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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

We present a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic…

Surface and color codes are two forms of topological quantum error correction in two spatial dimensions with complementary properties. Surface codes have lower-depth error detection circuits and well-developed decoders to interpret and…

Quantum Physics · Physics 2016-10-18 Jonathan E. Moussa

We provide a simplified, yet rigorous presentation of the ideas from Bomb\'{i}n's paper "Gauge Color Codes" [arXiv:1311.0879v3]. Our presentation is self-contained, and assumes only basic concepts from quantum error correction. We provide…

Quantum Physics · Physics 2015-04-08 Aleksander Kubica , Michael E. Beverland

Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the…

Quantum Physics · Physics 2013-03-12 Tomas Jochym-O'Connor , Yafei Yu , Bassam Helou , Raymond Laflamme

Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal…

Quantum Physics · Physics 2013-10-14 Markus Grassl , Martin Roetteler

We present a scalable architecture for fault-tolerant topological quantum computation using networks of voltage-controlled Majorana Cooper pair boxes, and topological color codes for error correction. Color codes have a set of transversal…

Mesoscale and Nanoscale Physics · Physics 2017-09-25 Daniel Litinski , Markus S. Kesselring , Jens Eisert , Felix von Oppen

The non-local interactions in several quantum device architectures allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will…

Quantum Physics · Physics 2025-06-05 Shubham P. Jain , Victor V. Albert

The threshold theorem promises a path to fault-tolerant quantum computation, provided the physical error rate is below a critical threshold. While transversal gates efficiently implement logical operations, they propagate errors and can…

Quantum Physics · Physics 2026-04-21 Yichen Xu , Yiqing Zhou , James P. Sethna , Eun-Ah Kim

Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing…

We show how to perform a fault-tolerant universal quantum computation in 2D architectures using only transversal unitary operators and local syndrome measurements. Our approach is based on a doubled version of the 2D color code. It enables…

Quantum Physics · Physics 2015-09-11 Sergey Bravyi , Andrew Cross

In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an…

Quantum Physics · Physics 2021-08-20 Narayanan Rengaswamy , Robert Calderbank , Michael Newman , Henry D. Pfister

A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-K\"onig bound for $n$-dimensional topological stabilizer codes. In this work, we extend…

Quantum Physics · Physics 2026-05-21 Ryohei Kobayashi , Guanyu Zhu , Po-Shen Hsin

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…

Quantum Physics · Physics 2014-02-14 Ashley M. Stephens

Topological quantum codes are intrinsically fault-tolerant to local noise, and underlie the theory of topological phases of matter. We explore geometry to enhance the performance of topological quantum codes by rotating the four dimensional…

Certain quantum codes allow logic operations to be performed on the encoded data, such that a multitude of errors introduced by faulty gates can be corrected. An important class of such operations are {\em transversal}, acting bitwise…

Quantum Physics · Physics 2007-09-11 Bei Zeng , Andrew Cross , Isaac L. Chuang

Steane code is one of the most widely studied quantum error-correction codes, which is a natural choice for fault-tolerant quantum computation (FTQC). However, the original Steane code is not fault-tolerant because the CNOT gates in an…

Quantum Physics · Physics 2024-03-08 Qiqing Xia , Huiqin Xie , Li Yang

Continuous-variable cluster states allow for fault-tolerant measurement-based quantum computing when used in tandem with the Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode. For quad-rail-lattice macronode cluster…

Quantum Physics · Physics 2022-01-05 Blayney W. Walshe , Rafael N. Alexander , Nicolas C. Menicucci , Ben Q. Baragiola

Quantum error-correcting codes with high encoding rate are good candidates for large-scale quantum computers as they use physical qubits more efficiently than codes of the same distance that encode only a few logical qubits. Some logical…

Quantum Physics · Physics 2025-03-26 Theerapat Tansuwannont , Yugo Takada , Keisuke Fujii

Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. Motivated by this goal, we study several classes of fault-tolerant…

Quantum Physics · Physics 2026-02-27 Aranya Chakraborty , Daniel Gottesman