Quantum Approximate $k$-Minimum Finding

TL;DR

This paper introduces an almost optimal quantum $k$-minimum finding algorithm that operates with approximate values, enabling more practical applications in quantum computing such as identifying smallest expectation values and lowest ground state energies.

Contribution

It extends previous quantum $k$-minimum algorithms to work with approximate values for all $k \, \geq \, 1$, broadening their applicability.

Findings

Efficient quantum algorithms for identifying the $k$ smallest expectation values.

Quantum algorithms for determining the $k$ lowest ground state energies of a Hamiltonian.

Extension of prior work to approximate value settings for all $k \geq 1$.

Abstract

Quantum $k$-minimum finding is a fundamental subroutine with numerous applications in combinatorial problems and machine learning. Previous approaches typically assume oracle access to exact function values, making it challenging to integrate this subroutine with other quantum algorithms. In this paper, we propose an (almost) optimal quantum $k$-minimum finding algorithm that works with approximate values for all $k \geq 1$, extending a result of van Apeldoorn, Gily\'{e}n, Gribling, and de Wolf (FOCS 2017) for $k=1$. As practical applications of this algorithm, we present efficient quantum algorithms for identifying the $k$ smallest expectation values among multiple observables and for determining the $k$ lowest ground state energies of a Hamiltonian with a known eigenbasis.