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Class of quantum error-correcting codes saturating the quantum Hamming bound
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I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode k=n-j-2 quantum bits (qubits) in n=${2}^{\mathit{j}}$ qubits and correct t=1 error. \textcopyright{} 1996 The American Physical Society.
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“Without error corrections, the effect of noise will accumulate and ruin the entire computation[72, 17, 54, 52, 7], and hence the computation must be protected.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [52]
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Cites
Scheme for reducing decoherence in quantum computer memory
Shor, P. W. (1995) · Physical Review A · doi:10.1103/physreva.52.r2493 · Cited by 4,479
Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review Letters · doi:10.1103/physrevlett.76.722 · Cited by 2,961
Good quantum error-correcting codes exist
Calderbank, A. R., & Shor, P. W. (1996) · Physical Review A · doi:10.1103/physreva.54.1098 · Cited by 2,516
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Improvement of Stabilizer-Based Entanglement Distillation Protocols by Encoding OperatorsPreprint
Watanabe, S., Matsumoto, R., & Uyematsu, T. (2005) · Cited by 7
Magic state distillation at intermediate size
Haah, J., Hastings, M. B., Poulin, D., & Wecker, D. (2018) · Quantum Information and Computation · doi:10.26421/qic18.1-2-6 · Cited by 8
Quantum state transmission over partially corrupted quantum information network
Hayashi, M., & Song, S. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033079 · Cited by 6
Higher rank matricial ranges and hybrid quantum error correction
Cao, N., Kribs, D. W., Li, C.-K., Nelson, M. I., Poon, Y., & Zeng, B. (2020) · Linear and Multilinear Algebra · doi:10.1080/03081087.2020.1748852 · Cited by 10
“m error correction (QEC) originated at the interface between quantum theory and coding theory in classical information transmission and is at the heart of designing those fault-tolerant architectures [16, 17, 18, 19, 20]. It was recognized early on during these investigations that the simultaneous transmission of both…”
BackgroundPerformance Analysis of Quantum Error-Correcting Surface Codes over Asymmetric Channels
Valentini, L., Forlivesi, D., & Chiani, M. (2023) · International Conference on Communications · doi:10.1109/icc45041.2023.10279096 · Cited by 6
Quantum data-syndrome codes: Subsystem and impure code constructions
Nemec, A. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-04166-z · Cited by 6
“We also protect the X-type syndrome and the Z-type syndromes of the Bacon-Shor code with [17, 2, 11] and [18, 2, 12] Cordaro-Wagner codes, so each protection scheme makes 204 or 216 total measurements.”
BackgroundInfluentialStabilizer Formalism for Operator Algebra Quantum Error Correction
Dauphinais, G., Kribs, D. W., & Vasmer, M. (2024) · Quantum · doi:10.22331/q-2024-02-21-1261 · Cited by 6
“Then and dim C = 2 n − s so that C can encode n − s qubits.”
BackgroundThermodynamic Interpretation of the Quantum Error Correcting Criterion
Korepin, V. E., & Terilla, J. (2002) · Quantum Information Processing · doi:10.1023/a:1022135318321 · Cited by 9
Operator Algebraic Formulation of the Stabilizer Formalism for Quantum Error Correction
Johnston, N., Kribs, D. W., & Teng, C.-W. (2009) · Acta Applicandae Mathematicae · doi:10.1007/s10440-008-9421-1 · Cited by 6
“The classic stabilizer formalism of Gottesman [1, 2] provides the simplest technique to generate codes for the standard model of error correction (QEC) in quantum computing [3–6].”
MethodMultiple System-Decomposition Method for Avoiding Quantum Decoherence
Jeknić-Dugić, J., & Dugić, M. (2008) · Chinese Physics Letters · doi:10.1088/0256-307x/25/2/006 · Cited by 6
Design for dependability in emerging technologies
Prodan, L., Udrescu, M., Boncalo, O., & Vlăduţiu, M. (2007) · ACM Journal on Emerging Technologies in Computing Systems · doi:10.1145/1265949.1265952 · Cited by 6
Discrete phase-space mappings, tomographic condition and permutation invariance
Muñoz, C., & Klimov, A. B. (2017) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aa5fb5 · Cited by 6
Improved quantum LDPC decoding strategies for the misidentified quantum depolarization channel
Xie, Y., Li, J., Malaney, R., & Yuan, J. (2016) · European Signal Processing Conference (EUSIPCO) · doi:10.1109/eusipco.2016.7760297 · Cited by 6
“Since the discovery of CSS (Calderbank, Shor and Steane) codes [9] [10] and stabilizer codes [11], it has been known ho w quantum error-correction codes can be developed in a simila r manner to classical codes.”
