Transversal Clifford gates on folded surface codes

Abstract

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 correct errors, while color codes have transversal Clifford gates and better code efficiency in the number of physical qubits needed to achieve a given code distance. A formal equivalence exists between color codes and folded surface codes, but it does not guarantee the transferability of any of these favorable properties. However, the equivalence does imply the existence of constant-depth circuit implementations of logical Clifford gates on folded surface codes. We achieve and improve this result by constructing two families of folded surface codes with transversal Clifford gates. This construction is presented generally for qudits of any dimension. The specific application of these codes to universal quantum computation based on qubit fusion is also discussed.

14 pages, 10 figures, major revision and expansion of the original version

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• Magic Mirror on the Wall, How to Benchmark Quantum Error Correction Codes, Overall ?
• Unconditionally separating noisy ${\mathsf{QNC}}^0$ from bounded polynomial threshold circuits of constant depth
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• Lattice Surgery Compilation Beyond the Surface Code
• Efficient simulation of logical magic state preparation protocols
• Interactive quantum advantage with noisy, shallow Clifford circuits