# Analysis of quantum Krylov algorithms with errors

**Authors:** William Kirby  
**arXiv:** [2401.01246](https://arxiv.org/abs/2401.01246)  
**Date:** 2024-01-02  
**Summary page:** https://www.summarizepaper.com/en/paper/2401.01246v1/

## Key points

- Quantum Krylov algorithms are used for estimating ground state energies in quantum systems
- Errors can affect the accuracy of these algorithms' outputs
- The author introduces a collective noise rate to quantify the overall impact of errors on the algorithm's results
- The main objective is to determine how errors affect the estimation of ground state energy
- Rigorous mathematical analysis shows that errors have a linear dependence on the noise rate
- This resolves a discrepancy between previous theoretical analyses and numerical simulations
- Generic errors are expressed in terms of an effective target Hamiltonian studied within an effective Krylov space
- This approach allows for comprehensive characterization and quantification of error effects on ground state energy estimates
- The research contributes to establishing a theoretical framework for understanding and managing errors in quantum Krylov algorithms

## Summary

This work by William Kirby presents a thorough analysis of quantum Krylov algorithms and their performance in the presence of errors. These algorithms are commonly used for estimating ground state energies in quantum systems through real-time evolutions. However, generic errors can affect the accuracy of their outputs. To address this issue, the author introduces a collective noise rate that quantifies the overall impact of errors on the algorithm's results. The main objective is to determine how these errors affect the estimation of ground state energy. Through rigorous mathematical analysis, it is proven that the resulting errors have a leading-order linear dependence on the noise rate. This finding resolves a discrepancy between previous theoretical analyses and known numerical simulations, providing a more accurate understanding of quantum Krylov errors. The key technique used in this analysis involves expressing generic errors in terms of an effective target Hamiltonian studied within an effective Krylov space. This approach allows for a comprehensive characterization and quantification of error effects on ground state energy estimates. Overall, this research contributes to establishing a theoretical framework for comprehending and managing errors in quantum Krylov algorithms, providing insights that can inform future developments and optimizations to enhance their robustness and accuracy in practical applications.