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Journal articles on the topic 'Quantum Error Correction'

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Author: Grafiati

Published: 4 June 2021

Last updated: 14 September 2024

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1

khan, Md Khalik, and Dr Sapna jain. "ERROR CORRECTION USING QUANTUM COMPUTING." International Journal of Engineering Applied Sciences and Technology 8, no. 1 (May 1, 2023): 78–85. http://dx.doi.org/10.33564/ijeast.2023.v08i01.014.

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Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweep like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.

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2

Khan, Khalik, and Sapna Jain. "Error Correction Using Quantum Computation." Journal of Digital Science 5, no. 1 (June 25, 2023): 12–22. http://dx.doi.org/10.33847/2686-8296.5.1_2.

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Quantum Error Correction (QEC) is an important technique for protecting quantum information against decoherence and errors. This involves the design and implementation of algorithms and techniques to minimize error rates and increase the stability of quantum circuits. One of the key parameters in QEC is the distance of the error- correcting code, which determines the number of errors that can be corrected. Another important parameter is the error probability, which quantifies the likelihood of errors occurring in the quantum system. In this context, the goal of a simulation sweeps like the one performed in the code is to study the performance of the QEC code for different values of the distance and error probability, and to optimize the code for maximum accuracy. By varying these parameters and observing the performance of the code, researchers can gain insights into how to design better codes and improve the reliability of quantum computing systems. We also discuss the challenges that need to be addressed for quantum computing to realize its potential in solving practical Error-correction problems.

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3

Khalifa, Othman O., Nur Amirah bt Sharif, Rashid A. Saeed, S. Abdel-Khalek, Abdulaziz N. Alharbi, and Ali A. Alkathiri. "Digital System Design for Quantum Error Correction Codes." Contrast Media & Molecular Imaging 2021 (December 15, 2021): 1–8. http://dx.doi.org/10.1155/2021/1101911.

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Quantum computing is a computer development technology that uses quantum mechanics to perform the operations of data and information. It is an advanced technology, yet the quantum channel is used to transmit the quantum information which is sensitive to the environment interaction. Quantum error correction is a hybrid between quantum mechanics and the classical theory of error-correcting codes that are concerned with the fundamental problem of communication, and/or information storage, in the presence of noise. The interruption made by the interaction makes transmission error during the quantum channel qubit. Hence, a quantum error correction code is needed to protect the qubit from errors that can be caused by decoherence and other quantum noise. In this paper, the digital system design of the quantum error correction code is discussed. Three designs used qubit codes, and nine-qubit codes were explained. The systems were designed and configured for encoding and decoding nine-qubit error correction codes. For comparison, a modified circuit is also designed by adding Hadamard gates.

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4

Locher, David F., Lorenzo Cardarelli, and Markus Müller. "Quantum Error Correction with Quantum Autoencoders." Quantum 7 (March 9, 2023): 942. http://dx.doi.org/10.22331/q-2023-03-09-942.

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Active quantum error correction is a central ingredient to achieve robust quantum processors. In this paper we investigate the potential of quantum machine learning for quantum error correction in a quantum memory. Specifically, we demonstrate how quantum neural networks, in the form of quantum autoencoders, can be trained to learn optimal strategies for active detection and correction of errors, including spatially correlated computational errors as well as qubit losses. We highlight that the denoising capabilities of quantum autoencoders are not limited to the protection of specific states but extend to the entire logical codespace. We also show that quantum neural networks can be used to discover new logical encodings that are optimally adapted to the underlying noise. Moreover, we find that, even in the presence of moderate noise in the quantum autoencoders themselves, they may still be successfully used to perform beneficial quantum error correction and thereby extend the lifetime of a logical qubit.

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5

ARIMITSU, T., T. HAYASHI, S. KITAJIMA, and F. SHIBATA. "QUANTUM ERROR-CORRECTION FOR SPATIALLY CORRELATED ERRORS." International Journal of Quantum Information 06, supp01 (July 2008): 575–80. http://dx.doi.org/10.1142/s0219749908003803.

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It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises. A model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of non-equilibrium thermo field dynamics that is a canonical operator formalism for dissipative quantum systems.

