Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes

By H. Bombin. 2013

267 citations recorded by the source corpus.

Color codes are topological stabilizer codes with unusual transversality properties. Here I show that their group of transversal gates is optimal and only depends on the spatial dimension, not the local geometry. I also introduce a generalized, subsystem version of color codes. In 3D they allow the transversal implementation of a universal set of gates by gauge fixing, while error-dectecting measurements involve only four or six qubits.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c5cc0e40ba920faff466cf7a09f247a74c4f7e50

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In Computational Theory and Mathematics

Citations in the corpus

Quantum advantage with shallow circuits (2017, background): “⊥ Then for any valid secret string z , and any y ∈ L q , z ⊕ y is also a valid secret string (since 2( z ⊕ y ) T x = 2 z T x + 2 y T x = 2 z T x for x ∈ L q ).”
Magic state cultivation: growing T states as cheap as CNOT gates (2024, background): “See [Bom13] for more discussion of adjusting signs to ensure gates are transversal.”
Logical quantum processor based on reconfigurable atom arrays (2023, methodology): “This transversal gate set can realize any Clifford circuit fault-tolerantly.”
Noise tailoring for scalable quantum computation via randomized compiling (2015, background): “However, recent ‘nogo’ theorems establish that for any fault-tolerant scheme, there exist some operations that cannot be performed in such a manner [30, 31] and so must be implemented via other means, such as magic-state injection [32] or gauge fixing [33].”
Quantum error correction for quantum memories (2013, unknown): “A possibly even more attractive family of 3D codes are the gauge color codes (Bombin, 2013) for which the Hadamard gate is transversal.”
A quantum processor based on coherent transport of entangled atom arrays (2021, unknown): “In addition, these techniques also enable implementation of higher-dimensional or nonlocal error correcting codes with more favorable properties [Bombin2015a, Breuckmann2021].”
Single-Shot Fault-Tolerant Quantum Error Correction (2014, methodology): “Once the gauge fixing is completed, a transversal gate can be applied that completes the universal gate set together with the Hadamard and CNOT [14].”