Quantum simulation of quantum phase transitions using the convex geometry of reduced density matrices

Warren, Samuel; Sager-Smith, LeeAnn M.; Mazziotti, David A.

doi:10.1103/physreva.106.012434

Title: Quantum simulation of quantum phase transitions using the convex geometry of reduced density matrices

Journal Article · Tue Jul 26 00:00:00 EDT 2022 · Physical Review A

DOI:https://doi.org/10.1103/physreva.106.012434· OSTI ID:1979623

^[1];

^[1]

University of Chicago, IL (United States)

Transitions of many-particle quantum systems between distinct phases at absolute-zero temperature, known as quantum phase transitions, require an exacting treatment of particle correlations. Here in this work, we present a general quantum-computing approach to quantum phase transitions that exploits the geometric structure of reduced density matrices. While typical approaches to quantum phase transitions examine discontinuities in the order parameters, the origin of phase transitions—their order parameters and symmetry breaking—can be understood geometrically in terms of the set of two-particle reduced density matrices (2-RDMs). The convex set of 2-RDMs provides a comprehensive map of the quantum system including its distinct phases as well as the transitions connecting these phases. Because 2-RDMs can potentially be computed on quantum computers at nonexponential cost, even when the quantum system is strongly correlated, they are ideally suited for a quantum-computing approach to quantum phase transitions. We compute the convex set of 2-RDMs for a Lipkin-Meshkov-Glick spin model on IBM superconducting-qubit quantum processors. Even though computations are limited to few-particle models due to device noise, comparisons with a classically solvable 1000-particle model reveal that the finite-particle quantum solutions capture the key features of the phase transitions including the strong correlation and the symmetry breaking.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Purdue Univ., West Lafayette, IN (United States); University of Chicago, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES); National Science Foundation (NSF)

Grant/Contract Number:: SC0019215; DGE-1746045; CHE-1565638; CHE-2035876; DMR-2037783

OSTI ID:: 1979623

Journal Information:: Physical Review A, Vol. 106, Issue 1; ISSN 2469-9926

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (58)

