Almost-Optimal Computational Basis State Transpositions

TL;DR

This paper presents an explicit method for transposing any n-qubit computational basis state with nearly optimal gate complexity, matching theoretical lower bounds and advancing quantum circuit design efficiency.

Contribution

It introduces a construction achieving a(n) gates for basis state transpositions, nearly matching the proven lower bounds for such operations.

Findings

Transpositions can be performed with a(n) gates.

The method nearly matches the theoretical lower bound.

The results optimize quantum circuit complexity for basis state manipulations.

Abstract

We give an explicit construction to perform any $n$-qubit computational basis state transposition using $\Theta(n)$ gates. This nearly coincides with the lower bound $\Omega(n/\log(nd))$ on worst-case and average-case gate complexity to perform transpositions using a $d$-element gate-set, which we also prove.