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## Nobel Prize : Quantum Entanglement Unveiled

7 October 2022: We have replaced our initial one-paragraph announcement with a full-length Focus story.

The Nobel Prize in Physics this year recognizes efforts to take quantum weirdness out of philosophy discussions and to place it on experimental display for all to see. The award is shared by Alain Aspect, John Clauser, and Anton Zeilinger, all of whom showed a mastery of entanglement—a quantum relationship between two particles that can exist over long distances. Using entangled photons, Clauser and Aspect performed some of the first “Bell tests,” which confirmed quantum mechanics predictions while putting to bed certain alternative theories based on classical physics. Zeilinger used some of those Bell-test techniques to demonstrate entanglement control methods that can be applied to quantum computing, quantum cryptography, and other quantum information technologies.

Since its inception, quantum mechanics has been wildly successful at predicting the outcomes of experiments. But the theory assumes that some properties of a particle are inherently uncertain—a fact that bothered many physicists, including Albert Einstein. He and his colleagues expressed their concern in a paradox they described in 1935 [ 1 ]: Imagine creating two quantum mechanically entangled particles and distributing them between two separated researchers, characters later named Alice and Bob. If Alice measures her particle, then she learns something about Bob’s particle—as if her measurement instantaneously changed the uncertainty about the state of his particle. To avoid such “spooky action at a distance,” Einstein proposed that lying underneath the quantum framework is a set of classical “hidden variables” that determine precisely how a particle will behave, rather than providing only probabilities.

The hidden variables were unmeasurable—by definition—so most physicists deemed their existence to be a philosophical issue, not an experimental one. That changed in 1964 when John Bell of the University of Wisconsin-Madison, proposed a thought experiment that could directly test the hidden variable hypothesis [ 2 ]. As in Einstein’s paradox, Alice and Bob are each sent one particle of an entangled pair. This time, however, the two researchers measure their respective particles in different ways and compare their results. Bell showed that if hidden variables exist, the experimental results would obey a mathematical inequality. However, if quantum mechanics was correct, the inequality would be violated.

Bell’s work showed how to settle the debate between quantum and classical views, but his proposed experiment assumed detector capabilities that weren’t feasible. A revised version using photons and polarizers was proposed in 1969 by Clauser, then at Columbia University, along with his colleagues [ 3 ]. Three years later, Clauser and Stuart Freedman (both at the University of California, Berkeley) succeeded in performing that experiment [ 4 ].

The Freedman-Clauser experiment used entangled photons obtained by exciting calcium atoms. When a calcium atom de-excites, it can emit two photons whose polarizations are aligned. The researchers installed two detectors (Alice and Bob) on opposite sides of the calcium source and measured the rate of coincidences—two photons hitting the detectors simultaneously. Each detector was equipped with a polarizer that could be rotated to an arbitrary orientation.

Freedman and Clauser showed theoretically that quantum mechanics predictions diverge strongly from hidden variable predictions when Alice and Bob’s polarizers are offset from each other by 22.5° or 67.5°. The researchers collected 200 hours of data and found that the coincidence rates violated a revamped Bell’s inequality, proving that quantum mechanics is right.

The results of the first Bell test were a blow to hidden variables, but there were “loopholes” that hidden-variable supporters could claim to rescue their theory. One of the most significant loopholes was based on the idea that the setting of Alice’s polarizer could have some influence on Bob’s polarizer or on the photons that are created at the source. Such effects could allow the elements of a hidden-variable system to “conspire” together to produce measurement outcomes that mimic quantum mechanics.

To close this so-called locality loophole, Aspect and his colleagues at the Institute of Optics Graduate School in France performed an updated Bell test in 1982, using an innovative method for randomly changing the polarizer orientations [ 5 ]. The system worked like a railroad switch, rapidly diverting photons between two separate “tracks,” each with a different polarizer. The changes were made as the photons were traveling from the source to the detectors, so there was not enough time for coordination between supposed hidden variables.

Zeilinger, who is now at the University of Vienna, has also worked on removing loopholes from Bell tests (see Viewpoint: Closing the Door on Einstein and Bohr’s Quantum Debate , written by Aspect). In 2017, for example, he and his collaborators devised a way to use light from distant stars as a random input for setting polarizer orientations (see Synopsis: Cosmic Test of Quantum Mechanics ).

Zeilinger also used the techniques of entanglement control to explore practical applications, such as quantum teleportation and entanglement swapping. For the latter, he and his team showed in 1998 that they could create entanglement between two photons that were never in contact [ 6 ]. In this experiment, two sets of entangled photon pairs are generated at two separate locations. One from each pair is sent to Alice and Bob, while the other two photons are sent to a third person, Cecilia. Cecilia performs a Bell-like test on her two photons, and when she records a particular result, Alice’s photon winds up being entangled with Bob's. This swapping could be used to send entanglement over longer distances than is currently possible with optical fibers (see Research News: The Key Device Needed for a Quantum Internet ).

