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Logical error rate scaling of the toric code

NEW JOURNAL OF PHYSICS 2014年第9期16卷 1-20页

作者： Watson, Fern H. E. Barrett, Sean D. Univ London Imperial Coll Sci Technol & Med Dept Phys London SW7 2AZ England

To date, a great deal of attention has focused on characterizing the performance of quantum error correcting codes via their thresholds, the maximum correctable physical error rate for a given noise model and decoding strategy. Practical quantum computers will necessarily operate below these thresholds meaning that other performance indicators become important. In this work we consider the scaling of the logical error rate of the toric code and demonstrate how, in turn, this may be used to calculate a key performance indicator. We use a perfect matching decoding algorithm to find the scaling of the logical error rate and find two distinct operating regimes. The first regime admits a universal scaling analysis due to a mapping to a statistical physics model. The second regime characterizes the behaviour in the limit of small physical error rate and can be understood by counting the error configurations leading to the failure of the decoder. We present a conjecture for the ranges of validity of these two regimes and use them to quantify the overhead-the total number of physical qubits required to perform error correction.

关键词： overhead toric code topological quantum error correction

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LOCALIZED ENDOMORPHISMS IN KITAEV'S toric code ON THE PLANE

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REVIEWS IN MATHEMATICAL PHYSICS 2011年第4期23卷 347-373页

作者： Naaijkens, Pieter Radboud Univ Nijmegen Inst Math Astrophys & Particle Phys NL-6500 GL Nijmegen Netherlands

We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher-Haag-Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of D(Z(2)), i.e. Drinfel'd's quantum double of the group algebra of Z(2).

关键词： Localized endomorphisms superselection sectors anyons toric code

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From doubled Chern-Simons-Maxwell lattice gauge theory to extensions of the toric code

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ANNALS OF PHYSICS 2015年 361卷 303-329页

作者： Olesen, T. Z. Vlasii, N. D. Wiese, U-J Univ Bern Inst Theoret Phys Albert Einstein Ctr Fundamental Phys CH-3012 Bern Switzerland Natl Acad Sci Ukraine Bogolyubov Inst Theoret Phys UA-03680 Kiev Ukraine

We regularize compact and non-compact Abelian Chem-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group lit, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space R-2 in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus U(1)(2) threaded by k units of quantized "magnetic" flux, with k being the level of the Chern-Simons theory. The holonomies of the torus U(1)(2) give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a Z(k)-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle 2 pi/k. Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Quantum information toric code Chern-Simons theory Lattice gauge theory Self-adjoint extension Berry gauge field

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Haag Duality and the Distal Split Property for Cones in the toric code

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LETTERS IN MATHEMATICAL PHYSICS 2012年第3期101卷 341-354页

作者： Naaijkens, Pieter Radboud Univ Nijmegen Inst Math Astrophys & Particle Phys NL-6500 GL Nijmegen Netherlands

We prove that Haag duality holds for cones in the toric code model. That is, for a cone I >, the algebra of observables localized in I > and the algebra of observables localized in the complement I > (c) generate each other's commutant as von Neumann algebras. Moreover, we show that the distal split property holds: if are two cones whose boundaries are well separated, there is a Type I factor such that . We demonstrate this by explicitly constructing .

关键词： Haag duality distal split property toric code

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An Accurate Union Find Decoder for Quantum Error Correction on the toric code

An Accurate Union Find Decoder for Quantum Error Correction ...

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1st International Conference on Smart Energy Systems and Artificial Intelligence (SESAI)

作者： Valentino, Federico Branchini, Beatrice Conficconi, Davide Sciuto, Donatella Santambrogio, Marco D. Politecn Milan Dipartimento Elettron Informat & Bioingn Milan Italy

ISBN: (纸本)9798350364613;9798350364606

Quantum computing represents an exciting computing paradigm that promises to solve problems untractahle for a classical computer. One main limiting factor is the noise impacting qubits, which hinders the superpolynomial speedup promise. Thus, although Quantum Error Correction (QEC) mechanisms are paramount, QEC demands high speed and low latency to scale quantum computations. Within this context, hardware accelerators, such as FPGAs, represent a valuable approach to fulfilling the QEC requirements. Nevertheless, the literature falls short in proposing solutions optimized for the toric code, which is capable of encoding two logical qubits and, therefore, requires fewer qubits to correct errors. This manuscript presents QUEKUF, an FPGA-based QEC dataflow architecture dealing with toric code. QUEKUF disposes of parallel processing units to spatially parallelize QEC that a centralized controller orchestrates for data movement and operation decisions. Experimental results show that our design attains up to 20.14 x improvement in execution time over a C++ implementation while keeping a high accuracy, similar to the theoretical framework.

关键词： FPGA Quantum Error Correction toric code

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Machine learning quantum error correction codes: learning the toric code

Machine learning quantum error correction codes: learning th...

