Scaling W state circuits in the qudit Clifford hierarchy
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We identify a novel qudit gate which we call the √(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+√(Z) gate set, d-qubit W states in the qudit { |0⟩ , |1⟩} subspace. For qutrits, this gives deterministic and fault-tolerant constructions for the qubit W state of sizes three with T count 3, six, and powers of three. 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.
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