“Quantum entanglement is not questioned anymore,” says quantum physicist Jean Dalibard from the College of France. “It has become a tool, in particular in the emerging field of quantum information processing, and the three nominated scientists can be considered as the godfathers of this new domain.”

Quantum information specialist Jian-Wei Pan of the University of Science and Technology of China in Hefei says the winners are fully deserving of the prize. He has worked with Zeilinger on several projects, including a quantum-based satellite link (see Focus: Intercontinental, Quantum-Encrypted Messaging and Video ). “Now, in China, we are putting a lot of effort into actually turning these dreams into reality, hoping to make the quantum technologies practically useful for our society.”

–Michael Schirber

Michael Schirber is a Corresponding Editor for Physics Magazine based in Lyon, France.

- A. Einstein et al. , “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47 , 777 (1935) .
- J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1 , 195 (1964) .
- J. F. Clauser et al. , “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23 , 880 (1969) .
- S. J. Freedman and J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28 , 938 (1972) .
- A. Aspect et al. , “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49 , 1804 (1982) .
- J. W. Pan et al. , “Experimental entanglement swapping: Entangling photons that never interacted,” Phys. Rev. Lett. 80 , 3891 (1998) .

## More Information

Research News: Hiding Secrets Using Quantum Entanglement

Research News: Diagramming Quantum Weirdness

APS press release

The Nobel Prize in Physics 2022 (Nobel Foundation)

## Experimental Test of Bell's Inequalities Using Time-Varying Analyzers

Alain Aspect, Jean Dalibard, and Gérard Roger

Phys. Rev. Lett. 49 , 1804 (1982)

Published December 20, 1982

## Experimental Entanglement Swapping: Entangling Photons That Never Interacted

Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger

Phys. Rev. Lett. 80 , 3891 (1998)

Published May 4, 1998

## Experimental Test of Local Hidden-Variable Theories

Stuart J. Freedman and John F. Clauser

Phys. Rev. Lett. 28 , 938 (1972)

Published April 3, 1972

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## Physical Review Research

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## Uncertainty principle of quantum processes

Yunlong xiao, kuntal sengupta, siren yang, and gilad gour, phys. rev. research 3 , 023077 – published 28 april 2021.

- Citing Articles (7)
- INTRODUCTION
- PRELIMINARIES
- MAASSEN–UFFINK UNCERTAINTY RELATIONS
- CONCLUSIONS AND DISCUSSIONS
- ACKNOWLEDGMENTS

Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics. The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space. However, many physical systems are dynamic in nature and described with the formalism of a quantum channel. In this paper, we show that the uncertainty principle can be reformulated to include process measurements that are performed on quantum channels. Since both the preparation of quantum states and the implementation of quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality. More specifically, we obtain expressions that generalize the Maassen–Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.

- Received 19 April 2020
- Accepted 29 March 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.023077

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

## Physics Subject Headings (PhySH)

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## Authors & Affiliations

- 1 School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639673, Singapore
- 2 Complexity Institute, Nanyang Technological University, Singapore 639673, Singapore
- 3 Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada
- 4 Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta, T2N 1N4, Canada
- * [email protected]
- † [email protected]

## Article Text

Vol. 3, Iss. 2 — April - June 2021

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Schematic illustration of the process positive-operator-valued measures (PPOVMs): (a) PPOVM T 1 and (b) PPOVM T 2 .

Schematic illustration of the lattice structure exhibited in example t17 excluding the GLB. Each point stands for an element, and the red line represents the binary relation “ ≺ ” between elements. In this plot, a lower point is majorized by the higher point whenever they are connected with a red line. Obviously, here, b S ↓ ≺ b S and b S ≺ b S ↓ , but b S ≠ b S ↓ .

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## Collective randomized measurements in quantum information processing

Phys. rev. lett., satoya imai, géza tóth, and otfried gühne.

The concept of randomized measurements on individual particles has proven to be useful for analyzing quantum systems and is central for methods like shadow tomography of quantum states. We introduce {} randomized measurements as a tool in quantum information processing. Our idea is to perform measurements of collective angular momentum on a quantum system and actively rotate the directions using simultaneous multilateral unitaries. Based on the moments of the resulting probability distribution, we propose systematic approaches to characterize quantum entanglement in a collective-reference-frame-independent manner. First, we show that existing spin-squeezing inequalities can be accessible in this scenario. Next, we present an entanglement criterion based on three-body correlations, going beyond spin-squeezing inequalities with two-body correlations. Finally, we apply our method to characterize entanglement between spatially-separated two ensembles.

