Scaling W state circuits in the qudit Clifford hierarchy

Lia Yeh

doi:10.1145/3594671.3594687

paper

Scaling W state circuits in the qudit Clifford hierarchy

arXiv:2304.12504 · doi:10.1145/3594671.3594687

Abstract

We identify a novel qudit gate which we call the $d Z$ 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+ $d Z$ 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.

11 pages. To be in published in proceedings of The First International Workshop on the Art, Science, and Engineering of Quantum Programming (QP2023)

Scaling W state circuits in the qudit Clifford hierarchy

Abstract

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