1
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Shaw AL, Chen Z, Choi J, Mark DK, Scholl P, Finkelstein R, Elben A, Choi S, Endres M. Benchmarking highly entangled states on a 60-atom analogue quantum simulator. Nature 2024; 628:71-77. [PMID: 38509372 PMCID: PMC10990925 DOI: 10.1038/s41586-024-07173-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2023] [Accepted: 02/07/2024] [Indexed: 03/22/2024]
Abstract
Quantum systems have entered a competitive regime in which classical computers must make approximations to represent highly entangled quantum states1,2. However, in this beyond-classically-exact regime, fidelity comparisons between quantum and classical systems have so far been limited to digital quantum devices2-5, and it remains unsolved how to estimate the actual entanglement content of experiments6. Here, we perform fidelity benchmarking and mixed-state entanglement estimation with a 60-atom analogue Rydberg quantum simulator, reaching a high-entanglement entropy regime in which exact classical simulation becomes impractical. Our benchmarking protocol involves extrapolation from comparisons against an approximate classical algorithm, introduced here, with varying entanglement limits. We then develop and demonstrate an estimator of the experimental mixed-state entanglement6, finding our experiment is competitive with state-of-the-art digital quantum devices performing random circuit evolution2-5. Finally, we compare the experimental fidelity against that achieved by various approximate classical algorithms, and find that only the algorithm we introduce is able to keep pace with the experiment on the classical hardware we use. Our results enable a new model for evaluating the ability of both analogue and digital quantum devices to generate entanglement in the beyond-classically-exact regime, and highlight the evolving divide between quantum and classical systems.
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Affiliation(s)
- Adam L Shaw
- California Institute of Technology, Pasadena, CA, USA.
| | - Zhuo Chen
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
- The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, Cambridge, MA, USA
| | - Joonhee Choi
- California Institute of Technology, Pasadena, CA, USA
- Department of Electrical Engineering, Stanford University, Stanford, CA, USA
| | - Daniel K Mark
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Pascal Scholl
- California Institute of Technology, Pasadena, CA, USA
| | | | - Andreas Elben
- California Institute of Technology, Pasadena, CA, USA
| | - Soonwon Choi
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA.
| | - Manuel Endres
- California Institute of Technology, Pasadena, CA, USA.
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2
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Naldesi P, Elben A, Minguzzi A, Clément D, Zoller P, Vermersch B. Fermionic Correlation Functions from Randomized Measurements in Programmable Atomic Quantum Devices. Phys Rev Lett 2023; 131:060601. [PMID: 37625073 DOI: 10.1103/physrevlett.131.060601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2022] [Accepted: 03/16/2023] [Indexed: 08/27/2023]
Abstract
We provide an efficient randomized measurement protocol to estimate two- and four-point fermionic correlations in ultracold atom experiments. Our approach is based on combining random atomic beam splitter operations, which can be realized with programmable optical landscapes, with high-resolution imaging systems such as quantum gas microscopes. We illustrate our results in the context of the variational quantum eigensolver algorithm for solving quantum chemistry problems.
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Affiliation(s)
- Piero Naldesi
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Andreas Elben
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA
- Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California 91125, USA
| | - Anna Minguzzi
- Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
| | - David Clément
- Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France
| | - Peter Zoller
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
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3
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Green AM, Elben A, Alderete CH, Joshi LK, Nguyen NH, Zache TV, Zhu Y, Sundar B, Linke NM. Experimental Measurement of Out-of-Time-Ordered Correlators at Finite Temperature. Phys Rev Lett 2022; 128:140601. [PMID: 35476480 DOI: 10.1103/physrevlett.128.140601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2021] [Revised: 02/03/2022] [Accepted: 03/09/2022] [Indexed: 06/14/2023]
Abstract
Out-of-time-ordered correlators (OTOCs) are a key observable in a wide range of interconnected fields including many-body physics, quantum information science, and quantum gravity. Measuring OTOCs using near-term quantum simulators will extend our ability to explore fundamental aspects of these fields and the subtle connections between them. Here, we demonstrate an experimental method to measure OTOCs at finite temperatures and use the method to study their temperature dependence. These measurements are performed on a digital quantum computer running a simulation of the transverse field Ising model. Our flexible method, based on the creation of a thermofield double state, can be extended to other models and enables us to probe the OTOC's temperature-dependent decay rate. Measuring this decay rate opens up the possibility of testing the fundamental temperature-dependent bounds on quantum information scrambling.
