Scaling W State Circuits in the qudit Clifford Hierarchy

We identify a novel qudit gate which we refer to as the (d)root Z gate. This is an alternate generalization of the qutrit T gate to any odd prime dimension d, in the d(th) level of the Clifford hierarchy. Using this gate which is efficiently realizable fault-tolerantly should a certain conjecture hold, we deterministically construct in the Clifford+ (d)root Z gate set, d-qubit W states in the qudit {|0 >, |1 >} subspace. For qutrits, we show deterministic and fault-tolerant constructions for the three-qubit W state with T count three, for the six-qubit W state, and for any power-of-three-qubit W state. Furthermore, we adapt these constructions to recursively scale the W state size to arbitrary size N, in O(N) gate count and O(log N) depth. This is moreover deterministic for any size qubit W state, and for any prime d-dimensional qudit W state, size a power of d. For these purposes, we devise constructions of the |0 >-controlled Pauli X gate and the controlled Hadamard gate in any prime qudit dimension. These decompositions, for which exact synthesis is unknown in Clifford+T for d > 3, may be of independent interest.