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A combinatorial approach to quantum error correcting codes

German Luna, Samuel Reid, Bianca De Sanctis, Vlad Gheorghiu
2014 Discrete Mathematics, Algorithms and Applications (DMAA)  
Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric n-vertices colourable graph and a group of operations (colouring rules) on  ...  We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code.  ...  Most known quantum error correcting codes belong to the class of stabilizer codes, which in turn are equivalent [5] to the so-called graph codes.  ... 
doi:10.1142/s1793830914500542 fatcat:ogilimz7djgzflcbln4vxfgv4a

A Parallel Simulator for Quantum Fault Tolerance in the Presence of Correlated Errors

Dan C. Marinescu, Gabriela M. Marinescu
2008 2008 22nd Workshop on Principles of Advanced and Distributed Simulation  
Correlated quantum errors occur with a lower probability than uncorrelated ones; the probability of system crashes when we use either traditional quantum error-correcting codes, or codes specially designed  ...  the [[7, 1, 3]] single-error correcting Hamming code and 22 non-failing error states per code block requires 1 GB of memory while a simulation of the [[23, 1, 5]] double-error correcting Golay code and  ...  Correlated quantum errors occur with a lower probability than uncorrelated ones; the probability of system crashes when we use either traditional quantum error-correcting codes, or codes specially designed  ... 
doi:10.1109/pads.2008.34 dblp:conf/pads/MarinescuM08 fatcat:vlcxvhbi7reuznaxisz5kez2jm

Quantum LDPC Codes from Balanced Incomplete Block Designs

Ivan Djordjevic
2008 IEEE Communications Letters  
We present a series of structured LDPC codes suitable for use in quantum error correction. Those codes belong to the class of dual-containing Calderbank-Shor-Steane (CSS) codes.  ...  The component CSS code is designed using the combinatorial object known as balanced incomplete block design (BIBD) with an even index. The quantum LDPC codes have the rate around 0.9.  ...  Decoherence introduces the errors, and someone has to rely on quantum error-correction.  ... 
doi:10.1109/lcomm.2008.080083 fatcat:35bcv3yiefeh5bxptnygomh65q

A Characterization of Entanglement-Assisted Quantum Low-Density Parity-Check Codes

Yuichiro Fujiwara, Vladimir D. Tonchev
2013 IEEE Transactions on Information Theory  
As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane  ...  While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation.  ...  ACKNOWLEDGMENT The authors would like to thank Mark M. Wilde, the anonymous referee and Associate Editor Jean-Pierre Tillich for their insightful comments and valuable suggestions.  ... 
doi:10.1109/tit.2013.2247461 fatcat:2husqunu6rfzvidtpcmhvpga2q

High-Rate Quantum Low-Density Parity-Check Codes Assisted by Reliable Qubits

Yuichiro Fujiwara, Alexander Gruner, Peter Vandendriessche
2015 IEEE Transactions on Information Theory  
Our approach to designing high-rate quantum error-correcting codes also allows for making direct use of other major syndrome decoding methods for linear codes, making it possible to deal with a situation  ...  A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code.  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous referees and Associate Editor Alexei Ashikhmin for their careful reading of the manuscript and valuable suggestions.  ... 
doi:10.1109/tit.2015.2398436 fatcat:h4kpezzcznh4zpxgkcqi6gqu44

Quantum computation, non-demolition measurements, and reflective control in living systems

Abir U. Igamberdiev
2004 Biosystems (Amsterdam. Print)  
The quantum device possesses its own potential internal quantum state (IQS), which is maintained for a prolonged time via reflective error-correction.  ...  codes into a single reflective set.  ...  program of error-correcting codes.  ... 
doi:10.1016/j.biosystems.2004.04.001 pmid:15527945 fatcat:4wzpon3cdjel7ntaxlrqzpe2ye

Deterministic error model for quantum computer simulation

Eric Chi, Stephen A. Lyon, Margaret Martonosi
2008 Physical Review A. Atomic, Molecular, and Optical Physics  
Quantum computers (QCs) must implement quantum error correcting codes (QECCs) to protect their logical qubits from errors, and modeling the effectiveness of QECCs on QCs is an important problem for evaluating  ...  The previously developed Monte Carlo (MC) error models may take days or weeks of execution to produce an accurate result due to their random sampling approach.  ...  For example, the ͓͓7,1,3͔͔ quantum error correction code ͑QECC͒ encodes one logical qubit into a block of seven physical qubits.  ... 
doi:10.1103/physreva.77.052315 fatcat:iobmkcbl65huzn5qqudq3z3ymq