BackgroundConstruction of optimal resources for concatenated quantum protocols
Pirker, A., Wallnöfer, J., Briegel, H. J., & Dür, W. (2017) · Physical Review A · doi:10.1103/physreva.95.062332 · Cited by 6
Fasihi, M. A. (2019) · Physica Scripta · doi:10.1088/1402-4896/ab1036 · Cited by 6
Maximal entanglement entanglement-assisted quantum codes of distance three
Guo, L., Fu, Q., Li, R., & Lü, L. (2015) · International Journal of Quantum Information · doi:10.1142/s0219749915500021 · Cited by 6
Controlling qubit transitions through quantum interference during nonadiabatic rapid passage
Li, R., & Gaitan, F. (2005) · Optics and Spectroscopy · doi:10.1134/1.2034612 · Cited by 6
The asymptotic probability distribution of the relative distance of additive quantum codes
Ma, Y. (2007) · Journal of Mathematical Analysis and Applications · doi:10.1016/j.jmaa.2007.08.023 · Cited by 6
Salas, P. J., & Sanz, A. L. (2004) · Physical Review A · doi:10.1103/physreva.69.052322 · Cited by 5
Optical implementation of a unitarily correctable code
Schreiter, K. M., Pasieka, A., Kaltenbaek, R., Resch, K. J., & Kribs, D. W. (2009) · Physical Review A · doi:10.1103/physreva.80.022311 · Cited by 6
ALGEBRAIC FORMULATION OF QUANTUM ERROR CORRECTION
Bény, C., Kribs, D. W., & Pasieka, A. (2008) · International Journal of Quantum Information · doi:10.1142/s0219749908003839 · Cited by 6
Optimized quantum error-correction codes for experiments
Nebendahl, V. (2015) · Physical Review A · doi:10.1103/physreva.91.022332 · Cited by 6
Noise tolerance of the BB84 protocol with random privacy amplification
Watanabe, S., Matsumoto, R., & Uyematsu, R. (2005) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit.2005.1523492 · Cited by 7
Quantum 2‐body Hamiltonian for topological color codes
Bombín, H., Kargarian, M., & Martin‐Delgado, M. A. (2009) · Fortschritte der Physik · doi:10.1002/prop.200900084 · Cited by 6
“rs: Toric Codes and Color Codes Some of the simplest quantum Hamiltonian models with topological order can be obtained from the formalism local stabilizer codes borrowed from quantum error correction [11] in quantum information. These are spin-1 2 local models of the form H = − X i S i, S i ∈ P n := hi,σ1 x,σz1,...,σx…”
BackgroundInfluentialDynamics of the entanglement witness for three qubits in common environment
Lu, H.-X. (2007) · Chinese Physics · doi:10.1088/1009-1963/16/7/012 · Cited by 6
Classical access structures of ramp secret sharing based on quantum stabilizer codes
Matsumoto, R. (2019) · Quantum Information Processing · doi:10.1007/s11128-019-2503-3 · Cited by 5
Quantum Synchronizable Codes From Augmentation of Cyclic Codes.Preprint
Xie, Y., & Yuan, J. (2014) · Cited by 6
Continuous Gaussian Measurements of the Free Boson CFT: An Exactly Solvable ModelPreprint
Minoguchi, Y., Rabl, P., & Buchhold, M. (2021) · arXiv (Cornell University) · Cited by 6
Ares, L., Pinske, J., Hinrichs, B., Kolb, M., & Sperling, J. (2025) · Physical Review A · doi:10.1103/hcj7-8zlg · Cited by 6
Quantum (r,δ)-locally recoverable codes
Galindo, C., Hernando, F., Martín-Cruz, H., & Matsumoto, R. (2025) · Finite Fields and Their Applications · doi:10.1016/j.ffa.2025.102785 · Cited by 5
“Only binary codes were considered in the seminal contributions to the study of QECCs [11,26,12,10 Subsequently, q -ary QECCs ( q being a power of a prime number p ) have been extensively studied and constructed with a wide variety of techniques, see for instance [8,5,30,1,29,33,19,2] among many…”
Designing quantum error correction codes for practical spin qudit systems
Lim, S., & Ardavan, A. (2025) · Physical Review A · doi:10.1103/7q6l-d4qh · Cited by 6
“While second-order error correction of qubit-based code-word require minimum 11 qubits [54] (i.e., a Hilbert space dimension of 2048) this 24-dimensional spin qudit can provide efficient encoding, and can provide robustness under any perturbation, because all significant practical perturbations on…”
Role of quantum state texture in probing resource theories and quantum phase transitions
Patra, A., Konar, T. K., Halder, P., & De, A. S. (2026) · Physical Review A · doi:10.1103/y2wz-ypln · Cited by 10
“The stabilizer formalism plays a crucial role in quantum information and computation, as many prototypical quantum error-correcting codes are built upon it [10, 54, 55], and Clifford operations within this framework can be implemented fault-tolerantly [56, 57].”
