We identify a novel qudit gate which we call the $\sqrt[d]{Z}$ gate. This is an alternate generalization of the qutrit $T$ gate to any odd prime dimension $d$, in the $d^{\text{th}}$ 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+$\sqrt[d]{Z}$ gate set, $d$-qubit $W$ states in the qudit $\{ |0\rangle , |1\rangle \}$ 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 $O(\text{log }N)$ 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\rangle $-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.

翻译：我们识别出一种新型量子比特门，称为$\sqrt[d]{Z}$门。这是对qutrit的$T$门在任意奇素数维度$d$上、处于克利福德层级第$d$级的一种替代推广。若特定猜想成立，该门可高效容错实现。利用该门，我们在克利福德+$\sqrt[d]{Z}$门集中确定性构造了量子比特$\{ |0\rangle , |1\rangle \}$子空间上的$d$量子比特$W$态。对于qutrit，这给出了大小为3、6及3的幂次的三量子比特$W$态的确定性和容错构造，其$T$门计数为3。此外，我们调整这些构造以递归地将$W$态规模扩展至任意大小$N$，门复杂度为$O(N)$，电路深度为$O(\text{log }N)$。这针对任意大小的量子比特$W$态以及任意素数$d$维量子比特$W$态（规模为$d$的幂次）均具有确定性。为此，我们设计了任意素数量子比特维度下的$ |0\rangle $受控泡利$X$门和受控Hadamard门构造。这些分解方法在$d>3$的克利福德+$T$门集中尚无精确合成路径，可能具有独立研究价值。