Entanglement-induced exponential advantage in amplitude estimation via state matrixization

TL;DR

This paper introduces a matrixization-based quantum amplitude estimation method that leverages entanglement properties to achieve exponential reductions in gate complexity compared to standard algorithms.

Contribution

The paper presents a novel matrixization framework for quantum amplitude estimation, enabling exponential complexity improvements based on entanglement characteristics.

Findings

Exponential gate complexity reduction when one state is low-entangled and the other is maximally entangled.

The matrixization approach can outperform standard amplitude estimation in specific entanglement regimes.

Generalization of results to broader quantum state regimes.

Abstract

Estimating quantum amplitude, or the overlap between two quantum states, is a fundamental task in quantum computing and underpins numerous quantum algorithms. In this work, we introduce a novel algorithmic framework for quantum amplitude estimation by transforming pure states into their matrix forms (Matrixization) and encoding them into non-diagonal blocks of density operators and diagonal blocks of unitary operators. Utilizing the construction details of state preparation circuits, we systematically reconstruct amplitude estimation algorithms within the novel matrixization framework through a technique known as channel block encoding. Compared with the standard approach, amplitude estimation through matrixization can have a different complexity that depends on the entanglement properties of the two quantum states. Specifically, our new algorithm can have exponentially smaller gate complexity when one of the two quantum states is prepared by a linear-depth quantum circuit that is below maximal entanglement under a certain bi-partition and the other state is maximally entangled. We later generalize this result to broader regimes and discuss implications. Our results demonstrate that the near-optimal performance of the standard amplitude estimation algorithm can be surpassed in specific cases.