A graph-based formalism for surface codes and twists
arXiv:2101.09349 · doi:10.22331/q-2024-07-18-1416
Abstract
Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to any graph embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce some new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width.
47 pages + appendix, 20 figures
References in corpus (28)
- Fault-tolerant quantum computation by anyons
- Anyons in an exactly solved model and beyond
- Unpaired Majorana fermions in quantum wires
- Topological quantum memory
- Models for gapped boundaries and domain walls
- Quantum orders in an exact soluble model
- Topological Order with a Twist: Ising Anyons from an Abelian Model
- Quantum codes on a lattice with boundary
- Tradeoffs for reliable quantum information storage in 2D systems
- Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study
- Correcting coherent errors with surface codes
- Constructions and Noise Threshold of Hyperbolic Surface Codes
- Majorana Fermion Surface Code for Universal Quantum Computation
- Majorana Fermion Codes
- The surface code with a twist
- Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage
- Structure of 2D Topological Stabilizer Codes
- The boundaries and twist defects of the color code and their applications to topological quantum computation
- Algebraic Methods for Quantum Codes on Lattices
- Improved quantum hypergraph-product LDPC codes
- Tradeoffs for reliable quantum information storage in surface codes and color codes
- Noise Thresholds for the [[4, 2, 2]]-concatenated Toric Code
- Low-complexity quantum codes designed via codeword-stabilized framework
- Topological wormholes
- Quantum Error Correction for Complex and Majorana Fermion Qubits
- Modular transformations through sequences of topological charge projections
- On sets of commuting and anticommuting Paulis
- Quantum Lego Expansion Pack: Enumerators from Tensor Networks