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Vaezi MS, Negari AR, Moharramipour A, Vaezi A. Amelioration for the Sign Problem: An Adiabatic Quantum Monte Carlo Algorithm. PHYSICAL REVIEW LETTERS 2021; 127:217003. [PMID: 34860094 DOI: 10.1103/physrevlett.127.217003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2020] [Revised: 03/29/2021] [Accepted: 10/11/2021] [Indexed: 06/13/2023]
Abstract
We introduce the adiabatic quantum Monte Carlo (AQMC) method, where we gradually crank up the interaction strength, as an amelioration of the sign problem. It is motivated by the adiabatic theorem and will approach the true ground state if the evolution time is long enough. We demonstrate that the AQMC algorithm enhances the average sign exponentially such that low enough temperatures can be accessed and ground-state properties probed. It is a controlled approximation that satisfies the variational theorem and provides an upper bound for the ground-state energy. We first benchmark the AQMC algorithm vis-à-vis the undoped Hubbard model on the square lattice which is known to be sign-problem-free within the conventional quantum Monte Carlo formalism. Next, we test the AQMC algorithm against the density-matrix-renormalization-group approach for the doped four-leg ladder Hubbard model and demonstrate its remarkable accuracy. As a nontrivial example, we apply our method to the Hubbard model at p=1/8 doping for a 16×8 system and discuss its ground-state properties. We finally utilize our method and demonstrate the emergence of U(1)_{2}∼SU(2)_{1} topological order in a strongly correlated Chern insulator.
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Affiliation(s)
- Mohammad-Sadegh Vaezi
- Pasargad Institute for Advanced Innovative Solutions (PIAIS), Tehran 19916-33361, Iran
| | - Amir-Reza Negari
- Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran
| | - Amin Moharramipour
- Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran
| | - Abolhassan Vaezi
- Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran
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Levy R, Clark BK. Mitigating the Sign Problem through Basis Rotations. PHYSICAL REVIEW LETTERS 2021; 126:216401. [PMID: 34114868 DOI: 10.1103/physrevlett.126.216401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Revised: 06/24/2020] [Accepted: 04/15/2021] [Indexed: 06/12/2023]
Abstract
Quantum Monte Carlo simulations of quantum many-body systems are plagued by the Fermion sign problem. The computational complexity of simulating Fermions scales exponentially in the projection time β and system size. The sign problem is basis dependent and an improved basis, for fixed errors, leads to exponentially quicker simulations. We show how to use sign-free quantum Monte Carlo simulations to optimize over the choice of basis on large two-dimensional systems. We numerically illustrate these techniques decreasing the "badness" of the sign problem by optimizing over single-particle basis rotations on one- and two-dimensional Hubbard systems. We find a generic rotation which improves the average sign of the Hubbard model for a wide range of U and densities for L×4 systems. In one example improvement, the average sign (and hence simulation cost at fixed accuracy) for the 16×4 Hubbard model at U/t=4 and n=0.75 increases by exp[8.64(6)β]. For typical projection times of β⪆100, this accelerates such simulation by many orders of magnitude.
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Affiliation(s)
- Ryan Levy
- Institute for Condensed Matter Theory and IQUIST and Department of Physics, University of Illinois at Urbana-Champaign, Champaign, Illinois 61801, USA
| | - Bryan K Clark
- Institute for Condensed Matter Theory and IQUIST and Department of Physics, University of Illinois at Urbana-Champaign, Champaign, Illinois 61801, USA
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Schnack J, Schulenburg J, Honecker A, Richter J. Magnon Crystallization in the Kagome Lattice Antiferromagnet. PHYSICAL REVIEW LETTERS 2020; 125:117207. [PMID: 32975976 DOI: 10.1103/physrevlett.125.117207] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2019] [Accepted: 08/10/2020] [Indexed: 06/11/2023]
Abstract
We present numerical evidence for the crystallization of magnons below the saturation field at nonzero temperatures for the highly frustrated spin-half kagome Heisenberg antiferromagnet. This phenomenon can be traced back to the existence of independent localized magnons or, equivalently, flatband multimagnon states. We present a loop-gas description of these localized magnons and a phase diagram of this transition, thus providing information for which magnetic fields and temperatures magnon crystallization can be observed experimentally. The emergence of a finite-temperature continuous transition to a magnon crystal is expected to be generic for spin models in dimension D>1 where flatband multimagnon ground states break translational symmetry.
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Affiliation(s)
- Jürgen Schnack
- Fakultät für Physik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
| | - Jörg Schulenburg
- Universitätsrechenzentrum, Universität Magdeburg, D-39016 Magdeburg, Germany
| | - Andreas Honecker
- Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université, F-95302 Cergy-Pontoise Cedex, France
| | - Johannes Richter
- Institut für Physik, Universität Magdeburg, P.O. Box 4120, D-39016 Magdeburg, Germany
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, D-01187 Dresden, Germany
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Hangleiter D, Roth I, Nagaj D, Eisert J. Easing the Monte Carlo sign problem. SCIENCE ADVANCES 2020; 6:eabb8341. [PMID: 32851184 PMCID: PMC7428338 DOI: 10.1126/sciadv.abb8341] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Accepted: 07/01/2020] [Indexed: 06/11/2023]
Abstract
Quantum Monte Carlo (QMC) methods are the gold standard for studying equilibrium properties of quantum many-body systems. However, in many interesting situations, QMC methods are faced with a sign problem, causing the severe limitation of an exponential increase in the runtime of the QMC algorithm. In this work, we develop a systematic, generally applicable, and practically feasible methodology for easing the sign problem by efficiently computable basis changes and use it to rigorously assess the sign problem. Our framework introduces measures of non-stoquasticity that-as we demonstrate analytically and numerically-at the same time provide a practically relevant and efficiently computable figure of merit for the severity of the sign problem. Complementing this pragmatic mindset, we prove that easing the sign problem in terms of those measures is generally an NP-complete task for nearest-neighbor Hamiltonians and simple basis choices by a reduction to the MAXCUT-problem.
