Quantum Circuit for Calculating Mobius-like Transforms Via Grover-like Algorithm

TL;DR

This paper introduces quantum circuits for efficiently computing Mobius-like transforms, achieving quadratic speedup over classical algorithms by leveraging Grover-like quantum algorithms.

Contribution

The paper presents the first quantum algorithms for Mobius-like transforms, reducing computational complexity from exponential to square root scale.

Findings

Quantum circuits for Mobius transforms are developed.

Quantum algorithms outperform classical by reducing complexity from O(2^n) to O(√2^n).

Demonstrates potential for quantum speedup in related linear transform computations.

Abstract

In this paper, we give quantum circuits for calculating two closely related linear transforms that we refer to jointly as Mobius-like transforms. The first is the Mobius transform of a function $f^{-}(S^-)\in \mathbb{C}$, where $S^-\subset \{0,1,\ldots,n-1\}$. The second is a marginal of a probability distribution $P(y^n)$, where $y^n\in Bool^n$. Known classical algorithms for calculating these Mobius-like transforms take ${\cal O}(2^n)$ steps. Our quantum algorithm is based on a Grover-like algorithm and it takes ${\cal O}(\sqrt{2^n})$ steps.