# Locality and Digital Quantum Simulation of Power-Law Interactions

## Abstract

The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance $r$ as a power law, $1/{r}^{\alpha}$. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah *et al*., FOCS’18. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah *et al.* quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when $\alpha >3D$ (where $D$ is the number of dimensions).

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Maryland, College Park, MD (United States); Duke Univ., Durham, NC (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Basic Energy Sciences (BES); US Army Research Office (ARO); National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); CIFAR; Heising-Simons Foundation; National Institute of Standards and Technology (NIST); ARCS Foundation

- OSTI Identifier:
- 1532796

- Alternate Identifier(s):
- OSTI ID: 1613091

- Grant/Contract Number:
- SC0019040; SC0019449; NSF PHY-1748958; DGE 1322106; PHY-1607611

- Resource Type:
- Published Article

- Journal Name:
- Physical Review. X

- Additional Journal Information:
- Journal Name: Physical Review. X Journal Volume: 9 Journal Issue: 3; Journal ID: ISSN 2160-3308

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 74 ATOMIC AND MOLECULAR PHYSICS; Physics

## Citation Formats

```
Tran, Minh C., Guo, Andrew Y., Su, Yuan, Garrison, James R., Eldredge, Zachary, Foss-Feig, Michael, Childs, Andrew M., and Gorshkov, Alexey V.. Locality and Digital Quantum Simulation of Power-Law Interactions. United States: N. p., 2019.
Web. doi:10.1103/PhysRevX.9.031006.
```

```
Tran, Minh C., Guo, Andrew Y., Su, Yuan, Garrison, James R., Eldredge, Zachary, Foss-Feig, Michael, Childs, Andrew M., & Gorshkov, Alexey V.. Locality and Digital Quantum Simulation of Power-Law Interactions. United States. https://doi.org/10.1103/PhysRevX.9.031006
```

```
Tran, Minh C., Guo, Andrew Y., Su, Yuan, Garrison, James R., Eldredge, Zachary, Foss-Feig, Michael, Childs, Andrew M., and Gorshkov, Alexey V.. Wed .
"Locality and Digital Quantum Simulation of Power-Law Interactions". United States. https://doi.org/10.1103/PhysRevX.9.031006.
```

```
@article{osti_1532796,
```

title = {Locality and Digital Quantum Simulation of Power-Law Interactions},

author = {Tran, Minh C. and Guo, Andrew Y. and Su, Yuan and Garrison, James R. and Eldredge, Zachary and Foss-Feig, Michael and Childs, Andrew M. and Gorshkov, Alexey V.},

abstractNote = {The propagation of information in nonrelativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1/rα. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al., FOCS’18. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α>3D (where D is the number of dimensions).},

doi = {10.1103/PhysRevX.9.031006},

journal = {Physical Review. X},

number = 3,

volume = 9,

place = {United States},

year = {2019},

month = {7}

}

https://doi.org/10.1103/PhysRevX.9.031006

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