Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation
Abstract
We study the behavior of errors in the quantum simulation of spin systems with long-range multibody interactions resulting from the Trotter-Suzuki decomposition of the time-evolution operator. We identify a regime where the Floquet operator underlying the Trotter decomposition undergoes sharp changes even for small variations in the simulation step size. This results in a time evolution operator that is very different from the dynamics generated by the targeted Hamiltonian, which leads to a proliferation of errors in the quantum simulation. These regions of sharp change in the Floquet operator, referred to as structural instability regions, appear typically at intermediate Trotter step sizes and in the weakly interacting regime, and are thus complementary to recently revealed quantum chaotic regimes of the Trotterized evolution [L. M. Sieberer et al. npj Quantum Inf. 5, 78 (2019); M. Heyl, P. Hauke, and P. Zoller, Sci. Adv. 5, eaau8342 (2019)]. We characterize these structural instability regimes in p-spin models, transverse-field Ising models with all-to-all p-body interactions, and analytically predict their occurrence based on unitary perturbation theory. We further show that the effective Hamiltonian associated with the Trotter decomposition of the unitary time-evolution operator, when the Trotter step size is chosen to be in the structuralmore »
- Authors:
-
- Univ. of New Mexico, Albuquerque, NM (United States). Center for Quantum Information and Control
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States); National Quantum Information Science (QIS) Research Centers (United States). Quantum Systems Accelerator (QSA)
- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)
- OSTI Identifier:
- 1969276
- Grant/Contract Number:
- AC02-05CH11231; PHY-1820679; PHY-2011582
- Resource Type:
- Accepted Manuscript
- Journal Name:
- PRX Quantum
- Additional Journal Information:
- Journal Volume: 3; Journal Issue: 1; Journal ID: ISSN 2691-3399
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; quantum simulation
Citation Formats
Chinni, Karthik, Muñoz-Arias, Manuel H., Deutsch, Ivan H., and Poggi, Pablo M. Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation. United States: N. p., 2022.
Web. doi:10.1103/prxquantum.3.010351.
Chinni, Karthik, Muñoz-Arias, Manuel H., Deutsch, Ivan H., & Poggi, Pablo M. Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation. United States. https://doi.org/10.1103/prxquantum.3.010351
Chinni, Karthik, Muñoz-Arias, Manuel H., Deutsch, Ivan H., and Poggi, Pablo M. Mon .
"Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation". United States. https://doi.org/10.1103/prxquantum.3.010351. https://www.osti.gov/servlets/purl/1969276.
@article{osti_1969276,
title = {Trotter Errors from Dynamical Structural Instabilities of Floquet Maps in Quantum Simulation},
author = {Chinni, Karthik and Muñoz-Arias, Manuel H. and Deutsch, Ivan H. and Poggi, Pablo M.},
abstractNote = {We study the behavior of errors in the quantum simulation of spin systems with long-range multibody interactions resulting from the Trotter-Suzuki decomposition of the time-evolution operator. We identify a regime where the Floquet operator underlying the Trotter decomposition undergoes sharp changes even for small variations in the simulation step size. This results in a time evolution operator that is very different from the dynamics generated by the targeted Hamiltonian, which leads to a proliferation of errors in the quantum simulation. These regions of sharp change in the Floquet operator, referred to as structural instability regions, appear typically at intermediate Trotter step sizes and in the weakly interacting regime, and are thus complementary to recently revealed quantum chaotic regimes of the Trotterized evolution [L. M. Sieberer et al. npj Quantum Inf. 5, 78 (2019); M. Heyl, P. Hauke, and P. Zoller, Sci. Adv. 5, eaau8342 (2019)]. We characterize these structural instability regimes in p-spin models, transverse-field Ising models with all-to-all p-body interactions, and analytically predict their occurrence based on unitary perturbation theory. We further show that the effective Hamiltonian associated with the Trotter decomposition of the unitary time-evolution operator, when the Trotter step size is chosen to be in the structural instability region, is very different from the target Hamiltonian, which explains the large errors that can occur in the simulation in the regions of instability. These results have implications for the reliability of near-term gate-based quantum simulators, and reveal an important interplay between errors and the physical properties of the system being simulated.},
doi = {10.1103/prxquantum.3.010351},
journal = {PRX Quantum},
number = 1,
volume = 3,
place = {United States},
year = {Mon Mar 28 00:00:00 EDT 2022},
month = {Mon Mar 28 00:00:00 EDT 2022}
}
Works referenced in this record:
Simulated Quantum Computation of Molecular Energies
journal, September 2005
- Aspuru-Guzik, A.