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## Quantum technology: from research to application

- Published: 27 April 2016
- Volume 122 , article number 130 , ( 2016 )

## Cite this article

- Wolfgang P. Schleich 1 ,
- Kedar S. Ranade 1 ,
- Christian Anton 2 ,
- Markus Arndt 3 ,
- Markus Aspelmeyer 4 ,
- Manfred Bayer 5 ,
- Gunnar Berg 6 ,
- Tommaso Calarco 7 ,
- Harald Fuchs 8 ,
- Elisabeth Giacobino 9 ,
- Markus Grassl 10 ,
- Peter Hänggi 11 ,
- Wolfgang M. Heckl 12 ,
- Ingolf-Volker Hertel 13 ,
- Susana Huelga 14 ,
- Fedor Jelezko 15 ,
- Bernhard Keimer 16 ,
- Jörg P. Kotthaus 17 ,
- Gerd Leuchs 10 ,
- Norbert Lütkenhaus 18 ,
- Ueli Maurer 19 ,
- Tilman Pfau 20 ,
- Martin B. Plenio 14 ,
- Ernst Maria Rasel 21 ,
- Ortwin Renn 22 ,
- Christine Silberhorn 23 ,
- Jörg Schiedmayer 24 ,
- Doris Schmitt-Landsiedel 25 ,
- Kurt Schönhammer 26 ,
- Alexey Ustinov 27 ,
- Philip Walther 28 ,
- Harald Weinfurter 29 ,
- Emo Welzl 19 ,
- Roland Wiesendanger 30 ,
- Stefan Wolf 31 ,
- Anton Zeilinger 4 &
- Peter Zoller 32

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The term quantum physics refers to the phenomena and characteristics of atomic and subatomic systems which cannot be explained by classical physics. Quantum physics has had a long tradition in Germany, going back nearly 100 years. Quantum physics is the foundation of many modern technologies. The first generation of quantum technology provides the basis for key areas such as semiconductor and laser technology. The “new” quantum technology, based on influencing individual quantum systems, has been the subject of research for about the last 20 years. Quantum technology has great economic potential due to its extensive research programs conducted in specialized quantum technology centres throughout the world. To be a viable and active participant in the economic potential of this field, the research infrastructure in Germany should be improved to facilitate more investigations in quantum technology research.

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## Quantum Technologies

## Scientists and citizens: getting to quantum technologies

http://www.wissenschaft.de/technik-kommunikation/physik/-/journal_content/56/12054/1119998/Welt-weit-erste-quantenkryptografisch-verschl%C3%BCs-selte-Bank%C3%BCberweisung/ (last accessed: 17 February 2015).

http://www.heise.de/tr/artikel/Photonen-als-Wahl-helfer-280423.html (last accessed: 17 February 2015).

Among these is the Institute for Quantum Computing in Waterloo, Canada, with start-up funding of around CAD 300 million. It is currently the largest centre for quantum information worldwide. The Center for Quantum Technologies in Singapore has been part of the national university since 2007 and is being funded for 10 years with start-up financing of SGD 158 million. In the US, the Joint Quantum Institute was established with the joint financing by the National Institute of Standards and Technology (NIST) and the University of Maryland (College Park). It is also the sponsor for a start-up in South Korea, with which the Max Planck Institute is also involved. In Japan, quantum cryptography is supported by a consortium which includes Toshiba, Mitsubishi and Japanese Telecom. In the UK at present, €270 million are allocated by the Engineering and Physical Sciences Research Council for a program intended to support the application of technologies based on quantum physics.

For example, former board members of Blackberry-RIM, Mike Lazaridis and Douglas Fregin, have established an investment fund of $100 million to support new businesses and spinoffs for quantum technology in Canada (“Quantum Valley”). In the US, the Quantum Wave Fund ( http://qwcap.com , last accessed: 18 February 2015) is a venture capital company making targeted investments in quantum technology.

Born et al. [ 2 ].

A further important advance in the understanding of the general structure of the Heisenberg approach was provided by Paul Dirac in 1925.

Schrödinger [ 12 ].

Schrödinger [ 13 ].

A. Einstein, B. Podolsky, N. Rosen “Can quantum-mechanical description be considered complete?”, Physical Review, 47 , p. 777 [ 5 ].

In this case, a semitransparent mirror, which partly allows a wave to pass and partly reflects it.

Named after Enrico Fermi (1901–1954) and Satyendranath Bose (1894–1974) respectively.