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6

Fowler, A. G., and K. Goyal. "Topological cluster state quantum computing." Quantum Information and Computation 9, no. 9&10 (September 2009): 721–38. http://dx.doi.org/10.26421/qic9.9-10-1.

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The quantum computing scheme described by Raussendorf et. al (2007), when viewed as a cluster state computation, features a 3-D cluster state, novel adjustable strength error correction capable of correcting general errors through the correction of Z errors only, a threshold error rate approaching 1% and low overhead arbitrarily long-range logical gates. In this work, we review the scheme in detail framing the discussion solely in terms of the required 3-D cluster state and its stabilizers.

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7

Gushanskiy, Sergey, Maxim Polenov, and Viktor Potapov. "Development of a Scheme for Correcting Arbitrary Errors and Averaging Noise in Quantum Computing." Cybernetics and Information Technologies 22, no. 2 (June 1, 2022): 26–35. http://dx.doi.org/10.2478/cait-2022-0014.

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Abstract Intensive research is currently being carried out to develop and create quantum computers and their software. This work is devoted to study of the influence of the environment on the quantum system of qubits. Quantum error correction is a set of methods for protecting quantum information and quantum state from unwanted interactions of the environment (decoherence) and other forms and types of noise. The article discusses the solution to the problem of research and development of corrective codes for rectifying several types of quantum errors that occur during computational processes in quantum algorithms and models of quantum computing devices. The aim of the work is to study existing methods for correcting various types of quantum errors and to create a corrective code for quantum error rectification. The scientific novelty is expressed in the exclusion of one of the shortcomings of the quantum computing process.

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8

Dauphinais, Guillaume, David W. Kribs, and Michael Vasmer. "Stabilizer Formalism for Operator Algebra Quantum Error Correction." Quantum 8 (February 21, 2024): 1261. http://dx.doi.org/10.22331/q-2024-02-21-1261.

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We introduce a stabilizer formalism for the general quantum error correction framework called operator algebra quantum error correction (OAQEC), which generalizes Gottesman's formulation for traditional quantum error correcting codes (QEC) and Poulin's for operator quantum error correction and subsystem codes (OQEC). The construction generates hybrid classical-quantum stabilizer codes and we formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code, generalizing the fundamental theorems for the QEC and OQEC stabilizer formalisms. We discover hybrid versions of the Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem to derive a result that gives the distance of such codes. We show how some recent hybrid subspace code constructions are captured by the formalism, and we also indicate how it extends to qudits.

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9

LIDAR, D. A., and K. KHODJASTEH. "ROBUST DYNAMICAL DECOUPLING: FEEDBACK-FREE ERROR CORRECTION." International Journal of Quantum Information 03, supp01 (November 2005): 41–52. http://dx.doi.org/10.1142/s0219749905001237.

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Dynamical decoupling is a feed-back free scheme for quantum error correction against noise and decoherence errors. An efficiency analysis of dynamical decoupling is performed. Furthermore we provide the basic concepts of dynamical decoupling and quantum error correcting codes, and give an example of a hybrid protection scheme. Some interesting extensions of dynamical decoupling are discussed at the end.

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10

Sharma, Sangat, Suresh Basnet, and Raju Khanal. "Implementation of Error Correction on IBM Quantum Computing Devices." Journal of Nepal Physical Society 8, no. 1 (December 13, 2022): 7–15. http://dx.doi.org/10.3126/jnphyssoc.v8i1.48278.

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Quantum noise cannot be avoided in the quantum computing devices due to unstable nature of qubits and signals. The error caused by quantum noise can be detected and corrected using different error correcting codes. In this work, we have tested the feasibility and accuracy of three qubit bit flip and phase flip error correcting code in quantum computer provided by International Business Machine Quantum Experience (IBM QX) cloud platform. Among five quantum processors, ibmq_ourense is found to have highest average accuracy 77.9% ± 3.09% on all qubits simultaneously. Three qubits bit flip error correction circuit gave correct output 89.9% ± 1.01% of the time on average. Similarly three qubits phase flip error correction circuit give 88.05% ±1.89%. The measurement error mitigation has improved the accuracy of bit flip and phase flip error correction code by 5.01% and 7.01% respectively on average. The error rate shows that the error in quantum computations are random in nature and can be corrected. IBM QX quantum computer are suitable for only small scale quantum computation and demonstrate purpose. Furthermore, the accuracy of error correction codes can be increased with the use of higher accuracy quantum qubits and quantum gates.

