Quantum Codes, Transversal Gates, and Representation Theory

Abstract

Recently an algorithm has been constructed that shows the binary icosahedral group2I together with a T-like gate forms the most efficient single-qubit universal gate set [54]. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. We fill this void by constructing a family of distance d = 3 codes that all implement 2I transversally [39, 42]. To do this, 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. Moreover, these methods produce many other quantum codes with interesting transver- sal gates. In particular, we use our methods to construct families of d ≥ 2 quantum codes realizing nearly all the possible transversal gate groups that are unitary 2-designs or better [41]. We also classify certain groups of two qubit gates that may occur as the transversal gate group of a quantum code with two logical qubits, describing the groups by their entanglement structure [40]. Finally, inspired by unitary 2-design groups, we consider the problem of finding Lie primitive subgroups of a simple Lie group.