For example, since 2012 industry has supported the Alcatel-Lucent’s Bell Labs guest professorship at the Friedrich-Alexander-University in Erlangen-Nuremberg, which researches practical applications together with the Max Planck Institute for the Science of Light.

The decomposition of a natural number into a product of prime numbers is of great importance for encryption methods.

Classical optics usually treats light as a wave. If light is considered at fundamental quantum level, this wave consists of discrete particles (photons).

As shown by the example of the Center for NanoScience (CeNS) at Ludwig-Maximilians-Universität Munich, this is a suitable way to facilitate technology spinoffs. http://www.cens.de/ (last accessed: 18 February 2015). accessed: 18 February 2015).

http://www.exist.de/exist-forschungstransfer/ (last accessed: 18 February 2015).

Superconductors are materials whose electrical resistance disappears below a transition temperature.

In September 2014, the European Telecommunication Standards Institute (ETSI) published an extensive report on the subject of quantum cryptography. The report “Quantum Safe Cryptography and Security” (ISBN 979-10-92620-03-0) is available at the following link: http://docbox.etsi.org/Workshop/2014/201410_CRYPTO/Quantum_Safe_Whitepaper_1_0_0.pdf (last accessed: 19 February 2015).

The key phrase “post-quantum cryptography” is used to describe investigations by computer scientists into conventional methods which remain secure if quantum computers are available to solve the factorization problem, for example. These methods do not achieve the fundamental security of quantum cryptography, but they are expected to be simpler to implement.

Named after Charles H. Bennett (*1943) and Gilles Brassard (*1955) and Artur Ekert (*1961) respectively. A protocol in this case is understood to be a series of handling instructions, at the conclusion of which success (in this context a key) or failure occurs.

The German Federal Ministry of Research and Technology supports the investigation of basic science for quantum communication as part of the IKT 2020 development scheme.

Named after David P. DiVincenzo (*1959); D. P. DiVincenzo [ 4 ]. Topics in Quantum Computers. In: L. Kowenhoven, G. Schön and L.L. Sohn (pub.): Mesoscopic Electron Transport. NATO ASI Series E. No. 345, Kluwer Academic Publishers, Dordrecht (1997), p. 657.

J. I. Cirac & P. Zoller. Quantum Computations with Cold Trapped Ions. Physical Review Letters [ 3 ], 4091–4094.

Unit of absolute temperature (0 °C = 273.15 K).

One femtosecond (fs) = 10 −15 s; one attosecond (as) = 10 −18 s.

A gyroscope is a device which determines spatial orientation.

These SI (French, Système international ) units include basic units such as the metre, kilogram, second, ampere and Kelvin.

GMR = Giant Magnetoresistance; magnetic field-dependent resistance.

The ZT value is a dimensionless variable which affects the efficiency of thermoelectric generators. The larger its value, the closer the efficiency is to the thermodynamic maximum.

The best materials known up to now have a ZT value of one.

C.H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing , vol. 175 (New York, 1984), pp. 175–179

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## Author information

Authors and affiliations.

Institut für Quantenphysik, Universität Ulm, 89069, Ulm, Germany

Wolfgang P. Schleich & Kedar S. Ranade

Department Science-Policy-Society, German National Academy of Sciences Leopoldina, Jägerberg 1, 06108, Halle, Germany

Christian Anton

Faculty of Physics, VCQ & QuNaBioS, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria

Markus Arndt

Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, 1090, Vienna, Austria

Markus Aspelmeyer & Anton Zeilinger

Experimentelle Physik 2, Technische Universität Dortmund, 44227, Dortmund, Germany

Manfred Bayer

Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 006120, Halle, Germany

Gunnar Berg

Institut für Komplexe Quantensysteme, Universität Ulm, 89069, Ulm, Germany

Tommaso Calarco

Physikalisches Institut, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm Str. 10, 48149, Münster, Germany

Harald Fuchs

Université de Paris, Paris, France

Elisabeth Giacobino

Max Planck Institute for the Science of Light, 91058, Erlangen, Germany

Markus Grassl & Gerd Leuchs

Institut für Physik, Universität Augsburg, Universitätsstr. 1, 86135, Augsburg, Germany

Peter Hänggi

Oskar-von-Miller Lehrstuhl für Wissenschaftskommunikation, School of Education & Physik Department, Technische Universität München, c/o Deutsches Museum, Museumsinsel 1, 80538, Munich, Germany

Wolfgang M. Heckl

Max-Born-Institut (MBI), im Forschungsverbund Berlin e.V, Max-Born-Institut Max-Born-Straße 2A, 12489, Berlin, Germany