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11

Choukroun, Yoni, and Lior Wolf. "Deep Quantum Error Correction." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 1 (March 24, 2024): 64–72. http://dx.doi.org/10.1609/aaai.v38i1.27756.

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Quantum error correction codes (QECC) are a key component for realizing the potential of quantum computing. QECC, as its classical counterpart (ECC), enables the reduction of error rates, by distributing quantum logical information across redundant physical qubits, such that errors can be detected and corrected. In this work, we efficiently train novel end-to-end deep quantum error decoders. We resolve the quantum measurement collapse by augmenting syndrome decoding to predict an initial estimate of the system noise, which is then refined iteratively through a deep neural network. The logical error rates calculated over finite fields are directly optimized via a differentiable objective, enabling efficient decoding under the constraints imposed by the code. Finally, our architecture is extended to support faulty syndrome measurement, by efficient decoding of repeated syndrome sampling. The proposed method demonstrates the power of neural decoders for QECC by achieving state-of-the-art accuracy, outperforming for small distance topological codes, the existing end-to-end neural and classical decoders, which are often computationally prohibitive.

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12

Kawabata, Shiro. "Quantum Interleaver: Quantum Error Correction for Burst Error." Journal of the Physical Society of Japan 69, no. 11 (November 15, 2000): 3540–43. http://dx.doi.org/10.1143/jpsj.69.3540.

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13

Chen, Zijun, Kevin J. Satzinger, Juan Atalaya, Alexander N. Korotkov, Andrew Dunsworth, Daniel Sank, Chris Quintana, et al. "Exponential suppression of bit or phase errors with cyclic error correction." Nature 595, no. 7867 (July 14, 2021): 383–87. http://dx.doi.org/10.1038/s41586-021-03588-y.

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AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

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14

Kribs, D. W., R. Laflamme, D. Poulin, and M. Lesosky. "Operator quantum error correction." Quantum Information and Computation 6, no. 4&5 (July 2006): 382–99. http://dx.doi.org/10.26421/qic6.4-5-6.

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This paper is an expanded and more detailed version of the work \cite{KLP04} in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques --- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method --- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems''.

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15

Greenbaum, Daniel, and Zachary Dutton. "Modeling coherent errors in quantum error correction." Quantum Science and Technology 3, no. 1 (December 20, 2017): 015007. http://dx.doi.org/10.1088/2058-9565/aa9a06.

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16

Lu, Feng, and Dan C. Marinescu. "Quantum Error Correction of Time-correlated Errors." Quantum Information Processing 6, no. 4 (July 7, 2007): 273–93. http://dx.doi.org/10.1007/s11128-007-0058-1.

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17

Magdalena de la Fuente, Julio Carlos, Nicolas Tarantino, and Jens Eisert. "Non-Pauli topological stabilizer codes from twisted quantum doubles." Quantum 5 (February 17, 2021): 398. http://dx.doi.org/10.22331/q-2021-02-17-398.

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It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.

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18

Fowler, Austin G., David S. Wang, and Lloyd C. L. Hollenberg. "Surface code quantum error correction incorporating accurate error propagation." Quantum Information and Computation 11, no. 1&2 (January 2011): 8–18. http://dx.doi.org/10.26421/qic11.1-2-2.

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The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is obtained by repeatedly measuring each syndrome qubit after appropriate interaction with its four nearest neighbor data qubits. Changes in the measurement value indicate the presence of chains of errors in space and time. The standard method of determining operations likely to return the code to its error-free state is to use the minimum weight matching algorithm to connect pairs of measurement changes with chains of corrections such that the minimum total number of corrections is used. Prior work has not taken into account the propagation of errors in space and time by the two-qubit interactions. We show that taking this into account leads to a quadratic improvement of the logical error rate.

