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Abstract In a recent paper, we defined a type of weighted unitary design called a twisted unitary 1-group and showed that such a design automatically induced error-detecting quantum codes. We also showed that twisted unitary 1-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of GAP computations and results from the mathematics literature on irreducible products of characters, we identify many new non-trivial quantum codes with unusual transversal gates. Transversal gates are of significant interest to the quantum information community for their central role in fault tolerant quantum computing. Most unitary$$\text {t}$$ $\text{t}$-designs have never been realized as the transversal gate group of a quantum code. We, for the first time, find nontrivial quantum codes realizing nearly every finite group which is a unitary 2-design or better as the transversal gate group of some error-detecting quantum code. 

Recently, an algorithm has been constructed that shows that the binary icosahedral group 2I together with a T-like gate forms the most efficient single-qubit universal gate set. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. However, no such code has ever been demonstrated in the literature. We fill this void by constructing a family of distance d ¼ 3 codes that all implement 2I transversally. A surprising feature of this family is that the codes can be deduced entirely from symmetry considerations that only 2I affords. 

We develop finite-dimensional versions of the quantum error-correcting codes proposed by Albert, Covey, and Preskill (ACP) for continuous-variable quantum computation on configuration spaces with non-Abelian symmetry groups. Our codes can be realized by a charged particle in a Landau level on a spherical geometry, in contrast to the planar Landau level realization of the qudit codes of Gottesman, Kitaev, and Preskill (GKP), or more generally by spin coherent states. Our quantum error-correction scheme is inherently approximate, and the encoded states may be easier to prepare than those of GKP or ACP.