Ingolf-Volker Hertel

Institute of Theoretical Physics, Universität Ulm, Albert-Einstein-Allee 11, 89069, Ulm, Germany

Susana Huelga & Martin B. Plenio

Institut für Quantenoptik, Universität Ulm, Albert-Einstein-Allee 11, 89081, Ulm, Germany

Fedor Jelezko

Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569, Stuttgart, Germany

Bernhard Keimer

Fakultät für Physik and Center for NanoScience (CeNS), Ludwig-Maximilians-Universität, Geschwister-Scholl-Platz 1, 80539, Munich, Germany

Jörg P. Kotthaus

Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, Waterloo, Canada

Norbert Lütkenhaus

Department of Computer Science, ETH Zurich, Zurich, Switzerland

Ueli Maurer & Emo Welzl

Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70550, Stuttgart, Germany

Tilman Pfau

Institut für Quantenoptik, Leibniz-Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

Ernst Maria Rasel

Institut für Sozialwissenschaften, Universität Stuttgart, Seidenstr. 36, 70174, Stuttgart, Germany

Ortwin Renn

Applied Physics, University of Paderborn, Warburger Strasse 100, 33098, Paderborn, Germany

Christine Silberhorn

Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020, Vienna, Austria

Jörg Schiedmayer

Lehrstuhl für Technische Elektronik, Technische Universität München, Arcisstr. 21, 80333, Munich, Germany

Doris Schmitt-Landsiedel

Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077, Göttingen, Germany

Kurt Schönhammer

Physikalisches Institut, Karlsruhe Institute of Technology, 76131, Karlsruhe, Germany

Alexey Ustinov

Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria

Philip Walther

Faculty of Physics, Ludwig-Maximilians-Universität, 80799, Munich, Germany

Harald Weinfurter

Department of Physics, University of Hamburg, Jungiusstraße 11, 20355, Hamburg, Germany

Roland Wiesendanger

Faculty of Informatics, Universita della Svizzera Italiana, Via G. Buffi 13, 6900, Lugano, Switzerland

Stefan Wolf

Institute for Theoretical Physics, University of Innsbruck, 6020, Innsbruck, Austria

Peter Zoller

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## Corresponding author

Correspondence to Christian Anton .

## Additional information

The following article is the re-publication of a text of previously published under the German National Academy of Sciences Leopoldina, acatech (the National Academy of Science and Engineering), the Union of the German Academies of Science and Humanities (ed.) (2015): Quantum Technology: From research to application. Halle (Saale), 64 pages. ISBN : 978-3-8047-3343-5. The German National Library lists this publication in the German National Bibliography; detailed bibliographic information can be accessed online at http://dnb.d-nb.de .

This paper is part of the topical collection “Quantum Repeaters: From Components to Strategies” guest edited by Manfred Bayer, Christoph Becher and Peter van Loock.

## 1.1 Funding schemes and projects

A large number of projects and research groups in the area of quantum technology receive funding in Germany. Sponsors include the German Research Foundation (DFG), the Max Planck Society, the German Federal Ministry of Education and Research (BMBF), as well as a number of other regional organisations. The EU also provided funding as part of its fifth, sixth and seventh Framework Programmes.

A good overview of these projects is provided by the QIPC (Quantum Information Processing and Communication) roadmap QUROPE/QUIET2, available online at http://qurope.eu/projects/ . The following is a list of some of these projects; the list serves as an example and is by no means exhaustive.

AQUTE | Atomic Quantum Technologies (EU Integrating project) |

CORNER | Correlated Noise Errors in Quantum Information Processing (EU STREP project), 2008–11 |

FINAQS | Future Inertial Atomic Quantum Sensors |

GOCE | Gravity Field and Steady-State Ocean Circulation Explorer |

HIP | Hybrid Information Processing (EU STREP project), 2008–11 |

ICT 2020 | Information and Communication Technologies (German Federal Ministry of Education and Research/BMBF) with the collaborative projects: |

• QuOReP (quantum repeater platform with quantum optical methods) | |

• QuaHL-Rep (quantum semiconductor repeaters) | |

• QUIMP (quantum interface between optical and microwave photons) | |

• IQuRe (quantum repeater information theory) | |

IQS | Inertial Atomic and Photonic Quantum Sensors: Ultimate Performance and Application |

LISA-II | Laser Interferometer Space Antenna II |

PICC | Physics of Ion Coulomb Crystals (EU project), 2010–2013 |

Q-ESSENCE | Quantum Interfaces, Sensors, and Communication based on Entanglement (EU Integrating project), 2010–2014 |

QNEMS | Quantum Nanoelectromechanical Systems, an FET STREP EU project 2009–2012 |