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19

Steinbach, J., and J. Twamley. "Motional quantum error correction." Journal of Modern Optics 47, no. 2-3 (February 2000): 453–85. http://dx.doi.org/10.1080/09500340008244053.

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20

Brun, Todd A., Igor Devetak, and Min-Hsiu Hsieh. "Catalytic Quantum Error Correction." IEEE Transactions on Information Theory 60, no. 6 (June 2014): 3073–89. http://dx.doi.org/10.1109/tit.2014.2313559.

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21

Barnes, Jeff P., and Warren S. Warren. "Automatic Quantum Error Correction." Physical Review Letters 85, no. 4 (July 24, 2000): 856–59. http://dx.doi.org/10.1103/physrevlett.85.856.

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22

Cory, D. G., M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo. "Experimental Quantum Error Correction." Physical Review Letters 81, no. 10 (September 7, 1998): 2152–55. http://dx.doi.org/10.1103/physrevlett.81.2152.

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23

Lloyd, Seth, and Jean-Jacques E. Slotine. "Analog Quantum Error Correction." Physical Review Letters 80, no. 18 (May 4, 1998): 4088–91. http://dx.doi.org/10.1103/physrevlett.80.4088.

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24

Salas, P. J. "Quality of a quantum error correcting scheme and memory error threshold estimation." Quantum Information and Computation 6, no. 6 (September 2006): 516–26. http://dx.doi.org/10.26421/qic6.6-4.

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The error correcting capabilities of the Calderbank-Shor-Steane [[7,1,3]] quantum code, together with a fault-tolerant syndrome extraction by means of several ancilla states, have been numerically studied. A simple probability expression to characterize the code ability for correcting an encoded qubit has been considered. This probability, as a correction quality criterion, permits the error correction capabilities among different recovery schemes to be compared. The memory error threshold is calculated by means of the best method of those considered.

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25

Takeda, Kenta, Akito Noiri, Takashi Nakajima, Takashi Kobayashi, and Seigo Tarucha. "Quantum error correction with silicon spin qubits." Nature 608, no. 7924 (August 24, 2022): 682–86. http://dx.doi.org/10.1038/s41586-022-04986-6.

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AbstractFuture large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation1,2. Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers3–5. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems6–8. However, the demonstration of QEC, which requires three or more coupled qubits1, and involves a three-qubit gate9–11 or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.

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26

El Allati, A., H. Amellal, and A. Meslouhi. "Improvement of quantum correlations by repetitive quantum error correction." International Journal of Quantum Information 17, no. 05 (August 2019): 1950044. http://dx.doi.org/10.1142/s0219749919500448.

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A quantum error-correcting code is established in entangled coherent states (CSs) with Markovian and non-Markovian environments. However, the dynamic behavior of these optical states is discussed in terms of quantum correlation measurements, entanglement and discord. By using the correcting codes, these correlations can be as robust as possible against environmental effects. As the number of redundant CSs increases due to the repetitive error correction, the probabilities of success also increase significantly. Based on different optical field parameters, the discord can withstand more than an entanglement. Furthermore, the behavior of quantum discord under decoherence may exhibit sudden death and sudden birth phenomena as functions of dimensionless parameters.

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27

Terhal, Barbara M. "Quantum error correction for quantum memories." Reviews of Modern Physics 87, no. 2 (April 7, 2015): 307–46. http://dx.doi.org/10.1103/revmodphys.87.307.

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28

Milburn, G. J., M. Sarovar, and C. Ahn. "Quantum control and quantum error correction." Australian Journal of Electrical and Electronics Engineering 2, no. 2 (January 2005): 151–57. http://dx.doi.org/10.1080/1448837x.2005.11464123.

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29

Weaver, Nik. "Quantum Graphs as Quantum Relations." Journal of Geometric Analysis 31, no. 9 (January 13, 2021): 9090–112. http://dx.doi.org/10.1007/s12220-020-00578-w.

