Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0703272 · Physical Review A

Optimal resources for topological two-dimensional stabilizer codes: Comparative study

Héctor Bombín, Miguel Angel Martin-Delgado

Published Jul 6, 2007• 177 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.76.012305

Related papers

Rescaling decoder for two-dimensional topological quantum color codes on 4.8.8 lattices

Pedro Parrado-Rodríguez, Manuel Rispler, Markus Müller

Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes, are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for two-dimensional topological color codes on the 4.8.8 lattice (also known as the square-octagon code), by building on the work of Sarvepalli and Raussendorf [Phys. Rev. A 85, 022317 (2012)], for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.

arXiv:2112.09584•Physical Review A•Sep 26, 2022•5 citations

Efficient fault-tolerant decoding of topological color codes

Ashley M. Stephens

Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.

arXiv:1402.3037•Feb 13, 2014•15 citations

Hardness of approximation for minimum-weight decoding of two-dimensional topological quantum codes

Louay Bazzi, Georges Khater

Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the $X$- and $Z$-syndromes. For color codes under independent $X$- and $Z$-error models, we consider Separate Minimum-Weight decoding. Assuming $P\neq NP$, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the $4.8.8$ color code on the torus, there exists a constant $c>0$ such that no polynomial-time algorithm can always produce a solution whose weight is within $cN^{1/14}$ of the optimum, where $N$ is the number of qubits, unless $P=NP$. For the planar surface code, we obtain an $Ω(N^{1/18})$ gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT. Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.

quant-pharXiv:2608.17109•Aug 19, 2026•0 citations

Encoder Circuit For Surface Code using Measurement-Based Quantum Computing Model

Priyam Srivastava, Vaibhav Katyal, Ankur Raina

Surface codes are one of the most important topological stabilizer codes in the theory of quantum error correction. In this paper, we provide an efficient way to obtain surface codes through Measurement-based quantum computation (MBQC) using cluster state as the resource state. Simple twodimensional surface codes are studied and analyzed using stabilizer formalism. We also present an algorithm to computationally obtain the stabilizer of the surface codes, through which we later determine the distance of the codes. We note the difference in the stabilizers of the surface codes obtained by Fowler et al. wherein they used CNOT entangling operation to create the resource state as opposed to the cluster state which is formed using CZ entangling operation. We provide a theoretical calculation to understand this difference. The obtained surface codes can be used practically as an encoder circuit to encode one logical qubit.

arXiv:2306.10267•Jun 17, 2023•0 citations

Optimal error correction in topological subsystem codes

Ruben S. Andrist, Héctor Bombín, Helmut G. Katzgraber +1 more

A promising approach to overcome decoherence in quantum computing schemes is to perform active quantum error correction using topology. Topological subsystem codes incorporate both the benefits of topological and subsystem codes, allowing for error syndrome recovery with only 2-local measurements in a two-dimensional array of qubits. We study the error threshold for topological subsystem color codes under very general external noise conditions. By transforming the problem into a classical disordered spin model, we estimate using Monte Carlo simulations that topological subsystem codes have an optimal error tolerance of $5.5(2)%$. This means there is ample space for improvement in existing error-correcting algorithms that typically find a threshold of approximately $2%$.

arXiv:1204.1838•Physical Review A•May 14, 2012•23 citations

Locality-preserving logical operators in topological stabilizer codes

Paul Webster, Stephen D. Bartlett

Locality-preserving logical operators in topological codes are naturally fault tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabilizer codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including color codes and toric codes, as well as more general codes such as Abelian quantum double models and codes with fermionic excitations in more than two dimensions.

arXiv:1709.00020•Physical Review A•Jan 24, 2018•36 citations
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:=n∕d2C\ensuremath{\mathrel{:=}}n∕{d}^{2}C\ensuremath:=n∕d2 for surface codes Cs{C}_{s}Cs​ and color codes Cc{C}_{c}Cc​ in several instances. On the torus, typical values are Cs=2{C}_{s}=2Cs​=2 and Cc=3∕2{C}_{c}=3∕2Cc​=3∕2, but we find that the optimal values are Cs=1{C}_{s}=1Cs​=1 and Cc=9∕8{C}_{c}=9∕8Cc​=9∕8. For planar codes, a typical value is Cs=2{C}_{s}=2Cs​=2, while we find that the optimal values are Cs=1{C}_{s}=1Cs​=1 and Cc=3∕4{C}_{c}=3∕4Cc​=3∕4. In general, a color code encodes twice as many logical qubits as does a surface code.
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes, are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for two-dimensional topological color codes on the 4.8.8 lattice (also known as the square-octagon code), by building on the work of Sarvepalli and Raussendorf [Phys. Rev. A 85, 022317 (2012)], for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.
Topological color codes defined by the 4.8.8 semiregular lattice feature geometrically local check operators and admit transversal implementation of the entire Clifford group, making them promising candidates for fault-tolerant quantum computation. Recently, several efficient algorithms for decoding the syndrome of color codes were proposed. Here, we modify one of these algorithms to account for errors affecting the syndrome, applying it to the family of triangular 4.8.8 color codes encoding one logical qubit. For a three-dimensional bit-flip channel, we report a threshold error rate of 0.0208(1), compared with 0.0305(4) previously reported for an integer-program-based decoding algorithm. When we account for circuit details, this threshold is reduced to 0.00143(1) per gate, compared with 0.00672(1) per gate for the surface code under an identical noise model.
Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the XXX- and ZZZ-syndromes. For color codes under independent XXX- and ZZZ-error models, we consider Separate Minimum-Weight decoding. Assuming P≠NPP\neq NPP=NP, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the 4.8.84.8.84.8.8 color code on the torus, there exists a constant c>0c>0c>0 such that no polynomial-time algorithm can always produce a solution whose weight is within cN1/14cN^{1/14}cN1/14 of the optimum, where NNN is the number of qubits, unless P=NPP=NPP=NP. For the planar surface code, we obtain an Ω(N1/18)Ω(N^{1/18})Ω(N1/18) gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT. Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.
Surface codes are one of the most important topological stabilizer codes in the theory of quantum error correction. In this paper, we provide an efficient way to obtain surface codes through Measurement-based quantum computation (MBQC) using cluster state as the resource state. Simple twodimensional surface codes are studied and analyzed using stabilizer formalism. We also present an algorithm to computationally obtain the stabilizer of the surface codes, through which we later determine the distance of the codes. We note the difference in the stabilizers of the surface codes obtained by Fowler et al. wherein they used CNOT entangling operation to create the resource state as opposed to the cluster state which is formed using CZ entangling operation. We provide a theoretical calculation to understand this difference. The obtained surface codes can be used practically as an encoder circuit to encode one logical qubit.
A promising approach to overcome decoherence in quantum computing schemes is to perform active quantum error correction using topology. Topological subsystem codes incorporate both the benefits of topological and subsystem codes, allowing for error syndrome recovery with only 2-local measurements in a two-dimensional array of qubits. We study the error threshold for topological subsystem color codes under very general external noise conditions. By transforming the problem into a classical disordered spin model, we estimate using Monte Carlo simulations that topological subsystem codes have an optimal error tolerance of 5.5(2)5.5(2)%5.5(2). This means there is ample space for improvement in existing error-correcting algorithms that typically find a threshold of approximately 22%2.
Locality-preserving logical operators in topological codes are naturally fault tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabilizer codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including color codes and toric codes, as well as more general codes such as Abelian quantum double models and codes with fermionic excitations in more than two dimensions.