Abstract
Qudit toric codes are a natural higher-dimensional generalization of the well-studied qubit toric code. However, standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyse their error correction thresholds and efficiency. The first decoder is a generalization of a ‘hard-decisions’ decoder due to Bravyi and Haah (arXiv:1112.3252). We modify this decoder to overcome a percolation effect which limits its threshold performance for many-level quantum systems. The second decoder is a generalization of a ‘soft-decisions’ decoder due to Poulin and Duclos-Cianci (2010 Phys. Rev. Lett. 104 050504), with a small cell size to optimize the efficiency of implementation in the high dimensional case. In each case, we estimate thresholds for the uncorrelated bit-flip error model and provide a comparative analysis of the performance of both these approaches to error correction of qudit toric codes.
Highlights
The study of quantum error correction and fault-tolerant quantum computation [1,2,3] for qubit systems is very well established, and the combination of topological codes [4] for robust error tolerance and magic state distillation [5] for universality has become a leading framework for fault-tolerant quantum computation [6,7,8,9].In contrast to qubit systems, fault-tolerant quantum computation with systems of dimension d higher than 2 are less well understood
In the limit of high dimensions this decoder reaches a saturating threshold of about 18%. We discover that this behaviour is due to a percolation effect, where thresholds achieved by this decoder are always upper bounded by the syndrome percolation threshold
We have summarized the thresholds obtained by our HDRG, enhanced-HDRG and SDRG decoders in figure 2
Summary
The algebra of the stabilizer group of the qudit toric code defined on the edges of a lattice is captured by homology. We will not give a detailed and formal introduction to homology here, but instead introduce the key concepts needed for understanding homology in toric codes, in terminology accessible to the general physicist. Homological equivalence of string-like operators supported on the edges of the toric code lattice, as used in the main text and in the literature, correspond to equivalence under multiplication by stabilizer operators—and two homologically equivalent operators are logically equivalent on the code-space of the toric code. While this definition will suffice for some readers, we invite those who would like a fuller introduction to homology to read on. For a formal introduction to homology as used in the topological code literature which does not take excursions into more general algebraic topology, we recommend chapter 3 of [63] or chapter 5 of [64]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
