Quantum Approximate $k$-Minimum Finding
Minbo Gao, Zhengfeng Ji, Qisheng Wang
TL;DR
The paper addresses quantum algorithms for finding the $k$ smallest values when only approximate function values are available, a setting that arises in practical quantum numerical tasks. It introduces two notions of output—weak and strong approximate minimum index sets—and designs almost optimal quantum algorithms with $\tilde{O}(\sqrt{nk})$ query complexity to an approximate oracle. The main contributions are two algorithms (for weak and strong sets) and two impactful applications: identifying the $k$ smallest expectations across multiple observables and locating the $k$ lowest energies of a Hamiltonian with a known eigenbasis, including a corrected solution to previous gaps in the literature. This work enables robust composition of quantum subroutines under approximate data and has potential impact on quantum SDP solvers, quantum ML pipelines, and quantum simulation tasks where precise oracle values are unavailable.
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, Gilyén, 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.