Circuit complexity across a topological phase transition
Abstract
We use Nielsen’s geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and nonequilibrium steady states of the Kitaev model exhibit nonanalytical behaviors at the critical points, and thus can be used to detect both equilibrium and dynamical topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen’s circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and we discuss generalizations to higher dimensions. Our result opens up an avenue for using circuit complexity as a tool to understand quantum many-body systems.
- Authors:
- Publication Date:
- Research Org.:
- Univ. of Maryland, College Park, MD (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC); National Science Foundation (NSF)
- OSTI Identifier:
- 1604852
- Alternate Identifier(s):
- OSTI ID: 1604796
- Grant/Contract Number:
- SC0019449; SC0020312; SC0019040; DGE-1322106; PHY-1748958
- Resource Type:
- Published Article
- Journal Name:
- Physical Review Research
- Additional Journal Information:
- Journal Name: Physical Review Research Journal Volume: 2 Journal Issue: 1; Journal ID: ISSN 2643-1564
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Long-range interactions; Quantum computation; Quantum quench; Topological phase transition; Bogoliubov-de Gennes equations; Computational complexity; Tight-binding model
Citation Formats
Liu, Fangli, Whitsitt, Seth, Curtis, Jonathan B., Lundgren, Rex, Titum, Paraj, Yang, Zhi-Cheng, Garrison, James R., and Gorshkov, Alexey V. Circuit complexity across a topological phase transition. United States: N. p., 2020.
Web. doi:10.1103/PhysRevResearch.2.013323.
Liu, Fangli, Whitsitt, Seth, Curtis, Jonathan B., Lundgren, Rex, Titum, Paraj, Yang, Zhi-Cheng, Garrison, James R., & Gorshkov, Alexey V. Circuit complexity across a topological phase transition. United States. https://doi.org/10.1103/PhysRevResearch.2.013323
Liu, Fangli, Whitsitt, Seth, Curtis, Jonathan B., Lundgren, Rex, Titum, Paraj, Yang, Zhi-Cheng, Garrison, James R., and Gorshkov, Alexey V. Mon .
"Circuit complexity across a topological phase transition". United States. https://doi.org/10.1103/PhysRevResearch.2.013323.
@article{osti_1604852,
title = {Circuit complexity across a topological phase transition},
author = {Liu, Fangli and Whitsitt, Seth and Curtis, Jonathan B. and Lundgren, Rex and Titum, Paraj and Yang, Zhi-Cheng and Garrison, James R. and Gorshkov, Alexey V.},
abstractNote = {We use Nielsen’s geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and nonequilibrium steady states of the Kitaev model exhibit nonanalytical behaviors at the critical points, and thus can be used to detect both equilibrium and dynamical topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen’s circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and we discuss generalizations to higher dimensions. Our result opens up an avenue for using circuit complexity as a tool to understand quantum many-body systems.},
doi = {10.1103/PhysRevResearch.2.013323},
journal = {Physical Review Research},
number = 1,
volume = 2,
place = {United States},
year = {2020},
month = {3}
}
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Works referenced in this record:
Complexity and entanglement for thermofield double states
journal, January 2019
- Chapman, Shira; Eisert, Jens; Hackl, Lucas
- SciPost Physics, Vol. 6, Issue 3
Topological insulators and superconductors
journal, October 2011
- Qi, Xiao-Liang; Zhang, Shou-Cheng
- Reviews of Modern Physics, Vol. 83, Issue 4
Extracting Entanglement Geometry from Quantum States
journal, October 2017
- Hyatt, Katharine; Garrison, James R.; Bauer, Bela
- Physical Review Letters, Vol. 119, Issue 14
Optimal Control for Unitary Preparation of Many-Body States: Application to Luttinger Liquids
journal, July 2011
- Rahmani, Armin; Chamon, Claudio
- Physical Review Letters, Vol. 107, Issue 1
One-dimensional Fermi liquids
journal, September 1995
- Voit, J.
- Reports on Progress in Physics, Vol. 58, Issue 9
Decay of Loschmidt Echo Enhanced by Quantum Criticality
journal, April 2006
- Quan, H. T.; Song, Z.; Liu, X. F.
- Physical Review Letters, Vol. 96, Issue 14
Kitaev Chains with Long-Range Pairing
journal, October 2014
- Vodola, Davide; Lepori, Luca; Ercolessi, Elisa
- Physical Review Letters, Vol. 113, Issue 15
Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
journal, August 2010
- Lutchyn, Roman M.; Sau, Jay D.; Das Sarma, S.
