Grover's algorithm is an approximation of imaginary-time evolution

TL;DR

This paper demonstrates that Grover's algorithm can be viewed as an approximation of imaginary-time evolution, offering a unified geometric and thermodynamic perspective that explains its variants and guides new algorithm development.

Contribution

It introduces a novel ITE-based framework for understanding Grover's algorithm, explaining existing variants and proposing a new, faster $\pi/2$-algorithm with practical advantages.

Findings

Grover's algorithm approximates imaginary-time evolution.

The framework explains the choice of angles in Grover's variants.

A new $\pi/2$-algorithm converges faster without overshooting.

Abstract

We reveal the power of Grover's algorithm from thermodynamic and geometric perspectives by showing that it is a product formula approximation of imaginary-time evolution (ITE), a Riemannian gradient flow on the special unitary group. This ITE formulation provides a unified perspective on Grover's algorithm, its variants and extensions to widely used quantum subroutines including amplitude amplification and oblivious amplitude amplification. Specifically, the framework explains the choice of angles in the original Grover's algorithm and $\pi/3$-algorithm. It also motivates a new $\pi/2$-algorithm, for cases a modest failure probability is acceptable, that converges faster than the $\pi/3$-algorithm without overshooting. Our analysis further provides a link between ITE and quantum signal processing, which yields a new implementation of the fixed-point quantum search algorithm. Moreover, the ITE formulation can systematically reproduce widely-used subroutines in modern quantum algorithms, such as (oblivious) amplitude amplification. These results collectively establish a deeper understanding of Grover's algorithm and suggest a potential role for thermodynamics and geometry in quantum algorithm design.