QUANTUS | Quantum Gases in Microgravity |

SECOQC | Secure Communication using Quantum Cryptography |

(Sixth EU Framework Programme) | |

SFB 450 | Collaborative Research Centre for the Analysis and Control of Ultrafast Photoinduced Reactions |

SFB 631 | Solid State Based Quantum Information Processing: Physical Concepts and Materials Aspects, 2003–2015) |

SFB/TRR 21 | Control of Quantum Correlations in Tailored Matter (CO.CO.MAT) |

In addition, support was provided for individual researchers through, for example, Alexander von Humboldt Professorships. These included David DiVincenzo (RWTH Aachen), Martin Plenio (Ulm) and Vahid Sandoghdar (Erlangen-Nürnberg).

## 1.2 Extended bibliography

1.2.1 books.

Audretsch, J.: Verschränkte Welt. Faszination der Quanten , Wiley–VCH, 2002.

Nielsen, M. A.; Chuang, I. L: Quantum Computation and Quantum Information , Cambridge University Press, 2000.

McManamon, P.; Willner, A. E. et al.: Optics and Photonics – Essential Technologies for Our Nation, The National Academies Press, 2013.

Peres, A.: Quantum Theory: Concepts and Methods, Springer-Verlag, 1995.

Renn, O.; Zwick, M. M.: Risiko - und Technikakzeptanz , Springer-Verlag, 1997.

Zeilinger, A.: Einsteins Schleier: Die neue Welt der Quantenphysik , Goldmann Verlag, 2005.

## 1.2.2 Review articles

Spektrum Dossier 4/2010: “Quanteninformation”.

“The Age of the Qubit: A new era of quantum information in science and technology”, Institute of Physics, 2011.

Cirac, J. I.; Zoller, P.: “New Frontiers in Quantum Information with Atoms and Ions”, Physics Today (2004), pp. 38–44.

Coffey, V. C.: “Next-Gen Quantum Networks”, Optics & Photonics News (March 2013), pp. 34–41.

Cronin, A. D.; Schmiedmayer, J.; Pritchard, D. E.: “Optics and interferometry with atoms and molecules”, Reviews of Modern Physics , volume 81 (2009), pp. 1051–1129.

Hänggi, P.: “Harvesting randomness”, Nature Materials , volume 10 (2011), pp. 6–7.

Ladd, T.D.; Jelezko, F.; Laflamme, R.; Nakamura, Y.; Monroe, C.; O’Brien, J.L.: “Quantum computers”, Nature , volume 464 (2010), pp. 45–53.

Leuchs, G.: “Wie viel Anschauung verträgt die Quantenmechanik?”, PdN – Physik in der Schule , volume 62 (2013), p. 5.

Monroe, C.: “Quantum Information Processing with Atoms and Photons”, Nature , volume 416 (2002), pp. 238–246.

Zoller, P. et al.: “Quantum information processing and communication”, The European Physical Journal D — Atomic, Molecular, Optical and Plasma Physics , volume 36 (2005), pp. 203–228.

## 1.2.3 Individual works

Aspect, A.; Dalibard, J.; Roger, G.: “Experimental Test of Bell’s Inequalities using time-varying Analyzers”, Physical Review Letters , volume 49 (1982), pp. 1804–1807.

Bell, J. S.: “On the Einstein–Podolsky–Rosen-Paradox”, Physics , volume 1 (1964), pp. 195–200.

Bennett, C.H.; Brassard, G.: “Quantum Cryptography: Public Key Distribution and Coin Tossing”, Proceedings of IEEE International Conference on Computers, Systems & Signal Processing, Bangalore, India , pp. 175–179 (1984).

Cirac, J.I.; Zoller, P.: “Quantum Computations with Cold Trapped Ions”, Physical Review Letters , volume 74 (1995), pp. 4091–4094.

Einstein, A.; Podolsky, B.; Rosen, N.: “Can quantum-mechanical description of physical reality be considered complete?”, Physical Review , volume 47 (1935), pp. 777–780.

Ekert, A.K.: “Quantum cryptography based on Bell’s Theorem”, Physical Review Letters , volume 67 (1991), pp. 661–663.

Feynman, R.P.: “Simulating physics with computers”, International Journal of Theoretical Physics , volume 21 (1982), pp. 467–488.

Heisenberg, W.: “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik”, Zeitschrift für Physik , volume 43 (1927), pp. 172–198.

Joy, B.: “Why the future doesn’t need us”, Wired , April 2000 (see http://www.wired.com/wired/archive/8.04/joy_pr.html )

Schrödinger, E.: “Die gegenwärtige Situation in der Quantenmechanik”, Die Naturwissenschaften , volume 23 (1935), pp.807–812, 823–828, 844–849.

