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Quantum Stabilizer Codes From Maximal Curves

Lingfei Jin
2014 IEEE Transactions on Information Theory  
A curve attaining the Hasse-Weil bound is called a maximal curve. Usually classical error-correcting codes obtained from a maximal curve have good parameters.  ...  It turns out that the stabilizer quantum codes produced from such Hermitian self-orthogonal classical codes have good parameters.  ...  to good quantum stabilizer codes.  ... 
doi:10.1109/tit.2013.2287694 fatcat:zys7uyibhjgsjpopdydafph7va

On certain self-orthogonal AG codes with applications to Quantum error-correcting codes [article]

Daniele Bartoli, Maria Montanucci, Giovanni Zini
2019 arXiv   pre-print
In this paper a construction of quantum codes from self-orthogonal algebraic geometry codes is provided.  ...  Examples are given by Castle curves, GK curves, generalized GK curves and the Abdon-Bezerra-Quoos maximal curves. Applications of our method to these curves are provided.  ...  Castle curves and AG codes from them [25] give rise to good quantum error-correcting codes.  ... 
arXiv:1912.08021v1 fatcat:c4ahb2bftza6zmxrvz6fnobp6m

Entanglement-assisted Quantum Codes from Algebraic Geometry Codes [article]

Francisco Revson F. Pereira, Ruud Pellikaan and Giuliano Gadioli La Guardia and Francisco Marcos de Assis
2019 arXiv   pre-print
For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via the traditional stabilizer formalism.  ...  Two of the families created have maximal entanglement and have quantum Singleton defect equal to zero or one.  ...  and dimension derived from quantum stabilizer codes [23] .  ... 
arXiv:1907.06357v2 fatcat:2bbzm6qcn5hyvmb6t7pecpdzfa

Page 6536 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
The recurrence protocol and the quantum privacy amplification protocol are equivalent to the protocols converted from [[2,1]] stabilizer codes.  ...  The distillable entanglement by the class of one-way protocols con- verted from stabilizer codes for a certain class of states is equal to or greater than the achievable rate of stabilizer codes over the  ... 

New families of quantum stabilizer codes from Hermitian self-orthogonal algebraic geometry codes [article]

Lin Sok
2021 arXiv   pre-print
There has been a lot of effort to construct good quantum codes from the classical error correcting codes.  ...  New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves.  ...  Hermitian self-orthogonal codes from maximal curves Curves having many rational points produce long codes.  ... 
arXiv:2110.00769v2 fatcat:7zqkaabpkjfivcx4ugl7c5yqpy

Channel Identification and its Impact on Quantum LDPC Code Performance [article]

Yixuan Xie, Jun Li, Robert Malaney, Jinhong Yuan
2012 arXiv   pre-print
In this work we probe the impact of channel estimation on the performance of quantum LDPC codes.  ...  Using state-of-the art quantum LDPC codes designed for the quantum depolarization channel, and utilizing various quantum probes with different entanglement properties, we show how the performance of such  ...  To summarize, the property of stabilizer codes can be directly inferred from classical codes.  ... 
arXiv:1202.0357v1 fatcat:beltd3mgynbs7k4k24rcipara4

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Zunaira Babar, Panagiotis Botsinis, Dimitrios Alanis, Soon Xin Ng, Lajos Hanzo
2015 IEEE Access  
Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence.  ...  These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code.  ...  quantum stabilizer code.  ... 
doi:10.1109/access.2015.2405533 fatcat:ejulfq3rdrcutdznccfstibeam

Decoherence Mitigation by Embedding a Logical Qubit in a Qudit [article]

Hideyuki Miyahara, Yiyou Chen, Vwani Roychowdhury, Louis-S. Bouchard
2022 arXiv   pre-print
Quantum information stored in a qubit is rapidly lost to the environment. The realization of robust qubits is one of the most important challenges in quantum computing.  ...  The retention of quantum information further improves with separation between the sublevels of the logical qubit.  ...  and stabilizer group Consider a 3-qudit repetition code and partial QEC for bit-flipping errors.  ... 
arXiv:2205.00418v1 fatcat:sofulworhjhedirwyychgjl4v4