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Affiliation(s)
- Alaina M Green
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - A Elben
- Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - C Huerta Alderete
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Lata Kh Joshi
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Nhung H Nguyen
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Torsten V Zache
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Yingyue Zhu
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Bhuvanesh Sundar
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- JILA, Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - Norbert M Linke
- Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA
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4
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Rath A, van Bijnen R, Elben A, Zoller P, Vermersch B. Importance Sampling of Randomized Measurements for Probing Entanglement. Phys Rev Lett 2021; 127:200503. [PMID: 34860040 DOI: 10.1103/physrevlett.127.200503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2021] [Accepted: 09/27/2021] [Indexed: 06/13/2023]
Abstract
We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of statistical errors is obtained using classical techniques of machine learning and tensor networks using partial information on the quantum state. In current experimental settings of engineered many-body quantum systems this significantly increases the (sub-)system sizes for which entanglement can be measured. In particular, we show an exponential reduction of the required number of measurements to estimate the purity of product states and GHZ states.
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Affiliation(s)
- Aniket Rath
- Université Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
| | - Rick van Bijnen
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Andreas Elben
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Peter Zoller
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Université Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
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5
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Kokail C, Sundar B, Zache TV, Elben A, Vermersch B, Dalmonte M, van Bijnen R, Zoller P. Quantum Variational Learning of the Entanglement Hamiltonian. Phys Rev Lett 2021; 127:170501. [PMID: 34739272 DOI: 10.1103/physrevlett.127.170501] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2021] [Revised: 08/20/2021] [Accepted: 09/01/2021] [Indexed: 06/13/2023]
Abstract
Learning the structure of the entanglement Hamiltonian (EH) is central to characterizing quantum many-body states in analog quantum simulation. We describe a protocol where spatial deformations of the many-body Hamiltonian, physically realized on the quantum device, serve as an efficient variational ansatz for a local EH. Optimal variational parameters are determined in a feedback loop, involving quench dynamics with the deformed Hamiltonian as a quantum processing step, and classical optimization. We simulate the protocol for the ground state of Fermi-Hubbard models in quasi-1D geometries, finding excellent agreement of the EH with Bisognano-Wichmann predictions. Subsequent on-device spectroscopy enables a direct measurement of the entanglement spectrum, which we illustrate for a Fermi Hubbard model in a topological phase.
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Affiliation(s)
- Christian Kokail
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
| | - Bhuvanesh Sundar
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- JILA, Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - Torsten V Zache
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
| | - Andreas Elben
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
| | - Benoît Vermersch
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
| | - Marcello Dalmonte
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea 265, 34136 Trieste, Italy
| | - Rick van Bijnen
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
| | - Peter Zoller
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
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6
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Cian ZP, Dehghani H, Elben A, Vermersch B, Zhu G, Barkeshli M, Zoller P, Hafezi M. Many-Body Chern Number from Statistical Correlations of Randomized Measurements. Phys Rev Lett 2021; 126:050501. [PMID: 33605765 DOI: 10.1103/physrevlett.126.050501] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2020] [Accepted: 12/01/2020] [Indexed: 06/12/2023]
Abstract
One of the main topological invariants that characterizes several topologically ordered phases is the many-body Chern number (MBCN). Paradigmatic examples include several fractional quantum Hall phases, which are expected to be realized in different atomic and photonic quantum platforms in the near future. Experimental measurement and numerical computation of this invariant are conventionally based on the linear-response techniques that require having access to a family of states, as a function of an external parameter, which is not suitable for many quantum simulators. Here, we propose an ancilla-free experimental scheme for the measurement of this invariant, without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wave function. Remarkably, our results apply to disklike geometries that are more amenable to current quantum simulator architectures.
