Arithmeticity, thinness and efficiency of qutrit Clifford+T gates

Abstract: The Clifford+T gate set is a topological generating set for PU(2), which has been well-studied from the perspective of quantum computation on a single qubit. The discovery that it generates a full S-arithmetic subgroup of PU(2) has led to a fruitful interaction between quantum computation and number theory, resulting in a proof that words in these gates cover PU(2) in an almost-optimal manner. In this paper we study the analogue gate set for PU(3). We show that in PU(3) the group generated by the Clifford+T gates is not arithmetic - in fact, it is a thin matrix group, namely a Zariski-dense group of infinite index in its ambient S-arithmetic group. On the other hand, we study a recently proposed extension of the Clifford+T gates, called Clifford+D, and show that these do generate a full S-arithmetic subgroup of PU(3), and satisfy a slightly weaker almost-optimal covering property than that of Clifford+T in PU(2). The proofs are different from those for PU(2): while both gate sets act naturally on a (Bruhat-Tits) tree, in PU(2) the generated group acts transitively on the vertices of the tree, and this is a main ingredient in proving both arithmeticity and efficiency. In the PU(3) Clifford+D case the action on the tree is far from being transitive. This makes the proof of arithmeticity considerably harder, and the study of efficiency by automorphic representation theory becomes more involved, and results in a covering rate which differs from the optimal one by a factor of $log_3(105)\approx 4.236$.