Optimal Resources for Topological 2D Stabilizer Codes: Comparative Study

Abstract

We study the resources needed to construct topological 2D stabilizer codes as a way to estimate in part their efficiency and this leads us to perform a comparative study of surface codes and color codes. This study clarifies the similarities and differences between these two types of stabilizer codes. We compute the error correcting rate $C:=n/d^2$ for surface codes $C_s$ and color codes $C_c$ in several instances. On the torus, typical values are $C_s=2$ and $C_c=3/2$, but we find that the optimal values are $C_s=1$ and $C_c=9/8$. For planar codes, a typical value is $C_s=2$, while we find that the optimal values are $C_s=1$ and $C_c=3/4$. In general, a color code encodes twice as much logical qubits as a surface code does.

revtex, 6 pages, 7 figures

References in corpus (14)

• Fault-tolerant quantum computation by anyons
• Topological quantum memory
• String-net condensation: A physical mechanism for topological phases
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• Fault-tolerant quantum computation with high threshold in two dimensions
• Topological Quantum Distillation
• Topological fault-tolerance in cluster state quantum computation
• Quantum codes on a lattice with boundary
• Topological characterization of quantum phase transitions in a S=1/2 spin model
• Topological Computation without Braiding
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Adiabatic Preparation of Topological Order
• Homological Error Correction: Classical and Quantum Codes
• Topological Quantum Error Correction with Optimal Encoding Rate

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