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 Z1/2, ₖ, where k is a positive integer that depends on the gate set and ₖ is a primitive 2ᵏ-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 3ᵏ by extending the classical qutrit gates X, CX, and CCX with the Hadamard gate H and the Tₖ gate Tₖ=diag (1, ₖ, ₖ²), where ₖ is a primitive 3ᵏ-th root of unity. This gate set is equivalent to the qutrit Toffoli+Hadamard gate set when k=1, and to the qutrit Clifford+Tₖ 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 3ᵏ if and only if the entries of U lie in the ring Z1/3, ₖ.