Separability Lindblad equation for dynamical open-system entanglement
Pinske, J., Ares, L., Hinrichs, B., Kolb, M., & Sperling, J. (2025) · Physical Review A · doi:10.1103/kd3b-bfxq · Cited by 6
“…thus refers to a lack of separation between subsystems and is considered to be the quintessential ingredient for quantum information processing tasks, including quantum teleportation [3, 4], quantum-computational speedup [5, 6], quantum communication [7, 8], and quantum-error correction [9, 10].”
Güngördü, U., Nepal, R., & Kovalev, A. A. (2014) · Physical Review A · doi:10.1103/physreva.90.042326 · Cited by 5
Unitarily inequivalent mutually unbiased bases for n qubits
Sehrawat, A., & Klimov, A. B. (2014) · Physical Review A · doi:10.1103/physreva.90.062308 · Cited by 5
Group structures and representations of graph states
Wu, J.-Y., Kampermann, H., & Bruß, D. (2015) · Physical Review A · doi:10.1103/physreva.92.012322 · Cited by 5
“These states can be exploited as quantum stabilizer codes [7, 14, 15, 24], which are linear codes and protect against errors.”
Background(2019) · Lecture notes in computer science · doi:10.1007/978-3-030-21363-3 · Cited by 5
Advances in quantum error correction based on superconducting quantum systems
Chen, Z.-J., Pan, X., Hua, Z., Wang, W., Ma, Y., Li, M., Zou, X., Sun, L., & Zou, C. (2022) · Acta Physica Sinica · doi:10.7498/aps.71.20221824 · Cited by 5
Fidelity-based distance bounds for N -qubit approximate quantum error correction
Fiusa, G., Soares-Pinto, D. O., & Pires, D. P. (2023) · Physical Review A · doi:10.1103/physreva.107.032422 · Cited by 5
On the construction of new toric quantum codes and quantum burst-error-correcting codes
Trinca, C. C., Interlando, J. C., Palazzo, R., de Andrade, A. A., & Watanabe, R. A. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-03931-4 · Cited by 5
“Thus, C = {|ψ〉 ∈ H | M |ψ〉 = |ψ〉, ∀ M ∈ S} [25].”
Zhang, Y., Liu, Y., Hou, X., & Gao, J. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-04013-1 · Cited by 5
“OQECCs generalize and unify the main methods of passive and active quantum error-correction: decoherence free subspaces [26], noiseless subsystems [42], and QECCs [8, 17, 38].”
BackgroundBulk and boundary entanglement transitions in the projective gauge-Higgs model
Sukeno, H., Ikeda, K., & Wei, T.-C. (2024) · Physical review. B./Physical review. B · doi:10.1103/physrevb.110.245102 · Cited by 7
“Some studies have focused on special classes of unitary gates, such as the Clifford gates or Majorana fermions, which can be efficiently simulated classically in combination with Pauli measurements due to the Gottesman-Knill theorem [18–20 , 23–39].”
Bubble Clustering Decoder for Quantum Topological Codes
Forlivesi, D., Valentini, L., & Chiani, M. (2025) · IEEE Transactions on Communications · doi:10.1109/tcomm.2025.3558012 · Cited by 5
“In this context, numerous short quantum error correcting codes (QECCs) have been proposed in the literature as early methods for encoding quantum information [3], [8], [9], [10], [11], [12], [13].”