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AbstractThe “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

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30

Cafaro, Carlo, and Peter van Loock. "A Simple Comparative Analysis of Exact and Approximate Quantum Error Correction." Open Systems & Information Dynamics 21, no. 03 (August 7, 2014): 1450002. http://dx.doi.org/10.1142/s1230161214500024.

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We present a comparative analysis of exact and approximate quantum error correction by means of simple unabridged analytical computations. For the sake of clarity, using primitive quantum codes, we study the exact and approximate error correction of the two simplest unital (Pauli errors) and nonunital (non-Pauli errors) noise models, respectively. The similarities and differences between the two scenarios are stressed. In addition, the performances of quantum codes quantified by means of the entanglement fidelity for different recovery schemes are taken into consideration in the approximate case. Finally, the role of self-complementarity in approximate quantum error correction is briefly addressed.

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31

WEN, KAI, and GUI LU LONG. "ONE-PARTY QUANTUM-ERROR-CORRECTING CODES FOR UNBALANCED ERRORS: PRINCIPLES AND APPLICATION TO QUANTUM DENSE CODING AND QUANTUM SECURE DIRECT COMMUNICATION." International Journal of Quantum Information 08, no. 04 (June 2010): 697–719. http://dx.doi.org/10.1142/s0219749910006289.

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In this article, we present unbalanced-quantum-error-correcting codes (one-party QECCs) — a novel idea for correcting unbalanced quantum errors. In some quantum communication tasks using entangled pairs, the error distributions between two parts of the pairs are unbalanced, and one party holds the whole entangled pairs at the final stage, and he or she is able to perform joint measurements on the pairs. In this situation the proposed one-party QECCs can improve error correction by allowing a higher-tolerated error rate. We have established the general correspondence between linear classical codes and the one-party QECCs, and we have given the general definition for these types of quantum-error-correcting codes. It has been shown that the one-party QECCs can correct errors as long as the error threshold is not larger than 0.5. They work even for fidelity less than 0.5 as long as it is larger than 0.25. We give several concrete examples of the one-party QECCs. We provide the applications of the one-party QECCs in quantum dense coding, so that it can function in noisy channels. As a result, a large number of quantum secure direct communication protocols based on dense coding are also able to be protected by this new type of one-party QECCs.

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32

Xue, Yun-Jia, Hao-Wen Wang, Yan-Bing Tian, Yi-Nuo Wang, Yu-Xuan Wang, and Shu-Mei Wang. "Quantum Information Protection Scheme Based on Reinforcement Learning for Periodic Surface Codes." Quantum Engineering 2022 (July 18, 2022): 1–9. http://dx.doi.org/10.1155/2022/7643871.

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Quantum information transfer is an information processing technology with high speed and high entanglement with the help of quantum mechanics principles. To solve the problem of quantum information getting easily lost during transmission, we choose topological quantum error correction codes as the best candidate codes to improve the fidelity of quantum information. The stability of topological error correction codes brings great convenience to error correction. The quantum error correction codes represented by surface codes have produced very good effects in the error correction mechanism. In order to solve the problem of strong spatial correlation and optimal decoding of surface codes, we introduced a reinforcement learning decoder that can effectively characterize the spatial correlation of error correction codes. At the same time, we use a double-layer convolutional neural network model in the confrontation network to find a better error correction chain, and the generation network can approach the best correction model, to ensure that the discriminant network corrects more nontrivial errors. To improve the efficiency of error correction, we introduced a double-Q algorithm and ResNet network to increase the error correction success rate and training speed of the surface code. Compared with the previous MWPM 0.005 decoder threshold, the success rate has slightly improved, which can reach up to 0.0068 decoder threshold. By using the residual neural network architecture, we saved one-third of the training time and increased the training accuracy to about 96.6%. Using a better training model, we have successfully increased the decoder threshold from 0.0068 to 0.0085, and the depolarized noise model being used does not require a priori basic noise, so that the error correction efficiency of the entire model has slightly improved. Finally, the fidelity of the quantum information has successfully improved from 0.2423 to 0.7423 by using the error correction protection schemes.

