A universal adiabatic quantum query algorithm

TL;DR

This paper presents a direct proof linking quantum adversary bounds to adiabatic quantum algorithms, establishing a universal approach in continuous-time quantum query complexity without spectral gap assumptions.

Contribution

It provides a new direct proof of the equivalence between adversary bounds and quantum query complexity in continuous-time models using adiabatic and Ehrenfest's theorems.

Findings

Established a universal adiabatic quantum query algorithm.

Connected adversary bounds with quantum mechanics theorems.

Proved lower bounds using Ehrenfest's theorem.

Abstract

Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern technique of theoretical computer science, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the adiabatic theorem, as it goes by constructing a universal adiabatic quantum query algorithm. Another originality is that we use for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap.