Quantum Complexity of Permutations

• URL: http://arxiv.org/abs/2207.14102v2
• Date: Fri, 12 Aug 2022 11:45:46 GMT
• Title: Quantum Complexity of Permutations
• Abstract summary: We study quantum complexity of permutations in $S_n$ using $sigma, tau, tau-1$ as logic gates. We show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $nrightarrow infty$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Let $S_n$ be the symmetric group of all permutations of $\{1, \cdots, n\}$ with two generators: the transposition switching $1$ with $2$ and the cyclic permutation sending $k$ to $k+1$ for $1\leq k\leq n-1$ and $n$ to $1$ (denoted by $\sigma$ and $\tau$). In this article, we study quantum complexity of permutations in $S_n$ using $\{\sigma, \tau, \tau^{-1}\}$ as logic gates. We give an explicit construction of permutations in $S_n$ with quadratic quantum complexity lower bound $\frac{n^2-2n-7}{4}$. We also prove that all permutations in $S_n$ have quadratic quantum complexity upper bound $3(n-1)^2$. Finally, we show that almost all permutations in $S_n$ have quadratic quantum complexity lower bound when $n\rightarrow \infty$.

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