- Science, Vol. 309, Issue 5741
Validity of many-body approximation methods for a solvable model
journal, February 1965
- Lipkin, H. J.; Meshkov, N.; Glick, A. J.
- Nuclear Physics, Vol. 62, Issue 2
Classical and quantum chaos for a kicked top
journal, September 1987
- Haake, F.; Kuś, M.; Scharf, R.
- Zeitschrift für Physik B Condensed Matter, Vol. 65, Issue 3
Excited-state quantum phase transitions in many-body systems with infinite-range interaction: Localization, dynamics, and bifurcation
journal, July 2016
- Santos, Lea F.; Távora, Marco; Pérez-Bernal, Francisco
- Physical Review A, Vol. 94, Issue 1
Energy gaps in quantum first-order mean-field–like transitions: The problems that quantum annealing cannot solve
journal, February 2010
- Jörg, T.; Krzakala, F.; Kurchan, J.
- EPL (Europhysics Letters), Vol. 89, Issue 4
Quantum simulation of strongly correlated condensed matter systems
journal, March 2018
- Hofstetter, W.; Qin, T.
- Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 51, Issue 8
Lyapunov exponent, stretching numbers, and islands of stability of the kicked top
journal, November 1997
- Constantoudis, V.; Theodorakopoulos, N.
- Physical Review E, Vol. 56, Issue 5
Nonlinear dynamics and quantum chaos of a family of kicked -spin models
journal, May 2021
- Muñoz-Arias, Manuel H.; Poggi, Pablo M.; Deutsch, Ivan H.
- Physical Review E, Vol. 103, Issue 5
Real-time dynamics of lattice gauge theories with a few-qubit quantum computer
journal, June 2016
- Martinez, Esteban A.; Muschik, Christine A.; Schindler, Philipp
- Nature, Vol. 534, Issue 7608
Quantum phases of matter on a 256-atom programmable quantum simulator
journal, July 2021
- Ebadi, Sepehr; Wang, Tout T.; Levine, Harry
- Nature, Vol. 595, Issue 7866
Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms
journal, July 2021
- Scholl, Pascal; Schuler, Michael; Williams, Hannah J.
- Nature, Vol. 595, Issue 7866
Curie–Weiss magnet—a simple model of phase transition
journal, October 2013
- Kochmański, M.; Paszkiewicz, T.; Wolski, S.
- European Journal of Physics, Vol. 34, Issue 6
Does Scrambling Equal Chaos?
journal, April 2020
- Xu, Tianrui; Scaffidi, Thomas; Cao, Xiangyu
- Physical Review Letters, Vol. 124, Issue 14
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
A variational eigenvalue solver on a photonic quantum processor
journal, July 2014
- Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter
- Nature Communications, Vol. 5, Issue 1
Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top
journal, September 2019
- Sieberer, Lukas M.; Olsacher, Tobias; Elben, Andreas
- npj Quantum Information, Vol. 5, Issue 1
Floquet Time Crystals
journal, August 2016
- Else, Dominic V.; Bauer, Bela; Nayak, Chetan
- Physical Review Letters, Vol. 117, Issue 9
Comparison of Different Methods for Computing Lyapunov Exponents
journal, May 1990
- Geist, K.; Parlitz, U.; Lauterborn, W.
- Progress of Theoretical Physics, Vol. 83, Issue 5
Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator
journal, November 2017
- Zhang, J.; Pagano, G.; Hess, P. W.
- Nature, Vol. 551, Issue 7682
Quantum supremacy using a programmable superconducting processor
journal, October 2019
- Arute, Frank; Arya, Kunal; Babbush, Ryan
- Nature, Vol. 574, Issue 7779
Quantum-annealing correction at finite temperature: Ferromagnetic -spin models
journal, February 2017
- Matsuura, Shunji; Nishimori, Hidetoshi; Vinci, Walter
- Physical Review A, Vol. 95, Issue 2
Dynamical Quantum Phase Transitions in Spin Chains with Long-Range Interactions: Merging Different Concepts of Nonequilibrium Criticality
journal, March 2018
- Žunkovič, Bojan; Heyl, Markus; Knap, Michael
- Physical Review Letters, Vol. 120, Issue 13
Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential
journal, June 1997
- Milburn, G. J.; Corney, J.; Wright, E. M.
