# Fault-tolerant quantum simulation of materials using Bloch orbitals

**Authors:** Nicholas C. Rubin, Dominic W. Berry, Fionn D. Malone, Alec F. White, Tanuj Khattar, A. Eugene DePrince III, Sabrina Sicolo, Michael Kühn, Michael Kaicher, Joonho Lee, Ryan Babbush  
**arXiv:** [2302.05531](https://arxiv.org/abs/2302.05531)  
**Date:** 2023-02-10  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2302.05531v1/

## Key points

- Simulation of chemistry using quantum computing has potential to revolutionize the field
- Previous research focused on finite-sized systems or required a large number of basis functions
- Authors propose extending quantum simulation using Bloch orbitals constructed from symmetry-adapted atom-centered orbitals
- Approach allows for modeling periodic ab initio Hamiltonians with modest number of basis functions
- Researchers adapt existing algorithms combining qubitization with tensor factorizations of Coulomb operator and introduce modifications for speedup leveraging symmetries
- Block encodings and new Bloch orbital form of tensor hypercontraction implemented
- Resources required to deploy algorithms estimated for modeling Lithium Nickel Oxide battery cathodes within surface code
- Paper divided into sections covering introduction, electronic structure Hamiltonian in Bloch orbitals, different representations for qubitizing materials Hamiltonians, scaling comparison and runtimes for diamond, classical and quantum simulations of LNO, and conclusion summarizing findings and contributions

## Summary

The simulation of chemistry using quantum computing holds great promise and has the potential to revolutionize the field. Previous research has mainly focused on modeling finite-sized systems or required a large number of plane wave basis functions. In this study, the authors propose a method to extend quantum simulation using Bloch orbitals constructed from symmetry-adapted atom-centered orbitals. This approach allows for modeling periodic ab initio Hamiltonians with a modest number of basis functions. The researchers focus on adapting existing algorithms that combine qubitization with tensor factorizations of the Coulomb operator and introduce modifications to achieve an asymptotic speedup leveraging translational (or Abelian) symmetries. They implement block encodings using known tensor factorizations and introduce a new Bloch orbital form of tensor hypercontraction. Additionally, they estimate the resources required to deploy their algorithms to model materials relevant to the chemistry of Lithium Nickel Oxide battery cathodes within the surface code. The paper is divided into several sections. It begins with an introduction highlighting the potential applications of quantum computing in simulating chemistry, followed by delving into electronic structure Hamiltonian in Bloch orbitals, discussing basis functions and matrix elements as well as second-quantized Hamiltonian. Different representations for qubitizing materials Hamiltonians are explored, including sparse representation, single-factorization representation, double-factorization representation, and tensor hypercontraction representation. A scaling comparison and runtimes for diamond are presented as an example case, followed by classical and quantum simulations of LNO (Lithium Nickel Oxide), providing background information on LNO and discussing correlated k-point calculations and single-shot density matrix embedding theory. Finally, the paper concludes by summarizing its findings and contributions. Overall, this research presents a novel approach to fault-tolerant quantum simulation of materials using Bloch orbitals and provides insights into resource requirements for challenging model materials like LNO.