Inverting a permutation is as hard as unordered search

TL;DR

This paper establishes the quantum query complexity of inverting permutations as asymptotically equivalent to unordered search, showing both problems share a complexity of Theta(sqrt(n)).

Contribution

It introduces a reduction that directly relates permutation inversion to unordered search, bypassing previous complex proof techniques.

Findings

Quantum query complexity of permutation inversion is Theta(sqrt(n)).

Permutation inversion and unordered search are essentially equivalent in quantum complexity.

The reduction simplifies the derivation of lower bounds on quantum query complexity.

Abstract

We show how an algorithm for the problem of inverting a permutation may be used to design one for the problem of unordered search (with a unique solution). Since there is a straightforward reduction in the reverse direction, the problems are essentially equivalent. The reduction we present helps us bypass the hybrid argument due to Bennett, Bernstein, Brassard, and Vazirani (1997) and the quantum adversary method due to Ambainis (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations. It directly implies that the quantum query complexity of the problem is asymptotically the same as that for unordered search, namely in Theta(sqrt(n)).