- Physical Review Letters, Vol. 105, Issue 7
Complexity for quantum field theory states and applications to thermofield double states
journal, March 2018
- Yang, Run-Qiu
- Physical Review D, Vol. 97, Issue 6
How Difficult is it to Prepare a Quantum State?
journal, January 2019
- Girolami, Davide
- Physical Review Letters, Vol. 122, Issue 1
Evolution of complexity following a quantum quench in free field theory
journal, June 2018
- Alves, Daniel W. F.; Camilo, Giancarlo
- Journal of High Energy Physics, Vol. 2018, Issue 6
Toward a Definition of Complexity for Quantum Field Theory States
journal, March 2018
- Chapman, Shira; Heller, Michal P.; Marrochio, Hugo
- Physical Review Letters, Vol. 120, Issue 12
More on complexity of operators in quantum field theory
journal, March 2019
- Yang, Run-Qiu; An, Yu-Sen; Niu, Chao
- Journal of High Energy Physics, Vol. 2019, Issue 3
Helical Liquids and Majorana Bound States in Quantum Wires
journal, October 2010
- Oreg, Yuval; Refael, Gil; von Oppen, Felix
- Physical Review Letters, Vol. 105, Issue 17
Detecting Equilibrium and Dynamical Quantum Phase Transitions in Ising Chains via Out-of-Time-Ordered Correlators
journal, July 2018
- Heyl, Markus; Pollmann, Frank; Dóra, Balázs
- Physical Review Letters, Vol. 121, Issue 1
Quantum Quenches in Chern Insulators
journal, December 2015
- Caio, M. D.; Cooper, N. R.; Bhaseen, M. J.
- Physical Review Letters, Vol. 115, Issue 23
New directions in the pursuit of Majorana fermions in solid state systems
journal, June 2012
- Alicea, Jason
- Reports on Progress in Physics, Vol. 75, Issue 7
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
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
Complexity and shock wave geometries
journal, December 2014
- Stanford, Douglas; Susskind, Leonard
- Physical Review D, Vol. 90, Issue 12
Quantum Computation as Gravity
journal, June 2019
- Caputa, Paweł; Magan, Javier M.
- Physical Review Letters, Vol. 122, Issue 23
Unpaired Majorana fermions in quantum wires
journal, October 2001
- Kitaev, A. Yu
- Physics-Uspekhi, Vol. 44, Issue 10S
Multipartite Entanglement in Topological Quantum Phases
journal, December 2017
- Pezzè, Luca; Gabbrielli, Marco; Lepori, Luca
- Physical Review Letters, Vol. 119, Issue 25
On a soluble model of an antiferromagnetic chain with alternating interactions and magnetic moments
journal, January 1975
- Perk, J. H. H.; Capel, H. W.; Zuilhof, M. J.
- Physica A: Statistical Mechanics and its Applications, Vol. 81, Issue 3
Circuit complexity for coherent states
journal, October 2018
- Guo, Minyong; Hernandez, Juan; Myers, Robert C.
- Journal of High Energy Physics, Vol. 2018, Issue 10
Complexity of the AdS soliton
journal, March 2018
- Reynolds, Alan P.; Ross, Simon F.
- Classical and Quantum Gravity, Vol. 35, Issue 9
Generic New Platform for Topological Quantum Computation Using Semiconductor Heterostructures
journal, January 2010
- Sau, Jay D.; Lutchyn, Roman M.; Tewari, Sumanta
- Physical Review Letters, Vol. 104, Issue 4
Topological Quantum Liquids with Long-Range Couplings
journal, June 2017
- Patrick, Kristian; Neupert, Titus; Pachos, Jiannis K.
- Physical Review Letters, Vol. 118, Issue 26
Adiabatic preparation of many-body states in optical lattices
journal, June 2010
- Sørensen, Anders S.; Altman, Ehud; Gullans, Michael
- Physical Review A, Vol. 81, Issue 6
Non-Abelian statistics and topological quantum information processing in 1D wire networks
journal, February 2011
- Alicea, Jason; Oreg, Yuval; Refael, Gil
- Nature Physics, Vol. 7, Issue 5
Dynamical topological quantum phase transitions for mixed states
journal, November 2017
- Heyl, M.; Budich, J. C.
- Physical Review B, Vol. 96, Issue 18
Two soluble models of an antiferromagnetic chain
journal, December 1961
- Lieb, Elliott; Schultz, Theodore; Mattis, Daniel
- Annals of Physics, Vol. 16, Issue 3
Principles and symmetries of complexity in quantum field theory
journal, February 2019
- Yang, Run-Qiu; An, Yu-Sen; Niu, Chao
- The European Physical Journal C, Vol. 79, Issue 2
Circuit complexity in fermionic field theory
journal, December 2018
- Khan, Rifath; Krishnan, Chethan; Sharma, Sanchita
- Physical Review D, Vol. 98, Issue 12
Circuit complexity in quantum field theory
journal, October 2017
- Jefferson, Robert A.; Myers, Robert C.