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作者： Rodriguez Fernandez, Carlos Gustavo Universidade Estadual Paulista

Usamos métodos de aprendizagem supervisionada para estudar a decodiﬁcação de erros em códigos tóricos de diferentes tamanhos. Estudamos múltiplos modelos de erro, e obtemos ﬁguras da eﬁcácia de decodiﬁcação como uma função da taxa de erro de um único qubit. Também comentamos como o tamanho das redes neurais decodiﬁcadoras e seu tempo de treinamento aumentam com o tamanho do código tó*** use supervised learning methods to study the error decoding in toric codes of diﬀerent sizes. We study multiple error models, and obtain ﬁgures of the decoding eﬃcacy as a function of the single qubit error rate. We also comment on how the size of the decoding neural networks and their training time scales with the size of the toric code.

关键词： Código tórico Aprendizado de máquina toric code Quantum error correction Machine learning Correção de erros quânticos Código torico Dissertação de mestrado

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Modified toric code models with flux attachment from Hopf algebra gauge theory

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2023年第29期56卷 295302-295302页

作者： Conlon, A. Pellegrino, D. Slingerland, J. K. Dublin Inst Adv Studies Sch Theoret Phys 10 Burlington Rd Dublin Ireland Maynooth Univ Dept Theoret Phys Maynooth Ireland

Kitaev's toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be extended with the gauge group generalized to any finite-dimensional semisimple Hopf algebra. This also leads to extensions of the toric code. Here we consider the simple case where the gauge group is unchanged but furnished with a non-trivial quasitriangular structure (R-matrix), which modifies the construction of the gauge theory. This leads to some interesting phenomena;for example, the space of functions on the group becomes a non-commutative algebra. We also obtain simple Hamiltonian models generalizing the toric code, which are of the same overall topological type as the toric code, except that the various species of particles created by string operators in the model are permuted in a way that depends on the R-matrix. In the case of Z(N) gauge theory, we find that the introduction of a non-trivial R-matrix amounts to flux attachment.

关键词： Hopf algebra topological lattice models gauge theory toric code

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Calculating the Minimum Distance of a toric code via Algebraic Algorithms

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MATHEMATICS IN COMPUTER SCIENCE 2023年第3-4期17卷 20-20页

作者： Baldemir, Fadime Sahin, Mesut Cankiri Karatekin Univ Dept Math TR-18100 Cankiri Turkiye Middle East Tech Univ Dept Math TR-06800 Ankara Turkiye Hacettepe Univ Dept Math TR-06800 Ankara Turkiye

toric codes are examples of evaluation codes. They are produced by evaluating homogeous polynomials of a fixed degree at the Fq-rational points of a subset Y of a toric variety X. These codes reveal how algebraic geometry and coding theory are interrelated. The minimum distance of a code is the minimum number of nonzero entries in the codewords of the code. Let I(Y) be the ideal generated by all homogeneous polynomials vanishing at all the points of Y, which is also known as the vanishing ideal of Y. We give three algebraic algorithms computing the minimum distance by using commutative algebraic tools such as the multigraded Hilbert polynomials of ideals obtained from I(Y) and zero divisors f of I(Y), and primary decomposition of I(Y), for finding a homogeneous polynomial f among all homogeneous polynomials of the same degree which has the maximum number of roots on Y.

关键词： Minimum distance toric code Hilbert function

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Determination of quantum toric error correction code threshold using convolutional neural network decoders

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Chinese Physics B 2022年第1期31卷 136-142页

作者： Hao-Wen Wang Yun-Jia Xue Yu-Lin Ma Nan Hua Hong-Yang Ma School of Sciences Qingdao University of TechnologyQingdao 266033China School of Information and Control Engineering Qingdao University of TechnologyQingdao 266033China

Quantum error correction technology is an important solution to solve the noise interference generated during the operation of quantum *** order to find the best syndrome of the stabilizer code in quantum error correction,we need to find a fast and close to the optimal threshold *** this work,we build a convolutional neural network(CNN)decoder to correct errors in the toric code based on the system research of machine *** analyze and optimize various conditions that affect CNN,and use the RestNet network architecture to reduce the running *** is shortened by 30%-40%,and we finally design an optimized algorithm for CNN *** this way,the threshold accuracy of the neural network decoder is made to reach 10.8%,which is closer to the optimal threshold of about 11%.The previous threshold of 8.9%-10.3%has been slightly improved,and there is no need to verify the basic noise.

关键词： quantum error correction toric code convolutional neural network(CNN)decoder

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toric codes from Order Polytopes

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DISCRETE & COMPUTATIONAL GEOMETRY 2023年第3期69卷 834-848页

作者： Can, Mahir Bilen Hibi, Takayuki Tulane Univ New Orleans LA 70118 USA Osaka Univ Grad Sch Informat Sci & Technol Dept Pure & Appl Math Suita Osaka 5650871 Japan

We investigate a class of linear error correcting codes in relation with the order polytopes. In particular we consider the order polytopes of tree posets and bipartite posets. We calculate the parameters of the assoc... 详细信息

关键词： toric code Parameter Poset polytope Order polytope Shrub Bipartite poset

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