# 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. 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., & 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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Works referenced in this record:

##
The density-matrix renormalization group in the age of matrix product states

journal, January 2011

- Schollwöck, Ulrich
- Annals of Physics, Vol. 326, Issue 1

##
Light cone renormalization and quantum quenches in one-dimensional Hubbard models

journal, February 2012

- Enss, Tilman; Sirker, Jesko
- New Journal of Physics, Vol. 14, Issue 2

##
Quantum simulation of the transverse Ising model with trapped ions

journal, October 2011

- Kim, K.; Korenblit, S.; Islam, R.
- New Journal of Physics, Vol. 13, Issue 10

##
Optimal Hamiltonian Simulation by Quantum Signal Processing

journal, January 2017

- Low, Guang Hao; Chuang, Isaac L.
- Physical Review Letters, Vol. 118, Issue 1

##
Chemical basis of Trotter-Suzuki errors in quantum chemistry simulation

journal, February 2015

- Babbush, Ryan; McClean, Jarrod; Wecker, Dave
- Physical Review A, Vol. 91, Issue 2

##
An area law for one-dimensional quantum systems

journal, August 2007

- Hastings, M. B.
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2007, Issue 08

##
Spectral Gap and Exponential Decay of Correlations

journal, April 2006

- Hastings, Matthew B.; Koma, Tohru
- Communications in Mathematical Physics, Vol. 265, Issue 3

##
Towards the fast scrambling conjecture

journal, April 2013

- Lashkari, Nima; Stanford, Douglas; Hastings, Matthew
- Journal of High Energy Physics, Vol. 2013, Issue 4

##
*Colloquium* : Area laws for the entanglement entropy

journal, February 2010

- Eisert, J.; Cramer, M.; Plenio, M. B.
- Reviews of Modern Physics, Vol. 82, Issue 1

##
Efficient Approximation of the Dynamics of One-Dimensional Quantum Spin Systems

journal, October 2006

- Osborne, Tobias J.
- Physical Review Letters, Vol. 97, Issue 15

##
Interplay of soundcone and supersonic propagation in lattice models with power law interactions

journal, June 2015

- Storch, David-Maximilian; den Worm, Mauritz van; Kastner, Michael
- New Journal of Physics, Vol. 17, Issue 6

##
Quantum many-body models with cold atoms coupled to photonic crystals

journal, April 2015

- Douglas, J. S.; Habibian, H.; Hung, C. -L.
- Nature Photonics, Vol. 9, Issue 5

##
Lieb-Robinson bounds on the speed of information propagation

journal, June 2010

- Prémont-Schwarz, Isabeau; Hnybida, Jeff
- Physical Review A, Vol. 81, Issue 6

##
Supersonic Quantum Communication

journal, June 2009

- Eisert, J.; Gross, D.
- Physical Review Letters, Vol. 102, Issue 24

##
Lieb-Robinson bounds for commutator-bounded operators

journal, April 2010

- Prémont-Schwarz, Isabeau; Hamma, Alioscia; Klich, Israel
- Physical Review A, Vol. 81, Issue 4

##
Observation of dipolar spin-exchange interactions with lattice-confined polar molecules

journal, September 2013

- Yan, Bo; Moses, Steven A.; Gadway, Bryce
- Nature, Vol. 501, Issue 7468

##
Lieb-Robinson Bounds and the Exponential Clustering Theorem

journal, March 2006

- Nachtergaele, Bruno; Sims, Robert
- Communications in Mathematical Physics, Vol. 265, Issue 1

##
Lieb-Robinson Bounds for Spin-Boson Lattice Models and Trapped Ions

journal, December 2013

- Jünemann, J.; Cadarso, A.; Pérez-García, D.
- Physical Review Letters, Vol. 111, Issue 23

##
Light-cone matrix product

journal, September 2009

- Hastings, M. B.
- Journal of Mathematical Physics, Vol. 50, Issue 9

##
The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual

journal, May 2018

- Kitaev, Alexei; Suh, S. Josephine
- Journal of High Energy Physics, Vol. 2018, Issue 5

##
Toward the first quantum simulation with quantum speedup

journal, September 2018

- Childs, Andrew M.; Maslov, Dmitri; Nam, Yunseong
- Proceedings of the National Academy of Sciences, Vol. 115, Issue 38

##
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

journal, December 2006

- Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard
- Communications in Mathematical Physics, Vol. 270, Issue 2

##
Propagation of Correlations in Quantum Lattice Systems

journal, July 2006

- Nachtergaele, Bruno; Ogata, Yoshiko; Sims, Robert
- Journal of Statistical Physics, Vol. 124, Issue 1

##
Quantum information with Rydberg atoms

journal, August 2010

- Saffman, M.; Walker, T. G.; Mølmer, K.
- Reviews of Modern Physics, Vol. 82, Issue 3

##
Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems

journal, September 2008

- Nachtergaele, Bruno; Raz, Hillel; Schlein, Benjamin
- Communications in Mathematical Physics, Vol. 286, Issue 3

##
Entanglement Area Laws for Long-Range Interacting Systems

journal, July 2017

- Gong, Zhe-Xuan; Foss-Feig, Michael; Brandão, Fernando G. S. L.
- Physical Review Letters, Vol. 119, Issue 5

##
Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order

journal, July 2006

- Bravyi, S.; Hastings, M. B.; Verstraete, F.
- Physical Review Letters, Vol. 97, Issue 5

##
Light-cone-like spreading of correlations in a quantum many-body system

journal, January 2012

- Cheneau, Marc; Barmettler, Peter; Poletti, Dario
- Nature, Vol. 481, Issue 7382

