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Two-dimensional quantum colour codes hold significant promise for quantum error correction, offering advantages such as planar connectivity and low overhead logical gates. Despite their theoretical appeal, the practical deployment of these…

Quantum Physics · Physics 2025-08-22 Stergios Koutsioumpas , Tamas Noszko , Hasan Sayginel , Mark Webster , Joschka Roffe

Fault-tolerant logic gates will consume a large proportion of the resources of a two-dimensional quantum computing architecture. Here we show how to perform a fault-tolerant non-Clifford gate with the surface code; a quantum…

Quantum Physics · Physics 2021-04-20 Benjamin J. Brown

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…

Quantum Physics · Physics 2022-10-12 Pedro Parrado-Rodríguez , Manuel Rispler , Markus Müller

Error correcting codes protect quantum information and form the basis of fault tolerant quantum computing. Leading proposals for fault-tolerant quantum computation require codes with an exceedingly rare property, a transverse non-Clifford…

Quantum Physics · Physics 2015-10-12 Earl T. Campbell

One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the $\mathbb{Z}_2$ surface code. Logical qubits can be encoded in a variety of ways in the surface…

Quantum Physics · Physics 2019-01-11 Ali Lavasani , Maissam Barkeshli

The surface code represents a promising candidate for fault-tolerant quantum computation due to its high error threshold and experimental accessibility with nearest-neighbor interactions. However, current exact surface code threshold…

Quantum Physics · Physics 2025-10-29 SiYing Wang , ZhiXin Xia , Yue Yan , Xiang-Bin Wang

The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new…

Quantum Physics · Physics 2017-04-26 Theodore J. Yoder , Isaac H. Kim

Fast, reliable logical operations are essential for realizing useful quantum computers. By redundantly encoding logical qubits into many physical qubits and using syndrome measurements to detect and correct errors, one can achieve low…

Estimating and reducing the overhead of fault tolerance (FT) schemes is a crucial step toward realizing scalable quantum computers. Of particular interest are schemes based on two-dimensional (2D) topological codes such as the surface and…

Quantum Physics · Physics 2021-06-23 Michael E. Beverland , Aleksander Kubica , Krysta M. Svore

We compute the error threshold of color codes, a class of topological quantum codes that allow a direct implementation of quantum Clifford gates, when both qubit and measurement errors are present. By mapping the problem onto a…

Quantum Physics · Physics 2011-08-10 Ruben S. Andrist , Helmut G. Katzgraber , H. Bombin , M. A. Martin-Delgado

Quantum error correction becomes a practical possibility only if the physical error rate is below a threshold value that depends on a particular quantum code, syndrome measurement circuit, and decoding algorithm. Here we present an…

The constituent parts of a quantum computer are inherently vulnerable to errors. To this end we have developed quantum error-correcting codes to protect quantum information from noise. However, discovering codes that are capable of a…

Quantum Physics · Physics 2016-08-24 Benjamin J. Brown , Naomi H. Nickerson , Dan E. Browne

Color codes present distinct advantages for fault-tolerant quantum computing, such as high encoding rates and the transversal implementation of Clifford gates. However, existing matching-based decoders for the color codes such as the…

Quantum Physics · Physics 2025-11-18 Yantong Liu , Junjie Wu , Lingling Lao

Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but…

Quantum Physics · Physics 2026-01-09 Seok-Hyung Lee , Felix Thomsen , Nicholas Fazio , Benjamin J. Brown , Stephen D. Bartlett

The development of practical, high-performance decoding algorithms reduces the resource cost of fault-tolerant quantum computing. Here we propose a decoder for the surface code that finds low-weight correction operators for errors produced…

Quantum Physics · Physics 2025-02-19 Asmae Benhemou , Kaavya Sahay , Lingling Lao , Benjamin J. Brown

The surface code is designed to suppress errors in quantum computing hardware and currently offers the most believable pathway to large-scale quantum computation. The surface code requires a 2-D array of nearest-neighbor coupled qubits that…

Quantum Physics · Physics 2013-10-04 Austin G. Fowler

The states needed in a quantum computation are extremely affected by decoherence. Several methods have been proposed to control error spreading. They use two main tools: fault-tolerant constructions and concatenated quantum error correcting…

Quantum Physics · Physics 2007-05-23 Pedro J. Salas , Angel L. Sanz

Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes…

Quantum Physics · Physics 2024-02-02 Yugo Takada , Yusaku Takeuchi , Keisuke Fujii

We introduce and analyze a new type of decoding algorithm called General Color Clustering (GCC), based on renormalization group methods, to be used in qudit color codes. The performance of this decoder is analyzed under code capacity…

Quantum Physics · Physics 2017-11-20 Jacob Marks , Tomas Jochym-O'Connor , Vlad Gheorghiu

The surface code is unarguably the leading quantum error correction code for 2-D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and…

Quantum Physics · Physics 2012-05-03 Austin G. Fowler , Adam C. Whiteside , Lloyd C. L. Hollenberg