Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates

Abstract

With respect to the transversal gate group (an invariant of quantum codes), we demonstrate that non-additive codes can outperform stabilizer codes. We do this by constructing spin codes that correspond to permutation-invariant multiqubit codes that can implement generalized phase gates transversally. Of particular note, we construct permutation-invariant quantum codes that implement a transversal $T$ gate using fewer qubits and with a better minimum distance than is possible with the best known stabilizer codes.

arXiv commentChanged title and abstract to more accurately reflect the contents of the paper. Changed to IEEE transactions formatting. Added more complete proofs of all major results. Added more references

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• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry
• Qudit encoding in Rydberg blockaded arrays of atoms
• Flexible Fault Tolerant Gate Gadgets