A Realizable Distributed Ion-Trap Quantum Computer
Thaker, D., Metodi, T. S., & Chong, F. T. (2006) · Lecture notes in computer science · doi:10.1007/11945918_16 · Cited by 5
“rich theory of quantum fault-tolerance has been proposed [3, 4, 5 ]. Our previous work builds on these theories and the physical implementations of quantum computers, to design a scalable and reconfigurable Quantum Logic Array (QLA) architecture [2].”
BackgroundQuantitative Treatment of Decoherence
Fedichkin, L. E., & Vladimir Privman, A. (2009) · Topics in applied physics · doi:10.1007/978-3-540-79365-6_8 · Cited by 8
Transition of d -level quantum systems through quantum channels with correlated noise
Fahmi, A., & Golshani, M. (2007) · Physical Review A · doi:10.1103/physreva.75.042301 · Cited by 5
Non-orthogonal Wigner-like mapping
Sainz, I., Roa, L., & Klimov, A. B. (2012) · Journal of Mathematical Physics · doi:10.1063/1.4712303 · Cited by 5
Breeding quantum error-correcting codes
Dong, Y., Hu, D. M., & Yu, S. (2010) · Physical Review A · doi:10.1103/physreva.81.022322 · Cited by 4
Reliable Quantum LDPC Codes over GF(4)
Xie, Y., & Yuan, J. (2016) · IEEE Globecom Workshops (GC Wkshps) · doi:10.1109/glocomw.2016.7849021 · Cited by 5
“Ever since the first quantum error-correcting code was conceived by Shor in 1995 [1], various types of quantum error-correcting codes have been extensively studied including the famous Calderbank-Shor-Steane (CSS) codes [2], [3] and stabilizer codes [4].”
BackgroundOn quantum stabilizer codes derived from local Frobenius rings
Gluesing-Luerssen, H., & Pllaha, T. (2019) · Finite Fields and Their Applications · doi:10.1016/j.ffa.2019.04.004 · Cited by 5
“um equivalent to linear codes in classical coding theory. While the first examples of quantum error-correcting codes were in fact stabilizer codes, the general construction was introduced by Gottesman [5] and Calderbank et al. [2]. Quantum stabilizer codes are particularly appealing because of their close relation to c…”
BackgroundUniversal Framework for Quantum Error-Correcting Codes
Li, Z., & Xing, L. (2021) · Entropy · doi:10.3390/e23080937 · Cited by 5
“Almost all of the quantum codes constructed so far are stabilizer or additive codes [3].”
BackgroundDetecting entanglement of quantum channels
Li, C., Wang, B.-H., Wu, B., & Yuan, X. (2021) · Communications in Theoretical Physics · doi:10.1088/1572-9494/ac1da1 · Cited by 5
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Santos, R., Jebarathinam, C., & Augusiak, R. (2022) · Physical Review A · doi:10.1103/physreva.106.012431 · Cited by 5
An Anyon Model in a Toric Honeycomb Lattice
Liang, L., & Yu, Y. (2011) · Communications in Theoretical Physics · doi:10.1088/0253-6102/55/1/16 · Cited by 6
Xiao, H., Zhang, Z., & Chronopoulos, A. T. (2017) · The European Physical Journal C · doi:10.1140/epjc/s10052-017-5246-2 · Cited by 5
“In [28], Gottesman has proposed a class of stabilizer codes, which are one-qubit error correcting, saturate the quantum Hamming bound, and assign distinct “error syndrome” to each correctable error operator according to a certain rule.”
BackgroundRecovery of Distributed Quantum Information Using Graph States from a Node Failure
Nadkarni, P. J., Raina, A., & Srinivasa, S. G. (2017) · IEEE Globecom Workshops (GC Wkshps) · doi:10.1109/glocomw.2017.8269075 · Cited by 5
“We use the stabilizer formalism to construct our code to correct errors [3].”
BackgroundMethodInfluentialQuantum complementarity and operator structures
Kribs, D. W., Livick, J., Nelson, M. I., Perira, R., & Rahaman, M. (2019) · Quantum Information and Computation · doi:10.26421/qic19.1-2-5 · Cited by 6
“Quantum error correction as a subject is considerably more developed than the theory of private quantum codes and algebras, with origins going back over two decades to the beginnings of modern quantum information science [40, 41, 16, 5, 25, 26].”
