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

Abstract

Recent work on fault-tolerant quantum computation making use of topological error correction shows great potential, with the 2d surface code possessing a threshold error rate approaching 1% (NJoP 9:199, 2007), (arXiv:0905.0531). However, the 2d surface code requires the use of a complex state distillation procedure to achieve universal quantum computation. The colour code of (PRL 97:180501, 2006) is a related scheme partially solving the problem, providing a means to perform all Clifford group gates transversally. We review the colour code and its error correcting methodology, discussing one approximate technique based on graph matching. We derive an analytic lower bound to the threshold error rate of 6.25% under error-free syndrome extraction, while numerical simulations indicate it may be as high as 13.3%. Inclusion of faulty syndrome extraction circuits drops the threshold to approximately 0.1%.

20 pages, 19 figures

References in corpus (5)

• Topological Quantum Distillation
• Topological fault-tolerance in cluster state quantum computation
• Error Threshold for Color Codes and Random 3-Body Ising Models
• Threshold error rates for the toric and surface codes
• Latency in local, two-dimensional, fault-tolerant quantum computing

Cited by in corpus (8)

• Surface codes: Towards practical large-scale quantum computation
• Experimental Quantum Computations on a Topologically Encoded Qubit
• Tricolored Lattice Gauge Theory with Randomness: Fault-Tolerance in Topological Color Codes
• Topological color codes on Union Jack lattices: A stable implementation of the whole Clifford group
• Efficient color code decoders in $d\ge 2$ dimensions from toric code decoders
• Quantum error correction with the color-Gottesman-Kitaev-Preskill code
• Minimising surface-code failures using a color-code decoder
• A slightly smaller surface code S gate