Generalized Gradient Approximation Made Simple Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias Physical Review Letters, Vol. 77, Issue 18, p. 3865-3868 https://doi.org/10.1103/PhysRevLett.77.3865	journal	October 1996
Validity of many-body approximation methods for a solvable model Lipkin, H. J.; Meshkov, N.; Glick, A. J. Nuclear Physics, Vol. 62, Issue 2 https://doi.org/10.1016/0029-5582(65)90862-X	journal	February 1965
Non-Fermi-liquid behaviour in the heavy-fermion system Löhneysen, Hilbert von Journal of Physics: Condensed Matter, Vol. 8, Issue 48 https://doi.org/10.1088/0953-8984/8/48/003	journal	November 1996
Topological superconductivity in semiconductor–superconductor–magnetic-insulator heterostructures Maiani, A.; Seoane Souto, R.; Leijnse, M. Physical Review B, Vol. 103, Issue 10 https://doi.org/10.1103/PhysRevB.103.104508	journal	March 2021
A dissipatively stabilized Mott insulator of photons Ma, Ruichao; Saxberg, Brendan; Owens, Clai Nature, Vol. 566, Issue 7742 https://doi.org/10.1038/s41586-019-0897-9	journal	February 2019
Continuous quantum phase transitions Sondhi, S. L.; Girvin, S. M.; Carini, J. P. Reviews of Modern Physics, Vol. 69, Issue 1 https://doi.org/10.1103/RevModPhys.69.315	journal	January 1997
Spontaneous symmetry breaking in a solvable nuclear model Stout, D. B. Nuclear Physics A, Vol. 567, Issue 3 https://doi.org/10.1016/0375-9474(94)90024-8	journal	January 1994
Boson localization and the superfluid-insulator transition Fisher, Matthew P. A.; Weichman, Peter B.; Grinstein, G. Physical Review B, Vol. 40, Issue 1 https://doi.org/10.1103/PhysRevB.40.546	journal	July 1989
Quantum phases of matter on a 256-atom programmable quantum simulator Ebadi, Sepehr; Wang, Tout T.; Levine, Harry Nature, Vol. 595, Issue 7866 https://doi.org/10.1038/s41586-021-03582-4	journal	July 2021
Antiferromagnetic criticality at a heavy-fermion quantum phase transition Knafo, W.; Raymond, S.; Lejay, P. Nature Physics, Vol. 5, Issue 10 https://doi.org/10.1038/nphys1374	journal	September 2009
Onset of a quantum phase transition with a trapped ion quantum simulator Islam, R.; Edwards, E. E.; Kim, K. Nature Communications, Vol. 2, Issue 1 https://doi.org/10.1038/ncomms1374	journal	July 2011
Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator Zhang, J.; Pagano, G.; Hess, P. W. Nature, Vol. 551, Issue 7682 https://doi.org/10.1038/nature24654	journal	November 2017
Thermodynamical Limit of the Lipkin-Meshkov-Glick Model Ribeiro, Pedro; Vidal, Julien; Mosseri, Rémy Physical Review Letters, Vol. 99, Issue 5 https://doi.org/10.1103/PhysRevLett.99.050402	journal	August 2007
Quantum Criticality: Competing Ground States in Low Dimensions Sachdev, S. Science, Vol. 288, Issue 5465 https://doi.org/10.1126/science.288.5465.475	journal	April 2000
Computation of quantum phase transitions by reduced-density-matrix mechanics Gidofalvi, Gergely; Mazziotti, David A. Physical Review A, Vol. 74, Issue 1 https://doi.org/10.1103/PhysRevA.74.012501	journal	July 2006
Equivalence of Critical Scaling Laws for Many-Body Entanglement in the Lipkin-Meshkov-Glick Model Orús, Román; Dusuel, Sébastien; Vidal, Julien Physical Review Letters, Vol. 101, Issue 2 https://doi.org/10.1103/PhysRevLett.101.025701	journal	July 2008
Electron–electron interactions and the paired-to-nematic quantum phase transition in the second Landau level Schreiber, K. A.; Samkharadze, N.; Gardner, G. C. Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-018-04879-1	journal	June 2018
Symmetry breaking and the geometry of reduced density matrices Zauner, V.; Draxler, D.; Vanderstraeten, L. New Journal of Physics, Vol. 18, Issue 11 https://doi.org/10.1088/1367-2630/18/11/113033	journal	November 2016
Entanglement in a second-order quantum phase transition Vidal, Julien; Palacios, Guillaume; Mosseri, Rémy Physical Review A, Vol. 69, Issue 2 https://doi.org/10.1103/PhysRevA.69.022107	journal	February 2004
Ising Superconductivity and Quantum Phase Transition in Macro-Size Monolayer NbSe ₂ Xing, Ying; Zhao, Kun; Shan, Pujia Nano Letters, Vol. 17, Issue 11 https://doi.org/10.1021/acs.nanolett.7b03026	journal	October 2017
Broken translational symmetry at edges of high-temperature superconductors Holmvall, P.; Vorontsov, A. B.; Fogelström, M. Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-018-04531-y	journal	June 2018