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33

Windridge, David, Riccardo Mengoni, and Rajagopal Nagarajan. "Quantum error-correcting output codes." International Journal of Quantum Information 16, no. 08 (December 2018): 1840003. http://dx.doi.org/10.1142/s0219749918400038.

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Quantum machine learning is the aspect of quantum computing concerned with the design of algorithms capable of generalized learning from labeled training data by effectively exploiting quantum effects. Error-correcting output codes (ECOC) are a standard setting in machine learning for efficiently rendering the collective outputs of a binary classifier, such as the support vector machine, as a multi-class decision procedure. Appropriate choice of error-correcting codes further enables incorrect individual classification decisions to be effectively corrected in the composite output. In this paper, we propose an appropriate quantization of the ECOC process, based on the quantum support vector machine. We will show that, in addition to the usual benefits of quantizing machine learning, this technique leads to an exponential reduction in the number of logic gates required for effective correction of classification error.

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34

Raussendorf, Robert. "Key ideas in quantum error correction." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1975 (September 28, 2012): 4541–65. http://dx.doi.org/10.1098/rsta.2011.0494.

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In this introductory article on the subject of quantum error correction and fault-tolerant quantum computation, we review three important ingredients that enter known constructions for fault-tolerant quantum computation, namely quantum codes, error discretization and transversal quantum gates. Taken together, they provide a ground on which the theory of quantum error correction can be developed and fault-tolerant quantum information protocols can be built.

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35

BÉNY, CÉDRIC, DAVID W. KRIBS, and ARON PASIEKA. "ALGEBRAIC FORMULATION OF QUANTUM ERROR CORRECTION." International Journal of Quantum Information 06, supp01 (July 2008): 597–603. http://dx.doi.org/10.1142/s0219749908003839.

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We give a brief introduction to the algebraic formulation of error correction in quantum computing called operator algebra quantum error correction (OAQEC). Then we extend one of the basic results for subsystem codes in operator quantum error correction (OQEC) to the OAQEC setting: Every hybrid classical-quantum code is shown to be unitarily recoverable in an appropriate sense. The algebraic approach of the proof yields a new, less technical proof for the OQEC case.

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36

Kern, O., G. Alber, and D. L. Shepelyansky. "Quantum error correction of coherent errors by randomization." European Physical Journal D 32, no. 1 (January 2005): 153–56. http://dx.doi.org/10.1140/epjd/e2004-00196-9.

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37

Chen, Xiu-Bo, Li-Yun Zhao, Gang Xu, Xing-Bo Pan, Si-Yi Chen, Zhen-Wen Cheng, and Yi-Xian Yang. "Low-overhead fault-tolerant error correction scheme based on quantum stabilizer codes." Chinese Physics B 31, no. 4 (March 1, 2022): 040305. http://dx.doi.org/10.1088/1674-1056/ac3817.

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Fault-tolerant error-correction (FTEC) circuit is the foundation for achieving reliable quantum computation and remote communication. However, designing a fault-tolerant error correction scheme with a solid error-correction ability and low overhead remains a significant challenge. In this paper, a low-overhead fault-tolerant error correction scheme is proposed for quantum communication systems. Firstly, syndrome ancillas are prepared into Bell states to detect errors caused by channel noise. We propose a detection approach that reduces the propagation path of quantum gate fault and reduces the circuit depth by splitting the stabilizer generator into X-type and Z-type. Additionally, a syndrome extraction circuit is equipped with two flag qubits to detect quantum gate faults, which may also introduce errors into the code block during the error detection process. Finally, analytical results are provided to demonstrate the fault-tolerant performance of the proposed FTEC scheme with the lower overhead of the ancillary qubits and circuit depth.

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38

Layden, David, Louisa Ruixue Huang, and Paola Cappellaro. "Robustness-optimized quantum error correction." Quantum Science and Technology 5, no. 2 (March 17, 2020): 025004. http://dx.doi.org/10.1088/2058-9565/ab79b2.

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39

Chiaverini, J., D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, et al. "Realization of quantum error correction." Nature 432, no. 7017 (December 2004): 602–5. http://dx.doi.org/10.1038/nature03074.