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Affiliation(s)
- Dominik Hangleiter
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany
| | - Ingo Roth
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany
| | - Daniel Nagaj
- RCQI, Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia
| | - Jens Eisert
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany
- Helmholtz Center Berlin, Hahn-Meitner-Platz 1, 14109 Berlin, Germany
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Marvian M, Lidar DA, Hen I. On the computational complexity of curing non-stoquastic Hamiltonians. Nat Commun 2019; 10:1571. [PMID: 30952854 PMCID: PMC6450938 DOI: 10.1038/s41467-019-09501-6] [Citation(s) in RCA: 28] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2018] [Accepted: 03/14/2019] [Indexed: 11/09/2022] Open
Abstract
Quantum many-body systems whose Hamiltonians are non-stoquastic, i.e., have positive off-diagonal matrix elements in a given basis, are known to pose severe limitations on the efficiency of Quantum Monte Carlo algorithms designed to simulate them, due to the infamous sign problem. We study the computational complexity associated with 'curing' non-stoquastic Hamiltonians, i.e., transforming them into sign-problem-free ones. We prove that if such transformations are limited to single-qubit Clifford group elements or general single-qubit orthogonal matrices, finding the curing transformation is NP-complete. We discuss the implications of this result.
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Affiliation(s)
- Milad Marvian
- Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA. .,Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, 90089, USA. .,Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA, 90089, USA.
| | - Daniel A Lidar
- Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, 90089, USA.,Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA, 90089, USA.,Department of Physics and Astronomy, University of Southern California, Los Angeles, CA, 90089, USA.,Department of Chemistry, University of Southern California, Los Angeles, CA, 90089, USA
| | - Itay Hen
- Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, CA, 90089, USA.,Department of Physics and Astronomy, University of Southern California, Los Angeles, CA, 90089, USA.,Information Sciences Institute, University of Southern California, Marina del Rey, CA, 90292, USA
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Hen I. Resolution of the sign problem for a frustrated triplet of spins. Phys Rev E 2019; 99:033306. [PMID: 30999420 DOI: 10.1103/physreve.99.033306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2018] [Indexed: 06/09/2023]
Abstract
We propose a mechanism for solving the "negative sign problem"-the inability to assign non-negative weights to quantum Monte Carlo configurations-for a toy model consisting of a frustrated triplet of spin-1/2 particles interacting antiferromagnetically. The introduced technique is based on the systematic grouping of the weights of the recently developed off-diagonal series expansion of the canonical partition function [Phys. Rev. E 96, 063309 (2017)PREHBM2470-004510.1103/PhysRevE.96.063309]. We show that, although the examined model is easily diagonalizable, the sign problem it encounters can nonetheless be very pronounced, and we offer a systematic mechanism to resolve it. We discuss the prospects of generalizing the suggested scheme and the steps required to extend it to more general and larger spin models.
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Affiliation(s)
- Itay Hen
- Information Sciences Institute, University of Southern California, Marina del Rey, California 90292, USA and Department of Physics and Astronomy, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA
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Stapmanns J, Corboz P, Mila F, Honecker A, Normand B, Wessel S. Thermal Critical Points and Quantum Critical End Point in the Frustrated Bilayer Heisenberg Antiferromagnet. PHYSICAL REVIEW LETTERS 2018; 121:127201. [PMID: 30296119 DOI: 10.1103/physrevlett.121.127201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2018] [Indexed: 06/08/2023]
Abstract
We consider the finite-temperature phase diagram of the S=1/2 frustrated Heisenberg bilayer. Although this two-dimensional system may show magnetic order only at zero temperature, we demonstrate the presence of a line of finite-temperature critical points related to the line of first-order transitions between the dimer-singlet and -triplet regimes. We show by high-precision quantum Monte Carlo simulations, which are sign-free in the fully frustrated limit, that this critical point is in the Ising universality class. At zero temperature, the continuous transition between the ordered bilayer and the dimer-singlet state terminates on the first-order line, giving a quantum critical end point, and we use tensor-network calculations to follow the first-order discontinuities in its vicinity.
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Affiliation(s)
- J Stapmanns
- Institut für Theoretische Festkörperphysik, JARA-FIT and JARA-HPC, RWTH Aachen University, 52056 Aachen, Germany
| | - P Corboz
- Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
| | - F Mila
- Institute of Physics, Ecole Polytechnique Fédérale Lausanne (EPFL), 1015 Lausanne, Switzerland
| | - A Honecker
- Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, France
| | - B Normand
- Neutrons and Muons Research Division, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland
| | - S Wessel
- Institut für Theoretische Festkörperphysik, JARA-FIT and JARA-HPC, RWTH Aachen University, 52056 Aachen, Germany
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