Papers › Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
Andrew N. Glaudell, Neil J. Ross, John van de Wetering, Lia Yeh
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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,ζₖ], where k is a positive integer that depends on the gate set and ζₖ 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,ωₖ, ωₖ²), where ωₖ 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 ℤ[1/3,ωₖ].
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