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Affiliation(s)
- Ze-Pei Cian
- Joint Quantum Institute, College Park, Maryland 20742, USA
- The Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Hossein Dehghani
- Joint Quantum Institute, College Park, Maryland 20742, USA
- The Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Andreas Elben
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Université Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
| | - Guanyu Zhu
- IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA
| | - Maissam Barkeshli
- Joint Quantum Institute, College Park, Maryland 20742, USA
- Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
| | - Peter Zoller
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Mohammad Hafezi
- Joint Quantum Institute, College Park, Maryland 20742, USA
- The Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
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7
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Elben A, Kueng R, Huang HYR, van Bijnen R, Kokail C, Dalmonte M, Calabrese P, Kraus B, Preskill J, Zoller P, Vermersch B. Mixed-State Entanglement from Local Randomized Measurements. Phys Rev Lett 2020; 125:200501. [PMID: 33258654 DOI: 10.1103/physrevlett.125.200501] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2020] [Accepted: 10/20/2020] [Indexed: 06/12/2023]
Abstract
We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)SCIEAS0036-807510.1126/science.aau4963].
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Affiliation(s)
- Andreas Elben
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Richard Kueng
- Institute for Integrated Circuits, Johannes Kepler University Linz, Altenbergerstrasse 69, 4040 Linz, Austria
| | - Hsin-Yuan Robert Huang
- Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA
- Department of Computing and Mathematical Sciences, Caltech, Pasadena, California 91125, USA
| | - Rick van Bijnen
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Christian Kokail
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Marcello Dalmonte
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea 265, 34136 Trieste, Italy
| | - Pasquale Calabrese
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
- SISSA, via Bonomea 265, 34136 Trieste, Italy
- INFN, via Bonomea 265, 34136 Trieste, Italy
| | - Barbara Kraus
- Institute for Theoretical Physics, University of Innsbruck, A6020 Innsbruck, Austria
| | - John Preskill
- Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA
- Department of Computing and Mathematical Sciences, Caltech, Pasadena, California 91125, USA
- Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California 91125, USA
- AWS Center for Quantum Computing, Pasadena, California 91125, USA
| | - Peter Zoller
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Université Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
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8
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Joshi MK, Elben A, Vermersch B, Brydges T, Maier C, Zoller P, Blatt R, Roos CF. Quantum Information Scrambling in a Trapped-Ion Quantum Simulator with Tunable Range Interactions. Phys Rev Lett 2020; 124:240505. [PMID: 32639800 DOI: 10.1103/physrevlett.124.240505] [Citation(s) in RCA: 28] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2020] [Accepted: 05/29/2020] [Indexed: 06/11/2023]
Abstract
In ergodic many-body quantum systems, locally encoded quantum information becomes, in the course of time evolution, inaccessible to local measurements. This concept of "scrambling" is currently of intense research interest, entailing a deep understanding of many-body dynamics such as the processes of chaos and thermalization. Here, we present first experimental demonstrations of quantum information scrambling on a 10-qubit trapped-ion quantum simulator representing a tunable long-range interacting spin system, by estimating out-of-time ordered correlators (OTOCs) through randomized measurements. We also analyze the role of decoherence in our system by comparing our measurements to numerical simulations and by measuring Rényi entanglement entropies.
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Affiliation(s)
- Manoj K Joshi
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Andreas Elben
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- University Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
| | - Tiff Brydges
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Christine Maier
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Peter Zoller
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Rainer Blatt
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Christian F Roos
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
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9
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Elben A, Yu J, Zhu G, Hafezi M, Pollmann F, Zoller P, Vermersch B. Many-body topological invariants from randomized measurements in synthetic quantum matter. Sci Adv 2020; 6:eaaz3666. [PMID: 32300654 PMCID: PMC7148113 DOI: 10.1126/sciadv.aaz3666] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/04/2019] [Accepted: 01/09/2020] [Indexed: 06/11/2023]
Abstract
Many-body topological invariants, as quantized highly nonlocal correlators of the many-body wave function, are at the heart of the theoretical description of many-body topological quantum phases, including symmetry-protected and symmetry-enriched topological phases. Here, we propose and analyze a universal toolbox of measurement protocols to reveal many-body topological invariants of phases with global symmetries, which can be implemented in state-of-the-art experiments with synthetic quantum systems, such as Rydberg atoms, trapped ions, and superconducting circuits. The protocol is based on extracting the many-body topological invariants from statistical correlations of randomized measurements, implemented with local random unitary operations followed by site-resolved projective measurements. We illustrate the technique and its application in the context of the complete classification of bosonic symmetry-protected topological phases in one dimension, considering in particular the extended Su-Schrieffer-Heeger spin model, as realized with Rydberg tweezer arrays.
