# Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution

**Authors:** Mario Motta, Chong Sun, Adrian Teck Keng Tan, Matthew J. O' Rourke, Erika Ye, Austin J. Minnich, Fernando G. S. L. Brandao, Garnet Kin-Lic Chan  
**arXiv:** [1901.07653](https://arxiv.org/abs/1901.07653)  
**Date:** 2019-01-23  
**Summary page:** https://www.summarizepaper.com/en/paper/1901.07653v3/

## Key points

- Research focus: Accurate computation of Hamiltonian ground, excited, and thermal states on quantum computers
- Implications: Significant impact on fields such as quantum simulation and machine learning
- Challenges addressed: Constructing large-scale quantum computers efficiently
- Quantum imaginary time evolution
- Quantum Lanczos
- Exponentially less space and time per iteration compared to classical methods like phase estimation or variational algorithms
- Implementation without deep circuits, ancillae, or high-dimensional optimization processes
- Generation of Gibbs averages through minimally entangled typical thermal states analog
- Demonstrated potential through implementations using exact classical emulation and prototype circuits on platforms like Rigetti quantum virtual machine and Aspen-1 quantum processing unit

## Summary

In their paper titled "Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution," Mario Motta, Chong Sun, Adrian Teck Keng Tan, Matthew J. O' Rourke, Erika Ye, Austin J. Minnich, Fernando G. S. L. Brandao, and Garnet Kin-Lic Chan explore the accurate computation of Hamiltonian ground, excited, and thermal states on quantum computers. This research has significant implications for various fields such as quantum simulation and machine learning. The authors highlight the challenges associated with constructing large-scale quantum computers and emphasize the importance of carrying out these tasks in a resource-efficient manner. They introduce two novel algorithms - quantum imaginary time evolution and quantum Lanczos - which serve as analogues to classical algorithms for finding ground and excited states. These algorithms offer several advantages over existing techniques like phase estimation or variational algorithms. For instance, they require exponentially less space and time per iteration compared to their classical counterparts. Importantly, these new algorithms can be implemented without deep circuits and ancillae or high-dimensional optimization processes. Furthermore, the authors discuss how quantum imaginary time evolution can be used as a subroutine to generate Gibbs averages through an analog of minimally entangled typical thermal states. By demonstrating the potential of these algorithms through implementations using exact classical emulation and prototype circuits on platforms like the Rigetti quantum virtual machine and Aspen-1 quantum processing unit, the researchers showcase the practicality and efficiency of their approach. This study was published in Nature Physics (16:205-210) in 2020 and provides valuable insights into advancing quantum computing capabilities for solving complex problems in physics and computer science domains.