BackgroundQuantum Computations with Polarized Photons
D’Ariano, G. M., Macchiavello, C., & Maccone, L. (2000) · Fortschritte der Physik · doi:10.1002/(sici)1521-3978(200005)48:5/7 · Cited by 20
LINEAR QUANTUM CODES OF MINIMUM DISTANCE THREE
Li, R., Li, X., & Xu, Z. (2006) · International Journal of Quantum Information · doi:10.1142/s0219749906002298 · Cited by 5
Improvement of stabilizer-based entanglement distillation protocols by encoding operators
Watanabe, S., Matsumoto, R., & Uyematsu, T. (2006) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/39/16/013 · Cited by 5
“In this section, we review the non-binary generalization [19, 23] of the stabilizer code [5, 6, 10].”
MethodTwo Infinite Families of Nonadditive Quantum Error-Correcting Codes
Yu, S., Chen, Q., & Oh, C. H. (2015) · IEEE Transactions on Information Theory · doi:10.1109/tit.2015.2489645 · Cited by 5
“Two infinite families of nonadditive quantum error-correcting codes Sixia Yu1,2, Qing Chen1,2, and C. H. Oh2 1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics & Department of Modern Physics University of Science and Technology of China, Hefei 230026, P.R. China 2Physics De…”
BackgroundNon-stabilizer quantum codes from abelian subggroups of the error group
Arvind, V., Kururi, P. P., & Parthasarathy, K. R. (2004) · Quantum Information and Computation · doi:10.5555/2011593.2011595 · Cited by 5
Convolutional and Block Quantum Error-Correcting Codes
Grassl, M. (2006) · IEEE Information Theory Workshop (ITW) · doi:10.1109/itw2.2006.323775 · Cited by 5
“While the theory of quantum block codes [1, 4 ] is fairly advanced by now, the investigation of the quantum version of convolutional codes has only started more recently [2, 3, 7, 10].”
BackgroundAutomated searching for quantum subsystem codes
Crosswhite, G. M., & Bacon, D. (2011) · Physical Review A · doi:10.1103/physreva.83.022307 · Cited by 5
Quantum Error Correction and Fault-TolerancePreprint
Gottesman, D. (2005) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0507174 · Cited by 16
“An example of a stabilizer code is the 5-qubit code, a [[ 5 ,1,3]] code whose stabilizer can be generated by”
BackgroundSelf-citationConcatenated Conjugate CodesPreprint
Hamada, M. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0610194 · Cited by 5
“CSS codes are a class of algebraic quantum error-correcting codes, called symplectic codes, or stabilizer codes [10], [11], [12].”
BackgroundError correction for mutually interacting qubits
Gea-Banacloche, J. R. (2000) · Physical Review A · doi:10.1103/physreva.62.062313 · Cited by 5
General Decoherence Suppression in Three-Level Atom in V- and Ξ-Configurations
Wang, Y., Liu, X.-S., Ruan, D., & Gui-Lu, L. (2008) · Communications in Theoretical Physics · doi:10.1088/0253-6102/49/4/16 · Cited by 5
Minimising the Decoherence of Rare Earth Ion Solid State Spin Qubits
Fraval, E. (2005) · ANU Open Research (Australian National University) · doi:10.25911/5d7a2a1e58bf6 · Cited by 5
“These strategies can be broadly grouped as Quantum Error Correction Codes (QECC) [60, 64, 65, 66], Decoherence Free Subspaces (DFS) [67, 68, 69, 70] and Dynamic Decoherence Control [8, 10, 11, 12, 13, 71].”
BackgroundQuantum Communication and DecoherencePreprint
Aschauer, H., & Briegel, H. J. (2002) · Cited by 5
Decoherence and quantum error correction for quantum computing and communications.
