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Quantum computers promise to solve problems that are intractable for classical computers, but qubits are vulnerable to many sources of error, limiting the depth of the circuits that can be reliably executed on today's quantum hardware.…
We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in suitable LDPC quantum codes using transversal…
The topological color code and the toric code are two leading candidates for realizing fault-tolerant quantum computation. Here we show that the color code on a $d$-dimensional closed manifold is equivalent to multiple decoupled copies of…
Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically…
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…
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…
Color codes are promising quantum error correction (QEC) codes because they have an advantage over surface codes in that all Clifford gates can be implemented transversally. However, thresholds of color codes under circuit-level noise are…
Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity,…
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…
One of the leading quantum computing architectures is based on the two-dimensional (2D) surface code. This code has many advantageous properties such as a high error threshold and a planar layout of physical qubits where each physical qubit…
We investigate stabilizer codes with carrier qudits of equal dimension $D$, an arbitrary integer greater than 1. We prove that there is a direct relation between the dimension of a qudit stabilizer code and the size of its corresponding…
The development of quantum codes with good error correction parameters and useful sets of transversal gates is a problem of major interest in quantum error-correction. Abundant prior works have studied transversal gates which are restricted…
We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose codespace is stabilized by Clifford operators. We formulate a…
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…
Recent hardware demonstrations and advances in circuit compilation have made quantum computing with higher-dimensional systems (qudits) on near-term devices an attractive possibility. Some problems have more natural or optimal encodings…
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…
The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and 'nearly diagonal' semi-Clifford gates are particularly important:…
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…
Topological stabilizer codes with different spatial dimensions have complementary properties. Here I show that the spatial dimension can be switched using gauge fixing. Combining 2D and 3D gauge color codes in a 3D qubit lattice,…
Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing…