# Quantum eigenvalue processing

**Authors:** Guang Hao Low, Yuan Su  
**arXiv:** [2401.06240](https://arxiv.org/abs/2401.06240)  
**Date:** 2024-01-11  
**Summary page:** https://www.summarizepaper.com/en/paper/2401.06240v1/

## Key points

- Paper: "Quantum eigenvalue processing" by Guang Hao Low and Yuan Su
- Problem: Solving linear algebra problems on a quantum computer using eigenvalues of non-normal input matrices
- Proposed framework: Quantum EigenValue Transformation (QEVT)
- Allows arbitrary polynomial transformations on eigenvalues of block-encoded non-normal operators
- Algorithm: Quantum EigenValue Estimation (QEVE)
- For operators with real spectra
- Linear differential equation solver with strictly linear time query complexity
- Ground state preparation algorithm for diagonalizable matrices with real spectra
- Efficient method for preparing a quantum superposition of Faber polynomials
- Techniques for generating n Fourier coefficients using O(polylog(n)) gates

## Summary

The paper titled "Quantum eigenvalue processing" by Guang Hao Low and Yuan Su addresses the problem of solving linear algebra problems on a quantum computer by processing the eigenvalues of non-normal input matrices. The authors propose a new framework called Quantum EigenValue Transformation (QEVT), which allows for arbitrary polynomial transformations to be applied to the eigenvalues of block-encoded non-normal operators. They also introduce an algorithm called Quantum EigenValue Estimation (QEVE) for operators with real spectra. This paper presents significant advancements in quantum eigenvalue processing, providing efficient solutions for solving linear algebra problems on quantum computers. <br><br>

Some key contributions include a linear differential equation solver with strictly linear time query complexity, a ground state preparation algorithm that extends previous results to diagonalizable matrices with real spectra, and an efficient method for preparing a quantum superposition of Faber polynomials. Additionally, the authors develop techniques for generating n Fourier coefficients using O(polylog(n)) gates, improving upon previous approaches with linear cost.