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Schleich, W.P., Ranade, K.S., Anton, C. et al. Quantum technology: from research to application. Appl. Phys. B 122 , 130 (2016). https://doi.org/10.1007/s00340-016-6353-8

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DOI : https://doi.org/10.1007/s00340-016-6353-8

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## JSmol Viewer

Quantum physics education research over the last two decades: a bibliometric analysis.

## 1. Introduction

- the relevance of quantum physics education within science education research on the one hand, and
- given the upcoming tasks in teaching modern quantum technologies to a broad audience on the other hand,
- How has the scientific output in terms of research publications and citations of articles on quantum physics education has developed over time from 2000 to 2021 in science education research?
- Who are the most active authors and countries publishing articles on quantum physics education research from 2000 to 2021?
- What are the most relevant publishing venues in science education research through which the results on quantum physics education are disseminated from 2000 to 2021 and which are the most cited articles?
- Can a broad collaboration among researchers and countries in quantum physics education research be observed?
- What are the most relevant keywords, and which co-occurrence patterns exist in articles on quantum physics education research?
- Study design: Definition of research questions and database selection.
- Data collection: Search query and data export.
- Data analysis: Decision on bibliometric methods that can be used to clarify the research questions and selection of software to conduct the data analysis.
- Data visualisation: Selection of visualisation method and appropriate mapping software.
- Interpretation: Interpretation of bibliometric analysis’ results.

## 2.1. Study Design

2.2. data collection, 2.3. data analysis and visualisation, 2.4. limitations.

- The numbers of published papers (e.g., by author or country) on quantum physics education reported in this article only refer to the bibliographic data documented in Scopus and Web of Science, respectively. Reported values should therefore not be considered as fixed. The latter holds especially true for the exact number of citations, since not necessarily all citations of a given article are recorded in the databases. In this way, orders for the most frequently cited articles or authors could deviate from reality or articles or authors could even be missing unfairly in such orders. However, we argue that the relevance of this limitation is restricted by the well-justified data collection (cf. Section 2 ) based on two of the most relevant databases, Scopus and Web of Science.
- Some authors do not publish many scientific articles but are instead active in important projects or initiatives, for example, or have a strong influence on the research field in other ways. This cannot be taken into account in bibliometric studies.
- In this study, we only focused on articles published in scientific journals so that future studies can also consider other sources, e.g., books or conference proceedings.
- In our analysis, we investigated the number of citations for the articles included in our database. Although the role of self-citations in scientific communication has previously been analyzed across disciplines [ 38 ], there is an ongoing debate “on the principles of the role of author-self citation”, and “there is no real consensus concerning how this type of self-citations should be defined operatively” ([ 39 ], p. 64). We did not specifically analyze self-citations in the field of quantum physics education research in this study but this could be of interest for further research.
- Altmetrics are social web metrics for published articles that are increasingly used as estimates of publications’ impact, cf. [ 40 ]. They are not considered in this study. However, this could be a starting point for further research.

## 3.1. Development of the Scientific Output on Quantum Physics Education Research

3.2. most active authors and countries publishing articles on quantum physics education research, 3.3. most relevant journals and most cited articels on quantum physics education, 3.4. collaborations among researchers and countries in quantum physics education research, 3.5. keyword co-occurrence patterns in quantum physics education research, 4. discussion and conclusions, 4.1. discussion of performance analysis results (research questions 1–3).

- Main results on research question 1 : The number of published papers on quantum physics education research has increased steadily over the observation period from 31 articles in 2000 to 118 articles in 2020, with an annual growth rate of about 6.9 % .
- Main results on research question 2 : The research on quantum physics education is significantly driven by authors from the USA: more than 1/3 of the documents analysed were published by a corresponding author from the USA. Against this backdrop, it is not surprising that among the top ten most productive authors, seven are from the USA (led by Singh, C.). Furthermore, among the ten leading countries in the research field, five are from Europe (UK, Italy, Germany, Spain and France).
- Main results on research question 3 : The two journals American Journal of Physics and European Journal of Physics published the most papers on quantum physics education research and the number of publications in these journals increased more than in all other journals over the observation period. Among the top ten most cited papers on quantum physics education research, nine articles are published in American Journal of Physics —the latter is true regardless of whether one analyses global or local citations.

