Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits

It is known that the 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 the present paper, we establish an analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of degree 3^k by extending the classical qutrit gates X, CX, and CCX with the Hadamard gate H and the T_k gate T_k = diag(1,ω_k,ω_k²), 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 3ⁿ ×3ⁿ unitary matrix U can be represented by an n-qutrit circuit over the Clifford-cyclotomic gate set of degree 3^k if and only if the entries of U lie in the ring Z[1/3,ω_k].