Quantum Coding Bounds and a Closed-Form Approximation of the Minimum Distance Versus Quantum Coding Rate

Daryus Chandra, Zunaira Babar, Hung Viet Nguyen, Dimitrios Alanis, Panagiotis Botsinis, Soon Xin Ng, Lajos Hanzo
2017 IEEE Access  
Index Terms-Quantum error correction codes, quantum stabilizer codes, quantum coding bound The authors are with the  ...  Ultimately, our contributions can be used for the characterization of quantum stabilizer codes.  ...  maximally-entangled quantum stabilizer codes in conjunction with arbitrary entanglement ratios and relate them to those of unassisted quantum stabilizer codes.  ... 
doi:10.1109/access.2017.2716367 fatcat:crcyyf2eananviy7t52cvpx7iu

Fractal Properties of Magic State Distillation [article]

Patrick Rall
2017 arXiv   pre-print
Quantum error correction codes allow a quantum computer to protect data from decoherence.  ...  One-parameter distillation curves of the Steane code for several iterations.  ... 
arXiv:1708.09256v1 fatcat:7tvllr4zubeajdpkujt457hrem

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves [article]

Maria Montanucci, Marco Timpanella, Giovanni Zini
2017 arXiv   pre-print
We investigate several types of linear codes constructed from two families S̃_q and R̃_q of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves  ...  Plane models for such curves are provided, and the Weierstrass semigroup H(P) at an F_q-rational point P is shown to be symmetric.  ...  More recently, AG Codes from the GK curve have been constructed in [2, 4, 11] .  ... 
arXiv:1709.05979v1 fatcat:o2w7ewsf25hxrpcuwtk5bcbmt4

Quantum error-correcting codes from algebraic geometry codes of Castle type

Carlos Munuera, Wanderson Tenório, Fernando Torres
2016 Quantum Information Processing  
We study Algebraic Geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes.  ...  We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.  ...  Codes from maximal curves. To end this section, let us see some quantum codes arising from maximal curves as the ones considered in Section 3.6. Example 4.8.  ... 
doi:10.1007/s11128-016-1378-9 fatcat:jiaoexkbbnbjzcj7shdo67wp4a

Effects of decoherence and imperfections for quantum algorithms [article]

A.A. Pomeransky, O.V. Zhirov, D.L. Shepelyansky
2004 arXiv   pre-print
We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos  ...  The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix  ...  From top to bottom, curves show data without GYQEC, with time fluctuating couplings, with GYQEC at l g = 1 (practically coincides with the previous curve) and at l g = 5, 10, 20.  ... 
arXiv:quant-ph/0407264v1 fatcat:phifsn56wvfz5pmixs7nkzyvvq

Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction

Y. C. Cheng, R. J. Silbey
2005 Physical Review A. Atomic, Molecular, and Optical Physics  
Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three qubit bit-flip code and five-qubit code are studied.  ...  This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors.  ...  This code is a stabilizer code with two stabilizer operators g 1 = ZZI and g 2 = IZZ. The three qubit bit-flip code corrects single bit-flip error on any of the three data qubits.  ... 
doi:10.1103/physreva.72.012320 fatcat:spqlaror5ncghcmoeqrq6rcvzi

Quantum message authentication codes [article]

Howard Barnum
2001 arXiv   pre-print
The resulting theorem is still relevant for quantum message authentication, but the very strong result claimed in earlier versions of this paper does not hold.  ...  I benefited from discussions with Herbert Bernstein, David DiVincenzo, and Eric Rains about higher-dimensional stabilizer codes.  ...  Conclusion We defined codes for authenticated communication or storage of quantum messages or data, using sets of stabilizer codes derived from an algebraic curve over a Galois field.  ... 
arXiv:quant-ph/0103123v3 fatcat:6fgvrexnozc2fct6axhhs4u7dy
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