Quantum Phase Transitions Sachdev, Subir https://doi.org/10.1017/CBO9780511973765	book	January 2011
Quantum Chemistry without Wave Functions: Two-Electron Reduced Density Matrices Mazziotti, David A. Accounts of Chemical Research, Vol. 39, Issue 3 https://doi.org/10.1021/ar050029d	journal	March 2006
Quantum phase transitions in highly crystalline two-dimensional superconductors Saito, Yu; Nojima, Tsutomu; Iwasa, Yoshihiro Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-018-03275-z	journal	February 2018
Disorder Promotes Ferromagnetism: Rounding of the Quantum Phase Transition in ${Sr}_{1 - x} {Ca}_{x} {RuO}_{3}$ Demkó, L.; Bordács, S.; Vojta, T. Physical Review Letters, Vol. 108, Issue 18 https://doi.org/10.1103/PhysRevLett.108.185701	journal	May 2012
Quantum-classical hybrid algorithm using an error-mitigating $N$ -representability condition to compute the Mott metal-insulator transition Smart, Scott E.; Mazziotti, David A. Physical Review A, Vol. 100, Issue 2 https://doi.org/10.1103/PhysRevA.100.022517	journal	August 2019
Topology-Driven Magnetic Quantum Phase Transition in Topological Insulators Zhang, J.; Chang, C. -Z.; Tang, P. Science, Vol. 339, Issue 6127 https://doi.org/10.1126/science.1230905	journal	March 2013
Quantum Computing Without Wavefunctions: Time-Dependent Density Functional Theory for Universal Quantum Computation Tempel, David G.; Aspuru-Guzik, Alán Scientific Reports, Vol. 2, Issue 1 https://doi.org/10.1038/srep00391	journal	May 2012
Crossing a topological phase transition with a quantum computer Smith, Adam; Jobst, Bernhard; Green, Andrew G. Physical Review Research, Vol. 4, Issue 2 https://doi.org/10.1103/PhysRevResearch.4.L022020	journal	April 2022
The lower bound method for reduced density matrices Erdahl, R. M.; Jin, B. Journal of Molecular Structure: THEOCHEM, Vol. 527, Issue 1-3 https://doi.org/10.1016/S0166-1280(00)00494-2	journal	August 2000
Rashba Cavity QED: A Route Towards the Superradiant Quantum Phase Transition Nataf, Pierre; Champel, Thierry; Blatter, Gianni Physical Review Letters, Vol. 123, Issue 20 https://doi.org/10.1103/PhysRevLett.123.207402	journal	November 2019
Quantum phase transitions in disordered two-dimensional superconductors Fisher, Matthew P. A. Physical Review Letters, Vol. 65, Issue 7 https://doi.org/10.1103/PhysRevLett.65.923	journal	August 1990
Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian Wahlen-Strothman, Jacob M.; Henderson, Thomas M.; Hermes, Matthew R. The Journal of Chemical Physics, Vol. 146, Issue 5 https://doi.org/10.1063/1.4974989	journal	February 2017
Correlated charge noise and relaxation errors in superconducting qubits Wilen, C. D.; Abdullah, S.; Kurinsky, N. A. Nature, Vol. 594, Issue 7863 https://doi.org/10.1038/s41586-021-03557-5	journal	June 2021
Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for $N$ -representability Hammond, Jeff R.; Mazziotti, David A. Physical Review A, Vol. 71, Issue 6 https://doi.org/10.1103/PhysRevA.71.062503	journal	June 2005
Uniqueness of the solution of the contracted Schrödinger equation Yasuda, Koji Physical Review A, Vol. 65, Issue 5 https://doi.org/10.1103/PhysRevA.65.052121	journal	May 2002
Quantum Critical Behavior for a Model Magnet Bitko, D.; Rosenbaum, T. F.; Aeppli, G. Physical Review Letters, Vol. 77, Issue 5 https://doi.org/10.1103/PhysRevLett.77.940	journal	July 1996
Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices Soltan-Panahi, Parvis; Lühmann, Dirk-Sören; Struck, Julian Nature Physics, Vol. 8, Issue 1 https://doi.org/10.1038/nphys2128	journal	October 2011
Nonanalytic paramagnetic response of itinerant fermions away and near a ferromagnetic quantum phase transition Maslov, Dmitrii L.; Chubukov, Andrey V. Physical Review B, Vol. 79, Issue 7 https://doi.org/10.1103/PhysRevB.79.075112	journal	February 2009
Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions Mazziotti, David A. Physical Review A, Vol. 57, Issue 6 https://doi.org/10.1103/PhysRevA.57.4219	journal	June 1998
Simulating physics with computers Feynman, Richard P. International Journal of Theoretical Physics, Vol. 21, Issue 6-7 https://doi.org/10.1007/BF02650179	journal	June 1982
High-temperature superconductivity Maple, M. B. Journal of Magnetism and Magnetic Materials, Vol. 177-181 https://doi.org/10.1016/S0304-8853(97)00999-2	journal	January 1998
