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Optimal resources for topological two-dimensional stabilizer codes: Comparative study
Open accessAbstract
We study the resources needed to construct topological two-dimensional 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 code. We compute the topological error-correcting rate $C\ensuremath{\mathrel{:=}}n∕{d}^{2}$ 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 many logical qubits as does a surface code.
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“Depending on the scheme being used, typically between 7 [55] and 9 [56][57][58] physical qubits are needed to represent the state of one qubit.”
From Neutral atom quantum computing hardware: performance and end-user perspective (2023) · cites this paper as [57]
In the citing paper
“Colour codes10,35 allow for more efficient logical operations and require fewer qubits than the surface code to encode a logical qubit at a fixed code distance36.”
From Scaling and logic in the colour code on a superconducting quantum processor (2025) · cites this paper as [36]
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Metadata
- Authors
- Bombin, H., & Martin-Delgado, M. A.
- Affiliations
- Universidad Complutense de Madrid (ES)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2007-07-06
- Volume
- 76
- Issue
- 1
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2026181396
- Cited by
- 178
- References
- 21
- Citations by year
- 2026: 13 · 2025: 35 · 2024: 18 · 2023: 13 · 2022: 13 · 2021: 10 · 2020: 5 · 2019: 13 · 2018: 13 · 2017: 10 · 2016: 7 · 2015: 4 · 2014: 5 · 2013: 5 · 2012: 5
- FWCI
- 2.76
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- surface code, color codes, two-dimensional codes, logical qubit encoding, toric code, quantum error correction, resource efficiency
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