Graphical algorithms and threshold error rates for the 2d colour code

Wang, D. S.; Fowler, Austin G.; Hill, Charles D.; Hollenberg, L. C. L.

doi:10.48550/arxiv.0907.1708

Cited by 14 publications

(18 citation statements)

References 21 publications

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“…Interestingly enough, this type of mapping can be made more general and applied to TCCs yielding the same error threshold [28] while maintaining its enhanced quantum capabilities [29,30]. These results have been confirmed using different types of computation methods [31][32][33][34][35]. It is also possible to carry out certain computations by changing the code geometry over time, something called 'code deformation' [12,36,37] that allows us to perform quantum computation in a different way.…”

Section: Introductionmentioning

confidence: 71%

Generalized toric codes coupled to thermal baths

2012

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We have studied the dynamics of a generalized toric code based on qudits at finite temperature by finding the master equation coupling the code's degrees of freedom to a thermal bath. We find that in the case of qutrits, new types of anyons and thermal processes appear that are forbidden for qubits. These include creation, annihilation and diffusion throughout the system code. It is possible to solve the master equation in a short-time regime and find expressions for the decay rates as a function of the dimension d of the qudits. While we provide an explicit proof that the system relaxes to the Gibbs state for arbitrary qudits, we also prove that above a certain crossover temperature the qutrits' initial decay rate is smaller than the original case for qubits. Surprisingly, this behavior only happens for qutrits and not for other qudits with d > 3.3 the property of locality in error detection and correction is of great importance both theoretically and for practical implementations [12][13][14]. It is also possible to generalize these topological codes for units of quantum information based on multilevel systems known as qudits, i.e. d-level systems [15][16][17][18], and study their local stability [19]. An alternative scheme to manipulate topological quantum information is based not in the ground-state properties of the system but in its excitations [11]. These are non-Abelian anyons that can implement universal gates for quantum information [20]. However, being within the framework of topological codes based on ground state properties, it is possible to formulate new surface codes known as topological color codes (TCCs) [21] such that they have enhanced quantum computational capabilities while preserving nice locality properties [21][22][23]. TCCs in two-dimensional (2D) surfaces allow for the implementation of quantum gates in the whole Clifford group. This makes possible quantum teleportation, distillation of entanglement and dense coding in a fully topological scenario. Moreover, with TCCs in 3D spatial manifolds it is possible to implement the quantum gate π/8, thereby allowing for universal quantum computation [24,25]. Very nice applications of topological surface codes can be seen in other fields [26,27].Acting externally on topological codes, in order to cure the system from external noise and decoherence, produces benefits from the locality properties of these codes. Namely, a very important figure of merit is the error threshold of the topological code, i.e. the critical value of the external noise below which it is possible to perform quantum operations with arbitrary accuracy and time. For toric codes with qubits, the error threshold is very good, about 11% [12]. This value is obtained by mapping the process of error correction to a classical Ising model on a 2D lattice with random bonds. Interestingly enough, this type of mapping can be made more general and applied to TCCs yielding the same error threshold [28] while maintaining enhanced quantum capabilities [29,30]. These results have been confirm...

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Section: Introductionmentioning

confidence: 71%

Generalized toric codes coupled to thermal baths

2012

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“…Decoding algorithms are typically designed to run efficiently on a classical computer, but there is generally no guarantee that the noise and recovery processes should be classically simulable. Because of this, almost all of the sizable research effort on active quantum error correction for topological systems has focused on the case of abelian anyons [2,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], which can be efficiently simulated due to the fact that they cannot be used for quantum computation.…”

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confidence: 99%

Classical simulation of quantum error correction in a Fibonacci anyon code

2017

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Classically simulating the dynamics of anyonic excitations in two-dimensional quantum systems is likely intractable in general because such dynamics are sufficient to implement universal quantum computation. However, processes of interest for the study of quantum error correction in anyon systems are typically drawn from a restricted class that displays significant structure over a wide range of system parameters. We exploit this structure to classically simulate, and thereby demonstrate the success of, an error-correction protocol for a quantum memory based on the universal Fibonacci anyon model. We numerically simulate a phenomenological model of the system and noise processes on lattice sizes of up to 128 × 128 sites, and find a lower bound on the errorcorrection threshold of approximately 0.125 errors per edge, which is comparable to those previously known for abelian and (non-universal) nonabelian anyon models.

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“…are possible while preserving a high error threshold [13] (in comparison to the Kitaev model [1]). This result has been obtained numerically after mapping the error correction (q = 0) for TCCs onto a random three-body Ising model on a triangular lattice [13]; later confirmed by other numerical methods [14,15,16]. It is, however, unclear if quantum computations on TCCs can be performed reliably in the presence of faulty measurements (q = 0).…”

Section: Introductionmentioning

confidence: 61%

Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes

Andrist¹,

Katzgraber²,

Bombin³

et al. 2011

New J. Phys.

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We compute the error threshold of color codes-a class of topological quantum codes that allow a direct implementation of quantum Clifford gateswhen both qubit and measurement errors are present. By mapping the problem onto a statistical-mechanical three-dimensional disordered Ising lattice gauge theory, we estimate via large-scale Monte Carlo simulations that color codes are stable against 4.8(2)% errors. Furthermore, by evaluating the skewness of the Wilson loop distributions, we introduce a very sensitive probe to locate first-order phase transitions in lattice gauge theories.

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Graphical algorithms and threshold error rates for the 2d colour code

Cited by 14 publications

References 21 publications

Generalized toric codes coupled to thermal baths

Generalized toric codes coupled to thermal baths

Classical simulation of quantum error correction in a Fibonacci anyon code

Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes

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