Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

It is known that the unitary matrices that can be exactly represented by a multiqubit circuit over the Toffoli+Hadamard, Clifford+T, or, more generally, Clifford-cyclotomic gate set are precisely the unitary matrices with entries in the ring ℤ[1/2,ζ_k], where k is a positive integer that depends on the gate set and ζ_k is a primitive 2^k-th root of unity. In this paper, we establish the analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of order 3^k by extending the classical qutrit gates X, CX, and Toffoli with the Hadamard gate H and the single-qutrit gate T_k=diag(1,ω_k, ω_k^2), where ω_k is a primitive 3^k-th root of unity. This gate set is equivalent to the qutrit Toffoli+Hadamard gate set when k=1, and to the qutrit Clifford+T_k gate set when k>1. We then prove that a 3^n× 3^n unitary matrix U can be represented by an n-qutrit circuit over the Clifford-cyclotomic gate set of order 3^k if and only if the entries of U lie in the ring ℤ[1/3,ω_k].