Adiabatic Quantum Counting by Geometric Phase Estimation

TL;DR

This paper presents an adiabatic quantum algorithm that estimates the proportion of marked items in a database by measuring a geometric phase, offering a potentially noise-resilient alternative to circuit-based methods.

Contribution

It introduces a novel adiabatic quantum counting method utilizing Berry phase estimation, expanding the toolkit for quantum algorithms beyond circuit models.

Findings

Cost scales as (1/ε)^{3/2} for error ε

Potential robustness to decoherence and noise due to geometric phase

Better than classical random algorithms in efficiency

Abstract

We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, $\alpha$, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase $2\pi\alpha$. By estimating the Berry phase, we can approximate $\alpha$, and solve the problem. For an error bound $\epsilon$, the algorithm can solve the problem with cost of order $(\frac{1}{\epsilon})^{3/2}$, which is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result may be robust to decoherence and resilient to certain noise.