Tradeoffs for reliable quantum information storage in surface codes and color codes

Abstract

The family of hyperbolic surface codes is one of the rare families of quantum LDPC codes with non-zero rate and unbounded minimum distance. First, we introduce a family of hyperbolic color codes. This produces a new family of quantum LDPC codes with non-zero rate and with minimum distance logarithmic in the blocklength. Second, we study the tradeoff between the length n, the number of encoded qubits k and the distance d of surface codes and color codes. We prove that kd^2 is upper bounded by C(log k)^2n, where C is a constant that depends only on the row weight of the parity-check matrix. Our results prove that the best asymptotic minimum distance of LDPC surface codes and color codes with non-zero rate is logarithmic in the length.

10 pages

References in corpus (2)

• Topological Quantum Distillation
• Homological Error Correction: Classical and Quantum Codes

Cited by in corpus (26)

• Roads towards fault-tolerant universal quantum computation
• Almost-linear time decoding algorithm for topological codes
• Constructions and Noise Threshold of Hyperbolic Surface Codes
• Constant overhead quantum fault-tolerance with quantum expander codes
• A theory of single-shot error correction for adversarial noise
• Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage
• Quantum Expander Codes
• The boundaries and twist defects of the color code and their applications to topological quantum computation
• Combining hard and soft decoders for hypergraph product codes
• The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure
• Connectivity constrains quantum codes
• Limits on the storage of quantum information in a volume of space
• Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes
• Efficient color code decoders in $d\ge 2$ dimensions from toric code decoders
• Numerical and analytical bounds on threshold error rates for hypergraph-product codes
• A decoding algorithm for CSS codes using the X/Z correlations
• The role of entropy in topological quantum error correction
• Quantum Pin Codes
• Correcting spanning errors with a fractal code
• Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
• Minimal distances for certain quantum product codes and tensor products of chain complexes
• Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance
• A graph-based formalism for surface codes and twists
• Finite-rate sparse quantum codes aplenty
• Homology-changing percolation transitions on finite graphs
• A note on the minimum distance of quantum LDPC codes