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Affiliation(s)
- Andreas Elben
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Jinlong Yu
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Guanyu Zhu
- Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20740, USA
| | - Mohammad Hafezi
- Joint Quantum Institute, NIST/University of Maryland, College Park, MD 20740, USA
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA
| | - Frank Pollmann
- Department of Physics, Technical University of Munich, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology, Schellingstr. 4, D-80799 München, Germany
| | - Peter Zoller
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France
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10
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Elben A, Vermersch B, van Bijnen R, Kokail C, Brydges T, Maier C, Joshi MK, Blatt R, Roos CF, Zoller P. Cross-Platform Verification of Intermediate Scale Quantum Devices. Phys Rev Lett 2020; 124:010504. [PMID: 31976701 DOI: 10.1103/physrevlett.124.010504] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2019] [Indexed: 06/10/2023]
Abstract
We describe a protocol for cross-platform verification of quantum simulators and quantum computers. We show how to measure directly the overlap Tr[ρ_{1}ρ_{2}] and the purities Tr[ρ_{1,2}^{2}], and thus a fidelity of two, possibly mixed, quantum states ρ_{1} and ρ_{2} prepared in separate experimental platforms. We require only local measurements in randomized product bases, which are communicated classically. As a proof of principle, we present the measurement of experiment-theory fidelities for entangled 10-qubit quantum states in a trapped ion quantum simulator.
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Affiliation(s)
- Andreas Elben
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Benoît Vermersch
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Rick van Bijnen
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Christian Kokail
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Tiff Brydges
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Christine Maier
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Manoj K Joshi
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Rainer Blatt
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Christian F Roos
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - Peter Zoller
- Center for Quantum Physics, University of Innsbruck, Innsbruck A-6020, Austria and Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, Innsbruck A-6020, Austria
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Brydges T, Elben A, Jurcevic P, Vermersch B, Maier C, Lanyon BP, Zoller P, Blatt R, Roos CF. Probing Rényi entanglement entropy via randomized measurements. Science 2019; 364:260-263. [DOI: 10.1126/science.aau4963] [Citation(s) in RCA: 256] [Impact Index Per Article: 51.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2018] [Accepted: 03/19/2019] [Indexed: 11/02/2022]
Abstract
Entanglement is a key feature of many-body quantum systems. Measuring the entropy of different partitions of a quantum system provides a way to probe its entanglement structure. Here, we present and experimentally demonstrate a protocol for measuring the second-order Rényi entropy based on statistical correlations between randomized measurements. Our experiments, carried out with a trapped-ion quantum simulator with partition sizes of up to 10 qubits, prove the overall coherent character of the system dynamics and reveal the growth of entanglement between its parts, in both the absence and presence of disorder. Our protocol represents a universal tool for probing and characterizing engineered quantum systems in the laboratory, which is applicable to arbitrary quantum states of up to several tens of qubits.
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Affiliation(s)
- Tiff Brydges
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Andreas Elben
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Petar Jurcevic
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Benoît Vermersch
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Christine Maier
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Ben P. Lanyon
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Peter Zoller
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Rainer Blatt
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
| | - Christian F. Roos
- Center for Quantum Physics and Institute for Experimental Physics, University of Innsbruck, Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria
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Elben A, Vermersch B, Dalmonte M, Cirac JI, Zoller P. Rényi Entropies from Random Quenches in Atomic Hubbard and Spin Models. Phys Rev Lett 2018; 120:050406. [PMID: 29481179 DOI: 10.1103/physrevlett.120.050406] [Citation(s) in RCA: 36] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/15/2017] [Indexed: 06/08/2023]
Abstract
We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimensions. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered time-dependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in existing quantum simulators and used to measure, for instance, area law scaling of entanglement in two-dimensional spin models or the entanglement growth in many-body localized systems.
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Affiliation(s)
- A Elben
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - B Vermersch
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - M Dalmonte
- The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
| | - J I Cirac
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
| | - P Zoller
- Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
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