##
Simulating Hamiltonian Dynamics with a Truncated Taylor Series

journal, March 2015

- Berry, Dominic W.; Childs, Andrew M.; Cleve, Richard
- Physical Review Letters, Vol. 114, Issue 9

##
Emergent locality in systems with power-law interactions

journal, January 2019

- Luitz, David J.; Bar Lev, Yevgeny
- Physical Review A, Vol. 99, Issue 1

##
The finite group velocity of quantum spin systems

journal, September 1972

- Lieb, Elliott H.; Robinson, Derek W.
- Communications in Mathematical Physics, Vol. 28, Issue 3

##
Simulating Bosonic Baths with Error Bars

journal, September 2015

- Woods, M. P.; Cramer, M.; Plenio, M. B.
- Physical Review Letters, Vol. 115, Issue 13

##
Dynamical Localization in Disordered Quantum Spin Systems

journal, August 2012

- Hamza, Eman; Sims, Robert; Stolz, Günter
- Communications in Mathematical Physics, Vol. 315, Issue 1

##
Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins

journal, April 2012

- Britton, Joseph W.; Sawyer, Brian C.; Keith, Adam C.
- Nature, Vol. 484, Issue 7395

##
Properties of nitrogen-vacancy centers in diamond: the group theoretic approach

journal, February 2011

- Maze, J. R.; Gali, A.; Togan, E.
- New Journal of Physics, Vol. 13, Issue 2

##
Lieb-Robinson bounds on $n$ -partite connected correlation functions

journal, November 2017

- Tran, Minh Cong; Garrison, James R.; Gong, Zhe-Xuan
- Physical Review A, Vol. 96, Issue 5

##
Nearly Linear Light Cones in Long-Range Interacting Quantum Systems

journal, April 2015

- Foss-Feig, Michael; Gong, Zhe-Xuan; Clark, Charles W.
- Physical Review Letters, Vol. 114, Issue 15

##
Dynamical Phase Transitions in Sampling Complexity

journal, July 2018

- Deshpande, Abhinav; Fefferman, Bill; Tran, Minh C.
- Physical Review Letters, Vol. 121, Issue 3

##
Dynamical error bounds for continuum discretisation via Gauss quadrature rules—A Lieb-Robinson bound approach

journal, February 2016

- Woods, M. P.; Plenio, M. B.
- Journal of Mathematical Physics, Vol. 57, Issue 2

##
Universal power-law decay of electron-electron interactions due to nonlinear screening in a Josephson junction array

journal, September 2016

- Otten, Daniel; Rubbert, Sebastian; Ulrich, Jascha
- Physical Review B, Vol. 94, Issue 11

##
Propagation front of correlations in an interacting Bose gas

journal, May 2012

- Barmettler, Peter; Poletti, Dario; Cheneau, Marc
- Physical Review A, Vol. 85, Issue 5

##
Quantum quenches in the many-body localized phase

journal, November 2014

- Serbyn, Maksym; Papić, Z.; Abanin, D. A.
- Physical Review B, Vol. 90, Issue 17

##
Persistence of Locality in Systems with Power-Law Interactions

journal, July 2014

- Gong, Zhe-Xuan; Foss-Feig, Michael; Michalakis, Spyridon
- Physical Review Letters, Vol. 113, Issue 3

Works referencing / citing this record:

##
More current with less particles due to power-law hopping

journal, October 2019

- Saha, Madhumita; Purkayastha, Archak; Maiti, Santanu K.
- Journal of Physics: Condensed Matter, Vol. 32, Issue 2

##
Quantum kinetic perturbation theory for near-integrable spin chains with weak long-range interactions

journal, September 2019

- Duval, Clément; Kastner, Michael
- New Journal of Physics, Vol. 21, Issue 9

##
Locality and heating in periodically driven, power-law-interacting systems

journal, November 2019

- Tran, Minh C.; Ehrenberg, Adam; Guo, Andrew Y.
- Physical Review A, Vol. 100, Issue 5

##
Nonlocal emergent hydrodynamics in a long-range quantum spin system

journal, January 2020

- Schuckert, Alexander; Lovas, Izabella; Knap, Michael
- Physical Review B, Vol. 101, Issue 2

##
Many-body localization in spin chains with long-range transverse interactions: Scaling of critical disorder with system size

journal, January 2020

- Maksymov, Andrii O.; Burin, Alexander L.
- Physical Review B, Vol. 101, Issue 2

##
Exact large deviation statistics and trajectory phase transition of a deterministic boundary driven cellular automaton

journal, August 2019

- Buča, Berislav; Garrahan, Juan P.; Prosen, Tomaž
- Physical Review E, Vol. 100, Issue 2

##
Nearly Optimal Lattice Simulation by Product Formulas

journal, August 2019

- Childs, Andrew M.; Su, Yuan
- Physical Review Letters, Vol. 123, Issue 5

##
Using Matrix Product States to Study the Dynamical Large Deviations of Kinetically Constrained Models

journal, November 2019

- Bañuls, Mari Carmen; Garrahan, Juan P.
- Physical Review Letters, Vol. 123, Issue 20

##
Finite Speed of Quantum Scrambling with Long Range Interactions

journal, December 2019

- Chen, Chi-Fang; Lucas, Andrew
- Physical Review Letters, Vol. 123, Issue 25

##
Long-Range Prethermal Phases of Nonequilibrium Matter

journal, February 2020

- Machado, Francisco; Else, Dominic V.; Kahanamoku-Meyer, Gregory D.
- Physical Review X, Vol. 10, Issue 1

##
Classification of phases for mixed states via fast dissipative evolution

journal, August 2019

- Coser, Andrea; Pérez-García, David
- Quantum, Vol. 3