Cooper-pair condensates with nonclassical long-range order on quantum devices Sager, LeeAnn M.; Mazziotti, David A. Physical Review Research, Vol. 4, Issue 1 https://doi.org/10.1103/PhysRevResearch.4.013003	journal	January 2022
Correlated electrons in high-temperature superconductors Dagotto, Elbio Reviews of Modern Physics, Vol. 66, Issue 3 https://doi.org/10.1103/RevModPhys.66.763	journal	July 1994
The largeNbehaviour of the Lipkin model and exceptional points Heiss, W. D.; Scholtz, F. G.; Geyer, H. B. Journal of Physics A: Mathematical and General, Vol. 38, Issue 9 https://doi.org/10.1088/0305-4470/38/9/002	journal	February 2005
Experimental data from a quantum computer verifies the generalized Pauli exclusion principle Smart, Scott E.; Schuster, David I.; Mazziotti, David A. Communications Physics, Vol. 2, Issue 1 https://doi.org/10.1038/s42005-019-0110-3	journal	January 2019
Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming Mazziotti, David A. Physical Review Letters, Vol. 93, Issue 21 https://doi.org/10.1103/PhysRevLett.93.213001	journal	November 2004
Classical and quantum phase transitions in the Lipkin-Meshkov-Glick model Castaños, Octavio; López-Peña, Ramón; Hirsch, Jorge G. Physical Review B, Vol. 74, Issue 10 https://doi.org/10.1103/PhysRevB.74.104118	journal	September 2006
Quantum Phase Transitions and Bipartite Entanglement Wu, L. -A.; Sarandy, M. S.; Lidar, D. A. Physical Review Letters, Vol. 93, Issue 25 https://doi.org/10.1103/PhysRevLett.93.250404	journal	December 2004
Observation of a quantum phase transition in the quantum Rabi model with a single trapped ion Cai, M. -L.; Liu, Z. -D.; Zhao, W. -D. Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-021-21425-8	journal	February 2021
Kummer variety, geometry ofN-representability, and phase transitions Coleman, A. John Physical Review A, Vol. 66, Issue 2 https://doi.org/10.1103/PhysRevA.66.022503	journal	August 2002
Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms Greiner, Markus; Mandel, Olaf; Esslinger, Tilman Nature, Vol. 415, Issue 6867 https://doi.org/10.1038/415039a	journal	January 2002
Advances in the Physics of High-Temperature Superconductivity Orenstein, J. Science, Vol. 288, Issue 5465 https://doi.org/10.1126/science.288.5465.468	journal	April 2000
Finite-Size Scaling Exponents of the Lipkin-Meshkov-Glick Model Dusuel, Sébastien; Vidal, Julien Physical Review Letters, Vol. 93, Issue 23 https://doi.org/10.1103/PhysRevLett.93.237204	journal	December 2004
Electronic in-plane symmetry breaking at field-tuned quantum criticality in CeRhIn5 Ronning, F.; Helm, T.; Shirer, K. R. Nature, Vol. 548, Issue 7667 https://doi.org/10.1038/nature23315	journal	August 2017
Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices Smart, Scott E.; Mazziotti, David A. Physical Review Letters, Vol. 126, Issue 7 https://doi.org/10.1103/PhysRevLett.126.070504	journal	February 2021
Convex-set description of quantum phase transitions in the transverse Ising model using reduced-density-matrix theory Schwerdtfeger, Christine A.; Mazziotti, David A. The Journal of Chemical Physics, Vol. 130, Issue 22 https://doi.org/10.1063/1.3143403	journal	June 2009
Benchmarking near-term quantum devices with the variational quantum eigensolver and the Lipkin-Meshkov-Glick model Robbins, Kenneth; Love, Peter J. Physical Review A, Vol. 104, Issue 2 https://doi.org/10.1103/PhysRevA.104.022412	journal	August 2021

Similar Records

Many-fermion simulation from the contracted quantum eigensolver without fermionic encoding of the wave function

Journal Article · Wed Jun 15 00:00:00 EDT 2022 · Physical Review A · OSTI ID:1979623

Smart, Scott E.; Mazziotti, David A.

Entangled phase of simultaneous fermion and exciton condensations realized

Journal Article · Tue Mar 08 00:00:00 EST 2022 · Physical Review. B · OSTI ID:1979623

Sager, LeeAnn M.; Mazziotti, David A.

Dual-cone variational calculation of the two-electron reduced density matrix

Journal Article · Wed Nov 18 00:00:00 EST 2020 · Physical Review A · OSTI ID:1979623

Mazziotti, David A.

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
phase transitions
quantum simulation
density matrix methods
electron-correlation calculations

Title: Quantum simulation of quantum phase transitions using the convex geometry of reduced density matrices

Citation Formats

References (58)

Similar Records

Related Subjects