Page 8049 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
8049 94B Theory of error-correcting codes and error-detecting codes 99-110; MR 2002f:94073] presented a combinatorial construction for such codes and proved that a 7*(2,6,v)-code exists for all positive  ...  ) Yet another approach to the extended ternary Golay code.  ... 

Living Systems are Dynamically Stable by Computing Themselves at the Quantum Level

Abir Igamberdiev
2003 Entropy  
To avoid a collapse of the quantum information in the process of correcting errors, it is possible to make a partial measurement that extracts only the error-information and leaves the encoded state untouched  ...  codes into a single reflective set.  ...  The decoherence-free state is maintained by the error-correcting code from the quantum computer. Error-correcting code is concatenated by the encoding in genome.  ... 
doi:10.3390/e5020076 fatcat:6hdjz755yvevbds36upoi6otnu

Page 9831 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews  
9831 94B Theory of error-correcting codes and error-detecting codes merators of linear codes over F) + uF) with u* = 1, whose images under a Gray map are Type I and Type II binary codes.  ...  This paper analyzes surface codes, which are topological error- correcting codes introduced by Kitaev [in Quantum communication, computing, and measurement, 181-188, Plenum, New York, 1997].  ... 

Page 6537 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
Vazirani, A survey of quantum complexity theory (193-217). Quantum error correcting codes and quantum cryptography: Daniel Gottesman, An introduction to quantum error correction (221-235); Samuel J.  ...  Since there’s a whole two-sphere of values for a qubit state, not just two possible values, there’s going to be a whole new toolkit of error-correction required if reliable quantum computing devices are  ... 

Page 9504 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
The second part of the paper illustrates how a cluster state preparation procedure can aid in quantum error correction.  ...  Summary: “We establish a further connection between one-way communication where a sender conveys information to a receiver who has related information, and error-correction coding where a sender attempts  ... 

Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes

G. Alber, Th. Beth, Ch. Charnes, A. Delgado, M. Grassl, M. Mussinger
2001 Physical Review Letters  
A new class of error-correcting quantum codes is introduced capable of stabilizing qubits against spontaneous decay arising from couplings to statistically independent reservoirs.  ...  These quantum codes are based on the idea of using an embedded quantum code and exploiting the classical information available about which qubit has been affected by the environment.  ...  [15] have presented a one-error correcting embedded quantum code which applies to the important special case of equal decay rates of all the qubits.  ... 
doi:10.1103/physrevlett.86.4402 pmid:11328185 fatcat:rra22sevrfgltjxcj4tyklabny

Fault-tolerant architectures for superconducting qubits

David P DiVincenzo
2009 Physica Scripta  
The basics of quantum error correction codes, which I will briefly review, have not significantly changed since their introduction fifteen years ago.  ...  It is now understood that two dimensional surface codes, close relatives of the original toric code of Kitaev, can be adapted to effectively perform logical gate operations in a very simple planar architecture  ...  It definitely takes a lot more effort to correct one quantum error than it does to correct one classical error! Fig. 2 .  ... 
doi:10.1088/0031-8949/2009/t137/014020 fatcat:qcetjsu6uffqbgw2alu4p5xggu

Solving Multi-Coloring Combinatorial Optimization Problems Using Hybrid Quantum Algorithms [article]

Young-Hyun Oh, Hamed Mohammadbagherpoor, Patrick Dreher, Anand Singh, Xianqing Yu, Andy J. Rindos
2019 arXiv   pre-print
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach.  ...  Therefore, our modeling approach with hybrid quantum algorithms can be applicable for combinatorial problems in various fields to find an optimal solution in polynomial time.  ...  These hybrid approaches require no error correction techniques in order to run quantum circuits on the NISQ devices.  ... 
arXiv:1911.00595v2 fatcat:yfqrqd73pfgwrfllfykbd7rgwu
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