559 Google Scholar

[6] Young K C, Sarovar M, Blume-Kohout R, et al. Error suppression and error correction in adiabatic quantum computation: techniques and challenges. Phys Rev X, 2013, 3: 5326-5333 Google Scholar

[7] Lidar D, Brun T. Quantum Error Correction. Cambridge: Cambridge University Press, 2013. 181--199. Google Scholar

[8] Bowen G. Entanglement required in achieving entanglement-assisted channel capacities. Phys Rev A, 2002, 66: 357-364 Google Scholar

[9] Brun T, Devetak I, Hsieh M H, et al. Catalytic quantum error correction. IEEE Trans Inf Theory, 2006, 60: 3073-3089 Google Scholar

[10] Brun T, Devetak I, Hsieh M H, et al. Correcting quantum errors with entanglement. Science, 2006, 314: 436-439 CrossRef Google Scholar

[11] Wilde M M. Quantum coding with entanglement. Dissertation for Ph.D. Degree. Los Angeles: University of Southern California, 2008. 21--40. Google Scholar

[12] Lai C Y, Brun T. Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys Rev A, 2013, 88: 2343-2347 Google Scholar

[13] Lai C Y, Brun T A, Wilde M M, et al. Duality in entanglement-assisted quantum error correction. IEEE Trans Inf Theory, 2013, 59: 4020-4024 CrossRef Google Scholar

[14] Lai C Y, Brun T A, Wilde M M, et al. Dualities and identities for entanglement-assisted quantum codes. Quantum Inf Process, 2014, 13: 957-990 CrossRef Google Scholar

[15] Bennett C H, Brassard G, Crepeau C, et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys Rev Lett, 1993, 70: 1895-1899 CrossRef Google Scholar

[16] Bennett C H, Wiesner S J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys Rev Lett, 1992, 69: 2881-2884 CrossRef Google Scholar

[17] Blume-Kohout R, Caves C M, Deutsch I H, et al. Climbing mount scalable: physical-resource requirements for a scalable quantum computer. Found Phys, 2002, 32: 1641-1670 CrossRef Google Scholar

[18] Soderberg K A B, Monroe C. Phonon-mediated entanglement for trapped ion quantum computing. Rep Prog Phys, 2010, 73: 569-580 Google Scholar

[19] Bennett C H, Brassard G, Popescu S, et al. Purification of noisy entanglement and faithful teleportation via noisy channels. Phy Rev L, 1996, 76: 722-725 CrossRef Google Scholar

[20] Bennett C H, Divincenzo D P, Smolin J A, et al. Mixed-state entanglement and quantum error correction. Phy Rev A, 1996, 54: 3824-3851 CrossRef Google Scholar

[21] Gottesman D. Fault-tolerant quantum computation with higher-dimensional systems. Chaos Soliton Fract, 1998, 10: 302-313 Google Scholar

[22] Rains E M. Nonbinary quantum codes. IEEE Trans Inf Theory, 1999, 45: 1827-1832 CrossRef Google Scholar

[23] Ashikhmin A, Knill E. Nonbinary quantum stabilizer codes. IEEE Trans Inf Theory, 2001, 47: 3065-3072 CrossRef Google Scholar

[24] Grassl M, Roetteler M, Beth T, et al. Efficient quantum circuits for non-qubit quantum error-correcting codes. Int J Found Comput S, 2003, 14: 757-775 CrossRef Google Scholar

[25] Grassl M, Beth T, Rotteler M, et al. On optimal quantum codes. Int J Quantum Inf, 2004, 2: 757-775 Google Scholar

[26] Ketkar A, Klappenecker A, Kumar S, et al. Nonbinary stabilizer codes over finite fields. IEEE Trans Inf Theory, 2006, 52: 4892-4914 CrossRef Google Scholar

[27] Kim J, Walker J. Nonbinary quantum error-correcting codes from algebraic curves. Discrete Math, 2008, 308: 3115-3124 CrossRef Google Scholar

[28] Feng K Q, Chen H. Quantum Error-Correcting Codes. Beijing: Science Press, 2010. 103--106. Google Scholar

[29] Smith A, Anderson B E, Sosa-Martinez H, et al. Quantum control in the Cs 6S(1/2) ground manifold using radio-frequency and microwave magnetic fields. Phys Rev Lett, 2013, 111: 170502-3124 CrossRef Google Scholar

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