# Quantum eigenvalue estimation via time series analysis

## TL;DR

This paper introduces a resource-efficient quantum eigenvalue estimation method based on time series analysis, suitable for near-term quantum devices, avoiding complex quantum Fourier transforms.

## Contribution

It presents a novel eigenvalue estimation technique that does not require large ancillas or coherence, differing from traditional quantum phase estimation.

## Key findings

- Polynomial complexity in inverse precision $\\epsilon$
- No need for large ancillary systems or controlled operations
- Accurate spectral property computation within additive error

## Abstract

We present an efficient method for estimating the eigenvalues of a Hamiltonian $H$ from the expectation values of the evolution operator for various times. For a given quantum state $\rho$, our method outputs a list of eigenvalue estimates and approximate probabilities. Each probability depends on the support of $\rho$ in those eigenstates of $H$ associated with eigenvalues within an arbitrarily small range. The complexity of our method is polynomial in the inverse of a given precision parameter $\epsilon$, which is the gap between eigenvalue estimates. Unlike the well-known quantum phase estimation algorithm that uses the quantum Fourier transform, our method does not require large ancillary systems, large sequences of controlled operations, or preserving coherence between experiments, and is therefore more attractive for near-term applications. The output of our method can be used to compute spectral properties of $H$ and other expectation values efficiently, within additive error proportional to $\epsilon$.