- Journal of High Energy Physics, Vol. 2017, Issue 10
Remnant Geometric Hall Response in a Quantum Quench
journal, November 2016
- Wilson, Justin H.; Song, Justin C. W.; Refael, Gil
- Physical Review Letters, Vol. 117, Issue 23
Quantum phase transitions
journal, November 2003
- Vojta, Matthias
- Reports on Progress in Physics, Vol. 66, Issue 12
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
journal, October 2010
- Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang
- Physical Review B, Vol. 82, Issue 15
Locating topological phase transitions using nonequilibrium signatures in local bulk observables
journal, January 2017
- Roy, Sthitadhi; Moessner, Roderich; Das, Arnab
- Physical Review B, Vol. 95, Issue 4
Complexity of operators generated by quantum mechanical Hamiltonians
journal, March 2019
- Yang, Run-Qiu; Kim, Keun-Young
- Journal of High Energy Physics, Vol. 2019, Issue 3
Complexity as a Novel Probe of Quantum Quenches: Universal Scalings and Purifications
journal, February 2019
- Camargo, Hugo A.; Caputa, Pawel; Das, Diptarka
- Physical Review Letters, Vol. 122, Issue 8
Probing Ground-State Phase Transitions through Quench Dynamics
journal, September 2019
- Titum, Paraj; Iosue, Joseph T.; Garrison, James R.
- Physical Review Letters, Vol. 123, Issue 11
Colloquium: Topological insulators
journal, November 2010
- Hasan, M. Z.; Kane, C. L.
- Reviews of Modern Physics, Vol. 82, Issue 4, p. 3045-3067
Observation of dynamical vortices after quenches in a system with topology
journal, December 2017
- Fläschner, N.; Vogel, D.; Tarnowski, M.
- Nature Physics, Vol. 14, Issue 3
'Luttinger liquid theory' of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas
journal, July 1981
- Haldane, F. D. M.
- Journal of Physics C: Solid State Physics, Vol. 14, Issue 19
Complexity, action, and black holes
journal, April 2016
- Brown, Adam R.; Roberts, Daniel A.; Susskind, Leonard
- Physical Review D, Vol. 93, Issue 8
Topological classification of dynamical phase transitions
journal, April 2015
- Vajna, Szabolcs; Dóra, Balázs
- Physical Review B, Vol. 91, Issue 15
Holographic Complexity Equals Bulk Action?
journal, May 2016
- Brown, Adam R.; Roberts, Daniel A.; Susskind, Leonard
- Physical Review Letters, Vol. 116, Issue 19
Direct Observation of Dynamical Quantum Phase Transitions in an Interacting Many-Body System
journal, August 2017
- Jurcevic, P.; Shen, H.; Hauke, P.
- Physical Review Letters, Vol. 119, Issue 8
Circuit complexity for free fermions
journal, July 2018
- Hackl, Lucas; Myers, Robert C.
- Journal of High Energy Physics, Vol. 2018, Issue 7
Quantum circuit complexity of one-dimensional topological phases
journal, May 2015
- Huang, Yichen; Chen, Xie
- Physical Review B, Vol. 91, Issue 19
Dynamical preparation of Floquet Chern insulators
journal, October 2015
- D’Alessio, Luca; Rigol, Marcos
- Nature Communications, Vol. 6, Issue 1
Long-range Ising and Kitaev models: phases, correlations and edge modes
journal, December 2015
- Vodola, Davide; Lepori, Luca; Ercolessi, Elisa
- New Journal of Physics, Vol. 18, Issue 1
Topological marker currents in Chern insulators
journal, January 2019
- Caio, M. D.; Möller, G.; Cooper, N. R.
- Nature Physics, Vol. 15, Issue 3
Adiabatic quantum dynamics of a random Ising chain across its quantum critical point
journal, October 2007
- Caneva, Tommaso; Fazio, Rosario; Santoro, Giuseppe E.
- Physical Review B, Vol. 76, Issue 14
Topological blocking in quantum quench dynamics
journal, June 2014
- Kells, G.; Sen, D.; Slingerland, J. K.
- Physical Review B, Vol. 89, Issue 23
Statistical Mechanics of the Anisotropic Linear Heisenberg Model
journal, September 1962
- Katsura, Shigetoshi
- Physical Review, Vol. 127, Issue 5
Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics
journal, May 2017
- Wang, Ce; Zhang, Pengfei; Chen, Xin
- Physical Review Letters, Vol. 118, Issue 18