Martínez, J. E., & de Navarra, U. (2022) · Deposito Adademico Digital Universidad De Navarra (University of Navarra) · doi:10.15581/10171/63013 · Cited by 11
Quantum Information Processing
Lloyd, S. (2009) · Springer eBooks · doi:10.1007/978-0-387-30440-3_437 · Cited by 5
Fidelity of at-error-correcting quantum code with more thanterrors
Matsumoto, R. (2001) · Physical Review A · doi:10.1103/physreva.64.022314 · Cited by 5
Area laws and efficient descriptions of quantum many-body statesPreprint
Ge, Y., & Eisert, J. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1411.2995 · Cited by 5
Bondal, A. I., & Zhdanovskiy, I. Y. (2021) · Russian Mathematical Surveys · doi:10.1070/rm9983 · Cited by 4
“There are many other instances when mutually unbiased bases are used in physics (see [23], [48], [54], [22], [30], [53], [27], [15], and so on).”
BackgroundUsing a Deep Quantum Neural Network to Enhance the Fidelity of Quantum Convolutional Codes
Xiao, H., Chen, X., & Xu, J. S. Y. (2022) · Applied Sciences · doi:10.3390/app12115662 · Cited by 4
New Quantum Burst-Error Correcting Codes from Interleaving Technique
Trinca, C. C., de Albuquerque, C. D., Palazzo, R., Interlando, J. C., de Andrade, A. A., & Watanabe, R. A. (2022) · Globecom · doi:10.1109/globecom48099.2022.10000761 · Cited by 4
“Thus, C = {|ψ⟩ ∈ H | M |ψ⟩ = |ψ⟩, ∀ M ∈ S} [26].”
Lower Bounds for Quasi-Cyclic Codes and New Binary Quantum Codes
Liu, Y., Guan, C., Du, C., & Ma, Z. (2023) · Symmetry · doi:10.3390/sym15030643 · Cited by 4
“Some bounds are proposed to measure the merit of the parameters of QECCs, such as quantum Hamming bound [42,43], quantum Singleton bound [44,45], and quantum GV-bound [46,47].”
Several new families of MDS EAQECCs with much larger dimensions and related application to EACQCs
Cao, M. (2023) · Quantum Information Processing · doi:10.1007/s11128-023-04197-6 · Cited by 4
“From then on, the study of quantum codes has achieved great developments in both theory and application (see, e.g., [2, 21, 23, 31, 33, 45, 47]).”
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Skotiniotis, M., Dür, W., & Kraus, B. (2013) · Quantum Information and Computation · doi:10.26421/qic13.3-4-9 · Cited by 4
“Several methods are known to deal with such a situation, including teleportation-based communication utilizing entanglement purification [5], or encoding of quantum information using quantum error-correcting codes [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].”
MethodEfficient Quantum Circuit for Encoding and Decoding of the [[8,3,5]] Stabilizer Code
Dong, P., Liu, J., & Cao, Z. (2012) · International Journal of Theoretical Physics · doi:10.1007/s10773-012-1442-x · Cited by 4
Entropic measure and hypergraph states
Qu, R., Ma, Y.-P., Bao, Y.-R., Wang, J., & Li, Z.-S. (2013) · Quantum Information Processing · doi:10.1007/s11128-013-0646-1 · Cited by 4
“Although graph states can describe a large family of entangled states including cluster states [7], GHZ states, stabilizer states [8,9], etc.”
BackgroundConstruction of minimal trellises for quantum stabilizer codes
Xiao, F., & Chen, H. (2012) · Science China Information Sciences · doi:10.1007/s11432-012-4595-6 · Cited by 4
Construction of color codes from polygons
da Silva, E. B., & Soares, W. S. (2018) · Journal of Physics Communications · doi:10.1088/2399-6528/aad062 · Cited by 4
Modified graph-state codes for single-node recovery in quantum distributed storage
Nadkarni, P. J., Raina, A., & Garani, S. S. (2020) · Physical Review A · doi:10.1103/physreva.102.062430 · Cited by 4
Molecular Machines for Quantum Error Correction
Guerreiro, T. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.030336 · Cited by 4
“The quantum Hamming bound [47] implies that for single qubit errors, a minimum of five qubits are required to achieve complete fault tolerance for one logical qubit.”
BackgroundModel for fractons, fluxons, and free vertex excitations
Kim, J., & Han, J. H. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.115128 · Cited by 4
Advance Sharing of Quantum Shares for Classical Secrets
Miyajima, R., & Matsumoto, R. (2022) · IEEE Access · doi:10.1109/access.2022.3204389 · Cited by 4
“Matsumoto’s 111 secret sharing scheme [22] is based on quantum stabilizer 112 codes [23], [24], [25], [26], [27], [28], [29] and summarized 113 as follows: 114”
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Ahadiansyah, D., Anwar, K., & Budiman, G. (2022) · IEEE International Symposium on Future Telecommunication Technologies (SOFTT) · doi:10.1109/softt56880.2022.10010083 · Cited by 4
“Gottesman created the stabilizer formalism [11] and used it to investigate the properties of various quantum codes.”