- Physical Review A, Vol. 55, Issue 6
Quantum-classical computation of Schwinger model dynamics using quantum computers
journal, September 2018
- Klco, N.; Dumitrescu, E. F.; McCaskey, A. J.
- Physical Review A, Vol. 98, Issue 3
Route to Extend the Lifetime of a Discrete Time Crystal in a Finite Spin Chain without Disorder
journal, April 2021
- Choudhury, Sayan
- Atoms, Vol. 9, Issue 2
Quantum localization bounds Trotter errors in digital quantum simulation
journal, April 2019
- Heyl, Markus; Hauke, Philipp; Zoller, Peter
- Science Advances, Vol. 5, Issue 4
Classical and quantum phase transitions in the Lipkin-Meshkov-Glick model
journal, September 2006
- Castaños, Octavio; López-Peña, Ramón; Hirsch, Jorge G.
- Physical Review B, Vol. 74, Issue 10
Quantum phase transitions in fully connected spin models: An entanglement perspective
journal, February 2011
- Filippone, Michele; Dusuel, Sébastien; Vidal, Julien
- Physical Review A, Vol. 83, Issue 2
Quantifying the Sensitivity to Errors in Analog Quantum Simulation
journal, November 2020
- Poggi, Pablo M.; Lysne, Nathan K.; Kuper, Kevin W.
- PRX Quantum, Vol. 1, Issue 2
Saddle-point scrambling without thermalization
journal, March 2021
- Kidd, R. A.; Safavi-Naini, A.; Corney, J. F.
- Physical Review A, Vol. 103, Issue 3
Harnessing the Power of the Second Quantum Revolution
journal, November 2020
- Deutsch, Ivan H.
- PRX Quantum, Vol. 1, Issue 2
Theory of Trotter Error with Commutator Scaling
journal, February 2021
- Childs, Andrew M.; Su, Yuan; Tran, Minh C.
- Physical Review X, Vol. 11, Issue 1
Characteristic Lyapunov Exponents and Smooth Ergodic Theory
journal, August 1977
- Pesin, Ya B.
- Russian Mathematical Surveys, Vol. 32, Issue 4
Quantum simulations with ultracold quantum gases
journal, April 2012
- Bloch, Immanuel; Dalibard, Jean; Nascimbène, Sylvain
- Nature Physics, Vol. 8, Issue 4
Probing many-body dynamics on a 51-atom quantum simulator
journal, November 2017
- Bernien, Hannes; Schwartz, Sylvain; Keesling, Alexander
- Nature, Vol. 551, Issue 7682
Simulation of the complex dynamics of mean-field -spin models using measurement-based quantum feedback control
journal, August 2020
- Muñoz-Arias, Manuel H.; Deutsch, Ivan H.; Jessen, Poul S.
- Physical Review A, Vol. 102, Issue 2
Entanglement and the generation of random states in the quantum chaotic dynamics of kicked coupled tops
journal, October 2008
- Trail, Collin M.; Madhok, Vaibhav; Deutsch, Ivan H.
- Physical Review E, Vol. 78, Issue 4
On quantum mean-field models and their quantum annealing
journal, June 2012
- Bapst, Victor; Semerjian, Guilhem
- Journal of Statistical Mechanics: Theory and Experiment, Vol. 2012, Issue 06
Quantum Time Crystals
journal, October 2012
- Wilczek, Frank
- Physical Review Letters, Vol. 109, Issue 16
Floquet time crystal in the Lipkin-Meshkov-Glick model
journal, June 2017
- Russomanno, Angelo; Iemini, Fernando; Dalmonte, Marcello
- Physical Review B, Vol. 95, Issue 21
Quantum machine learning
journal, September 2017
- Biamonte, Jacob; Wittek, Peter; Pancotti, Nicola
- Nature, Vol. 549, Issue 7671
Ergodic theory of chaos and strange attractors
journal, July 1985
- Eckmann, J. -P.; Ruelle, D.
- Reviews of Modern Physics, Vol. 57, Issue 3
Discrete Time Crystals: Rigidity, Criticality, and Realizations
journal, January 2017
- Yao, N. Y.; Potter, A. C.; Potirniche, I. -D.
- Physical Review Letters, Vol. 118, Issue 3