Qubit T Gate Magic State Stabilizer Rank

Universal quantum computation can be achieved using the Clifford+T gateset where tensoring T gate magic states extends circuits to quantum universality. The T gate magic state’s stabilizer decomposition is often used to determine the classical simulation cost of this gateset since stabilizer state inner products produce useful Gauss sum primitives. Unfortunately, the T gate magic state stabilizer rank scaling, or more generally its Gauss sum rank scaling, is not formally known and has only been found numerically for up to seven qubits. We show an iterative dependence on reductions of the Gauss sum rank for the T gate magic state. This introduces the first algebraic formalism to explain and find the T gate Gauss sum rank scaling, which is a lower bound on its stabilizer rank scaling. We show that this bound is tight for currently known values of the T gate magic state stabilizer rank and then improve the asymptotic bound further.