# How to generate the following $n$-level $n$-particle singlet state?

Could you please give a direction/reference towards realising the following using any set of realisable quantum gates

$$\boxed{|S_{n}\rangle = \frac{1} {\sqrt{n!}} \sum_{S\in P_n^{n}} ( \,-1) \,^{\Gamma(S)}|s_{0}\rangle |s_{1}\rangle ....|s_{n-1}\rangle}$$

Here $$P_n^{n}$$ is the set of all permutations of $$Z_n := \{0,1,··· ,n−1\}$$, $$S$$ is a permutation (or sequence) in the form $$S = s_0 s_1 ···s_{n−1}$$. $$\Gamma(S)$$, named inverse number, is defined as the number of transpositions of pairs of elements of $$S$$ that must be composed to place the elements in canonical order, $$012 · · · n−1$$.

• It is an interesting question. May I ask what does the state represent? – Martin Vesely Dec 1 '19 at 7:20
• @MartinVesely The state enumerates all permutations of $Z_{n}$. I don't know the significance of inverse number here. – qcnoob Dec 1 '19 at 9:30
• – glS Dec 2 '19 at 11:54
• just want to point out that "any set of realisable quantum gates" is not a sensible requirement, otherwise I can just answer this by saying "use any gate that sends $|0\cdots 0\rangle$ to $|S_n\rangle$", which is a realisable quantum gate (any gate is "realisable"). You are probably thinking of using as "realisable quantum gates" single-qubit gates and CNOTs, or something similar – glS Dec 2 '19 at 11:56
• I just realised that this question is an exact duplicate of quantumcomputing.stackexchange.com/q/8668/55 – glS Dec 2 '19 at 15:07