Quantum Codes from Twisted Unitary $t$-groups

Abstract

We introduce twisted unitary $t$-groups, a generalization of unitary $t$-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary $t$-groups automatically correspond to quantum codes with distance $d=t+1$. By construction these codes have many transversal gates, which naturally do not spread errors and thus are useful for fault tolerance.

Updated to PRL version: changed the title, removed the word "free", added a non-additive codes disclaimer, reorganized the supplemental material

References in corpus (6)

• Restrictions on Transversal Encoded Quantum Gate Sets
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• Unitary designs and codes
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Cited by in corpus (7)

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• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry
• Bosonic quantum Fourier codes
• Quantum Codes and Irreducible Products of Characters
• Synthesis of Single Qutrit Circuits from Clifford+R
• Flexible Fault Tolerant Gate Gadgets