scholarly journals A renormalization group decoding algorithm for topological quantum codes

2010 ◽  
Author(s):  
Guillaume Duclos-Cianci ◽  
David Poulin
Keyword(s):  
Renormalization Group ◽  
Quantum Codes ◽  
Decoding Algorithm ◽  
Topological Quantum

scholarly journals Topological quantum codes from self-complementary self-dual graphs

2015 ◽  
Vol 14 (11) ◽  
pp. 4057-4066 ◽  
Author(s):  
Avaz Naghipour ◽  
Mohammad Ali Jafarizadeh ◽  
Sedaghat Shahmorad
Keyword(s):  
Quantum Codes ◽  
Dual Graphs ◽  
Topological Quantum

Families of codes of topological quantum codes from tessellations tessellations {4i+2,2i+1}, {4i,4i}, {8i-4,4} and ${12i-6,3}

2014 ◽  
Vol 14 (15&16) ◽  
pp. 1424-1440
Author(s):  
Clarice Dias de Albuquerque ◽  
Reginaldo Palazzo Jr. ◽  
Eduardo Brandani da Silva
Keyword(s):  
Specific Property ◽  
Bipartite Graphs ◽  
Quantum Codes ◽  
Singleton Bound ◽  
Topological Quantum ◽  
Complete Bipartite Graphs

In this paper we present some classes of topological quantum codes on surfaces with genus $g \geq 2$ derived from hyperbolic tessellations with a specific property. We find classes of codes with distance $d = 3$ and encoding rates asymptotically going to 1, $\frac{1}{2}$ and $\frac{1}{3}$, depending on the considered tessellation. Furthermore, these codes are associated with embedding of complete bipartite graphs. We also analyze the parameters of these codes, mainly its distance, in addition to show a class of codes with distance 4. We also present a class of codes achieving the quantum Singleton bound, possibly the only one existing under this construction.

New classes of TQC associated with self-dual, quasi self-dual and denser tessellations

2010 ◽  
Vol 10 (11&12) ◽  
pp. 956-970
Author(s):  
C. D. Albuquerque ◽  
R. Palazzo Jr. ◽  
E. B. Silva
Keyword(s):  
Minimum Distance ◽  
Bipartite Graphs ◽  
The Self ◽  
Quantum Codes ◽  
Complete Graphs ◽  
Mathematical Structures ◽  
Topological Quantum ◽  
Complete Bipartite Graphs ◽  
Compact Surfaces ◽  
The One

In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.

scholarly journals Complementary projection defects and decomposition

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Fabian Klos ◽  
Daniel Roggenkamp
Keyword(s):  
Renormalization Group ◽  
Fixed Points ◽  
Correlation Functions ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Renormalization Group Flows ◽  
Topological Quantum Field Theories

Abstract As put forward in [1] topological quantum field theories can be projected using so-called projection defects. The projected theory and its correlation functions can be completely realized within the unprojected one. An interesting example is the case of topological quantum field theories associated to IR fixed points of renormalization group flows, which by this method can be realized inside the theories associated to the UV. In this note we show that projection defects in triangulated defect categories (such as defects in 2d topologically twisted $$ \mathcal{N} $$ N = (2, 2) theories) always come with complementary projection defects, and that the unprojected theory decomposes into the theories associated to the two projection defects. We demonstrate this in the context of Landau-Ginzburg orbifold theories.

scholarly journals Fault-tolerant renormalization group decoder for abelian topological codes

2014 ◽  
Vol 14 (9&10) ◽  
pp. 721-740
Author(s):  
Guillaume Duclos-Cianci ◽  
David Poulin
Keyword(s):  
Renormalization Group ◽  
Failure Probability ◽  
Perfect Matching ◽  
Fault Tolerant ◽  
Three Dimensional ◽  
Space Time ◽  
Two Dimensions ◽  
Decoding Algorithm ◽  
Bit Flip ◽  
Toric Code

We present a three-dimensional generalization of a renormalization group decoding algorithm for topological codes with Abelian anyonic excitations that we introduced for two dimensions in \cite{DP09a,DP10a}. We also provide a complete detailed description of the structure of the algorithm, which should be sufficient for anyone interested in implementing it. This 3D implementation extends our previous 2D algorithm by incorporating a failure probability of the syndrome measurements, i.e., it enables fault-tolerant decoding. We report a fault-tolerant storage threshold of $\sim1.9(4)\%$ for Kitaev's toric code subject to a 3D bit-flip channel (i.e. including imperfect syndrome measurements). This number is to be compared with the $2.9\%$ value obtained via perfect matching \cite{H04a}. The 3D generalization inherits many properties of the 2D algorithm, including a complexity linear in the space-time volume of the memory, which can be parallelized to logarithmic time.

scholarly journals Cellular automaton decoders for topological quantum codes with noisy measurements and beyond

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael Vasmer ◽  
Dan E. Browne ◽  
Aleksander Kubica
Keyword(s):  
Cellular Automaton ◽  
Error Correction ◽  
Measurement Errors ◽  
High Performance ◽  
Three Dimensional ◽  
Noise Model ◽  
Quantum Codes ◽  
Correction Procedure ◽  
Topological Quantum ◽  
Wide Range

AbstractWe propose an error correction procedure based on a cellular automaton, the sweep rule, which is applicable to a broad range of codes beyond topological quantum codes. For simplicity, however, we focus on the three-dimensional toric code on the rhombic dodecahedral lattice with boundaries and prove that the resulting local decoder has a non-zero error threshold. We also numerically benchmark the performance of the decoder in the setting with measurement errors using various noise models. We find that this error correction procedure is remarkably robust against measurement errors and is also essentially insensitive to the details of the lattice and noise model. Our work constitutes a step towards finding simple and high-performance decoding strategies for a wide range of quantum low-density parity-check codes.

scholarly journals Scalable characterization of localizable entanglement in noisy topological quantum codes

2020 ◽  
Vol 22 (5) ◽  
pp. 053038
Author(s):  
David Amaro ◽  
Markus Müller ◽  
Amit Kumar Pal
Keyword(s):  
Quantum Codes ◽  
Topological Quantum
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