BackgroundMethodLearning to Decode Trapping Sets in QLDPC Codes
Pradhan, A. K., Raveendran, N., Rengaswamy, N., Xiao, X. W., & Vasic, B. V. (2023) · International Symposium on Turbo Codes and Iterative Information Processing · doi:10.1109/istc57237.2023.10273526 · Cited by 4
“A (n, k) quantum stabilizer code [14] is a 2-dimensional subspace C of the Hilbert space (C2)⊗n given by the common +1 eigenspace of stabilizer group S:”
BackgroundComplexity Considerations in the Heisenberg Uncertainty Principle
Nye, L. (2024) · Journal of High Energy Physics Gravitation and Cosmology · doi:10.4236/jhepgc.2024.104083 · Cited by 4
A small error-correction code for protecting three-qubit quantum information
Yang, C., Chu, S., & Han, S. (2004) · Journal of Experimental and Theoretical Physics Letters · doi:10.1134/1.1753423 · Cited by 4
Fidelity Lower Bounds for Stabilizer and CSS Quantum Codes
Ashikhmin, A. (2014) · IEEE Transactions on Information Theory · doi:10.1109/tit.2014.2303801 · Cited by 4
“See [8],[18],[5] for the exact definition of stabilizer codes.”
Background2-designs and redundant syndrome extraction for quantum error correction
Premakumar, V. N., Sha, H., Crow, D., Bach, E., & Joynt, R. (2021) · Quantum Information Processing · doi:10.1007/s11128-021-03015-1 · Cited by 4
Quantum teleportation with mutually unbiased bases
Chen, D., Zhang, L., & Zhang, J. (2020) · Quantum Information Processing · doi:10.1007/s11128-020-2621-y · Cited by 4
New Quantum Codes Constructed by Quantum Caps in PG(3,9) and PG(4,9)
Li, H., Li, R., & Fu, Q. (2020) · IEEE Access · doi:10.1109/access.2020.3045990 · Cited by 4
“Then, there exists a pure quantum code [[ n , k , d ]] q provided that Proposition 2 (Quantum Hamming Bound [18], [19]): For any pure quantum code [[ n , k , d ]] The rest of this paper is organized as follows: basic concepts related to the linear code and projective cap are recalled in Sect.”
BackgroundA generalization of cyclic code equivalence algorithm to constacyclic codes
Akre, D., Aydın, N., Harrington, M., & Pandey, S. (2022) · Designs Codes and Cryptography · doi:10.1007/s10623-022-01124-1 · Cited by 5
“All of these properties of codes have been used extensively to find optimal quantum error-correcting codes [20, 21, 22].”
BackgroundGlobal stabilizer quantum error correction with combinatorial arrays
Fujiwara, Y. (2015) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit.2015.7282628 · Cited by 4
“Stabilizer codes [2], [3] are quantum analogues of classical linear codes.”
BackgroundInformation-theoretical approach to control of quantum-mechanical systems
Kawabata, S. (2003) · Physical Review A · doi:10.1103/physreva.68.064302 · Cited by 4
Metadata
- Authors
- Gottesman, D.
- Affiliations
- California Institute of Technology (US)
- Corresponding author
- Gottesman, Daniel
- ORCID
- Gottesman, Daniel
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1996-09-01
- Volume
- 54
- Issue
- 3
- Pages
- 1862-1868
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2069711632
- Cited by
- 1,227
- References
- 3
- Citations by year
- 2026: 26 · 2025: 75 · 2024: 48 · 2023: 56 · 2022: 45 · 2021: 65 · 2020: 44 · 2019: 36 · 2018: 34 · 2017: 36 · 2016: 33 · 2015: 32 · 2014: 41 · 2013: 38 · 2012: 34
- FWCI
- 38.07
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy (FG0392ER40701)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Coding theory and cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum error-correcting codes, code construction, single error correction, qubit encoding, code rate, quantum information
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