Quantum Complexity of Permutations

TL;DR

This paper investigates the quantum complexity of permutations in the symmetric group using specific generators, establishing quadratic bounds and showing most permutations have high quantum complexity as n grows.

Contribution

It provides explicit quadratic bounds on the quantum complexity of permutations and demonstrates that almost all permutations have high complexity for large n.

Findings

Quadratic lower bound on quantum complexity for certain permutations

Quadratic upper bound applies to all permutations in S_n

Most permutations have high quantum complexity as n approaches infinity

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 $n\rightarrow \infty$.