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40

Schindler, P., J. T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, and R. Blatt. "Experimental Repetitive Quantum Error Correction." Science 332, no. 6033 (May 26, 2011): 1059–61. http://dx.doi.org/10.1126/science.1203329.

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41

Ekert, Artur, and Chiara Macchiavello. "Quantum Error Correction for Communication." Physical Review Letters 77, no. 12 (September 16, 1996): 2585–88. http://dx.doi.org/10.1103/physrevlett.77.2585.

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42

Taghavi, Soraya, Robert L. Kosut, and Daniel A. Lidar. "Channel-Optimized Quantum Error Correction." IEEE Transactions on Information Theory 56, no. 3 (March 2010): 1461–73. http://dx.doi.org/10.1109/tit.2009.2039162.

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43

Wootton, James R. "Quantum memories and error correction." Journal of Modern Optics 59, no. 20 (November 20, 2012): 1717–38. http://dx.doi.org/10.1080/09500340.2012.737937.

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44

Aoki, Takao, Go Takahashi, Tadashi Kajiya, Jun-ichi Yoshikawa, Samuel L. Braunstein, Peter van Loock, and Akira Furusawa. "Quantum error correction beyond qubits." Nature Physics 5, no. 8 (June 28, 2009): 541–46. http://dx.doi.org/10.1038/nphys1309.

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45

Devitt, Simon J., William J. Munro, and Kae Nemoto. "Quantum error correction for beginners." Reports on Progress in Physics 76, no. 7 (June 20, 2013): 076001. http://dx.doi.org/10.1088/0034-4885/76/7/076001.

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46

Knight, P. L., M. B. Plenio, and V. Vedral. "Decoherence and quantum error correction." Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 355, no. 1733 (December 15, 1997): 2381–85. http://dx.doi.org/10.1098/rsta.1997.0134.

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47

Steane, A. M. "Introduction to quantum error correction." Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 356, no. 1743 (August 15, 1998): 1739–58. http://dx.doi.org/10.1098/rsta.1998.0246.

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48

Cao, Qian, Hao-Wen Wang, Ying-Jie Qu, Yun-Jia Xue, and Shu-Mei Wang. "Quantum Teleportation Error Suppression Algorithm Based on Convolutional Neural Networks and Quantum Topological Semion Codes." Quantum Engineering 2022 (November 24, 2022): 1–10. http://dx.doi.org/10.1155/2022/6245336.

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Quantum error correction (QEC) is a key technique for building scalable quantum computers that can be used to mitigate the effects of errors on physical quantum bits. Since quantum states are more or less affected by noise, errors are inevitable. Traditional QEC codes face huge challenges. Therefore, designing an error suppression algorithm based on neural networks (NN) and quantum topological error correction (QTEC) codes is particularly important for quantum teleportation. In this paper, QTEC codes: semion codes—a greater than 2 dimensional (2D) error correction code based on the double semion model—are used to suppress errors during quantum teleportation, using a NN to build a decoder based on semion codes and to simulate the quantum information error suppression process and the suppression effect. The proposed convolutional neural network (CNN) decoder is suitable for small distance topological semion codes. The aim is to optimize the NN for better decoder performance while deriving the relationship between decoder performance and slope and pseudothreshold during training and calculate the thresholds for different noise areas when the code distances are the same, P t h r e s h o l d = 0.082 for A r e a < 0.007 d B and P t h r e s h o l d = 0.096 for A r e a < 0.01 d B . This paper demonstrates the ability of CNNs to suppress errors in quantum transmission information and the great potential of NNs in the field of quantum computing.

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Li, Ruihu, Luobin Guo, and Zongben Xu. "Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound." Quantum Information and Computation 14, no. 13&14 (October 2014): 1107–16. http://dx.doi.org/10.26421/qic14.13-14-4.

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We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

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Alber, G., A. Delgado, and M. Mussinger. "Quantum Error Correction and Quantum Computation with Detected-jump Correcting Quantum Codes." Fortschritte der Physik 49, no. 10-11 (October 2001): 901. http://dx.doi.org/10.1002/1521-3978(200110)49:10/11<901::aid-prop901>3.0.co;2-a.

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