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

Fast post processing algorithms for fault tolerant quantum computation using surface codes

Arman Zaribafiyan

Published Jan 1, 2014• 0 citations
Publisher

Abstract

Local architecture of qubits in two dimensional lattices is known as one of the candidates to run fault tolerant quantum computation with high threshold. Topological quantum error correcting codes, such as surface codes, are used to make this architecture robust against various quantum error models. In this research we present a fast, concise, operationally inexpensive and highly parallelizable decoding algorithm for surface codes without using concatenation which has a threshold range of 8.6 % to 10.5 % varying based on a parameter called OMSS. Thanks to parallelization of the proposed algorithm, the time complexity scales logarithmically in the lattice size for small OMSS. This can be compared to the work of \citep{Poulin-2010,Poulin-2010-IEEE} which has a threshold of 8 % for the bit-flip error channel within the same complexity order.

DOI
10.14288/1.0166069

Related papers

Fast fault-tolerant decoder for qubit and qudit surface codes

Fern H. E. Watson, Hussain Anwar, Dan E. Browne

The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than $8%$ can be achieved.

arXiv:1411.3028•Physical Review A•Sep 8, 2015•62 citations

Fault-tolerant quantum computing with three-dimensional surface codes

Michael Vasmer

Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.

•Dec 28, 2019•2 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

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

Quantum memory based on concatenating surface codes and quantum Hamming codes

Menglong Fang, Daiqin Su

Designing quantum error correcting codes that promise a high error threshold, low resource overhead, and efficient decoding algorithms is crucial to achieve large-scale fault-tolerant quantum computation. The concatenated quantum Hamming code is one of the potential candidates that allows for constant space overhead and efficient decoding. We study the concatenation of surface codes with quantum Hamming codes as a quantum memory, and estimate its error threshold, resource overhead, and decoding time. A high error threshold is achieved, which can in principle be pushed up to the threshold of the surface code. Furthermore, the concatenated codes can suppress logical errors to a much lower level than the surface codes, under the assumption of comparable amount of resource overhead. The advantage in suppressing errors starts to show for a quantum memory of intermediate scale. Concatenating surface codes with quantum Hamming codes therefore provides a promising avenue to demonstrate small-scale fault-tolerant quantum circuits in the near future, and also paves a way for large-scale fault-tolerant quantum computation.

arXiv:2407.16176•Physical Review A•Sep 4, 2025•0 citations

Towards Practical Classical Processing for the Surface Code

Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg

The surface code is unarguably the leading quantum error correction code for 2D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an n×n lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in O(n2) average time per round of error correction. We also describe how to parallelize the algorithm to achieve O(1) average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.

arXiv:1110.5133•Physical Review Letters•May 1, 2012•208 citations
Local architecture of qubits in two dimensional lattices is known as one of the candidates to run fault tolerant quantum computation with high threshold. Topological quantum error correcting codes, such as surface codes, are used to make this architecture robust against various quantum error models. In this research we present a fast, concise, operationally inexpensive and highly parallelizable decoding algorithm for surface codes without using concatenation which has a threshold range of 8.6 % to 10.5 % varying based on a parameter called OMSS. Thanks to parallelization of the proposed algorithm, the time complexity scales logarithmically in the lattice size for small OMSS. This can be compared to the work of \citep{Poulin-2010,Poulin-2010-IEEE} which has a threshold of 8 % for the bit-flip error channel within the same complexity order.
The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 88%8 can be achieved.
Quantum computers are far more error-prone than their classical counterparts. Therefore, to build a quantum computer capable of running large-scale quantum algorithms, we must use the techniques of quantum error correction to ensure that the computer produces the correct output even when its components are unreliable. However, the resource requirements of building such a fault-tolerant quantum computer are currently prohibitive. Here, we examine the utility of using three-dimensional (3D) surface codes in a fault-tolerant quantum computer. This family of topological error-correcting codes is a generalization of the well-known 2D surface code to three spatial dimensions. We show that certain 3D surface codes have a transversal logical non-Clifford gate. In a quantum computing architecture, a non-Clifford gate is required to achieve computational universality. Transversal gates do not entangle qubits in different codes, so they are naturally fault tolerant because they do not spread errors. Next, we consider the problem of decoding 3D surface codes. In a quantum error-correcting code, we cannot observe the qubits directly, so we measure parity-check operators to gain information about the state of the code. Decoding is the problem of estimating what error has occurred given a list of unsatisfied parity checks. We observe that 3D surface codes offer asymmetric protection against bit-flip and phase-flip errors, but in both cases, we find that a threshold error rate exists below which we can suppress logical errors by increasing the size of the code. We use our results about logical gates and decoding to propose two fault-tolerant quantum computing architectures that utilize 3D surface codes. Finally, we compare the resource requirements of our architectures with the requirements of leading quantum computing architectures based on topological codes. We find that one of our architectures may be competitive with the leading architectures, depending on the properties of the physical systems used to build the qubits.
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.
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.
Designing quantum error correcting codes that promise a high error threshold, low resource overhead, and efficient decoding algorithms is crucial to achieve large-scale fault-tolerant quantum computation. The concatenated quantum Hamming code is one of the potential candidates that allows for constant space overhead and efficient decoding. We study the concatenation of surface codes with quantum Hamming codes as a quantum memory, and estimate its error threshold, resource overhead, and decoding time. A high error threshold is achieved, which can in principle be pushed up to the threshold of the surface code. Furthermore, the concatenated codes can suppress logical errors to a much lower level than the surface codes, under the assumption of comparable amount of resource overhead. The advantage in suppressing errors starts to show for a quantum memory of intermediate scale. Concatenating surface codes with quantum Hamming codes therefore provides a promising avenue to demonstrate small-scale fault-tolerant quantum circuits in the near future, and also paves a way for large-scale fault-tolerant quantum computation.
The surface code is unarguably the leading quantum error correction code for 2D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an n×n lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in O(n2) average time per round of error correction. We also describe how to parallelize the algorithm to achieve O(1) average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.