## 4.2. Discussion of Science Mapping Results (Research Questions 4 and 5)

- Main results on research question 4 : The scientific community engaged with quantum physics education research has not formed well-established (international) collaborations yet. Instead, the community is characterised by several smaller and predominantly national collaborations (cf. research question 4).
- Main results on research question 5 : Quantum physics education research comprises two main areas, as a co-word analysis revealed. On the one hand, quantum physics education research is dedicated to reconstructing quantum physics content for teaching; on the other hand, it focuses on empirical research into learning and teaching quantum physics. These two pillars are by no means disconnected, but rather interconnected and are complemented by smaller research areas that primarily focus on quantum physics experiments for laboratory courses. During the observation period, a shift in the research focus from more content-specific work to empirical studies on the teaching and learning of quantum physics can be observed.

## Institutional Review Board Statement

Informed consent statement, data availability statement, conflicts of interest.

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Click here to enlarge figure

Database | Search Query | Refinements | Outcome |
---|---|---|---|

Scopus | SRCTITLE((physics OR science) AND education) AND SRCTYPE(j) AND (PUBYEAR > 1999 AND PUBYEAR < 2022) AND (TITLE-ABS-KEY(“quantum physics”) OR TITLE-ABS-KEY(“mechanics”) OR TITLE-ABS-KEY(“quantum”)) | - | 231 documents |

Web of Science | (TS = (physics) OR TS = (science)) AND TS = (education) AND PY = (2000–2021) AND TI = (“quantum physics”) OR TI = (“quantum mechanics”) OR TI = (“quantum”) OR AB = (“quantum physics”) OR AB = (“quantum mechanics”) OR AB = (“quantum”) OR AK = (“quantum physics”) OR AK = (“quantum mechanics”) OR AK = (“quantum”) | Restriction to articles published in journals and to the research area Education Educational Research | 1379 documents |

Rubric | Summary |
---|---|

Main information about data | |

Timespan | 2000–2021 |

Number of sources | 44 |

Number of documents | 1520 |

Average years from publication | 8.71 |

Average citations per document | 9.70 |

Average citations per year per document | 0.93 |

Total number of references (without duplicates) | 24,497 |

Total number of author keywords | 1660 |

Authors | |

Number of authors | 2607 |

Number of authors of single-authored documents | 422 |

Number of authors of multi-authored documents | 2185 |

Authors collaboration | |

Number of single-authored documents | 540 |

Authors per document | 1.72 |

Co-authors per document | 2.24 |

Research Question | Main Technique (Concrete Analysis) |
---|---|

1. How has the scientific output in terms of research publications and citations of articles on quantum physics education has developed over time from 2000 to 2021 in science education research? | Performance analysis (e.g., analysis of (a) the number of articles published per year and (b) the number of average article citations per year) |

2. Who are the most active authors and countries publishing articles on quantum physics education research from 2000 to 2021? | Performance analysis (e.g., identification of (a) the most productive authors inlcuding their scientific production over time and (b) the most productive countries) |

3. What are the most relevant publishing venues in science education research through which the results on quantum physics education are disseminated from 2000 to 2021 and which are the most cited articles? | Performance analysis (e.g., identification of (a) the articles most cited and (b) the most relevant sources in terms of the number of published articles and their temporal development) |

4. Can a broad collaboration among researchers and countries in quantum physics education research be observed? | Science mapping (e.g., co-authorship analysis) |

5. What are the most relevant keywords, and which co-occurrence patterns exist in articles on quantum physics education research? | Science mapping (e.g., co-word analysis) |

Most Productive Authors | # Articles |
---|---|

1. Singh, C. | 33 |

2. Marshman, E. | 16 |

3. Robinett, R. | 14 |

4. Marsiglio, F. | 13 |

5. Belloni, M. | 8 |

6. Kohnle, A. | 8 |

7. Passante, G. | 8 |

8. Shaffer, P. | 8 |

9. Shegelski, M. | 8 |

10. Emigh, P. | 7 |

Most Relevant Sources | # Articles |
---|---|

1. American Journal of Physics | 477 |

2. European Journal of Physics | 465 |

3. Journal of Chemical Education | 231 |

4. Physical Review (ST) Physics Education Research | 72 |

5. Physics Education | 57 |

6. Science & Education | 40 |

7. Chemistry Education Research and Practice | 22 |

8. International Journal of Science Education | 12 |

9. The Physics Teacher | 9 |

10. International Journal of Mathematical Education in Science and Technology | 7 |

Corresponding Author | Publication Year | Journal | TC | TC/Year | Reference |
---|---|---|---|---|---|

Bender, C.M. | 2003 | Am. J. Phys. | 268 | 14.11 | [ ] |

Novotny, L. | 2010 | Am. J. Phys. | 233 | 19.42 | [ ] |

Bonneau, G. | 2001 | Am. J. Phys. | 170 | 8.10 | [ ] |

Griffiths, D.J. | 2001 | Am. J. Phys. | 158 | 7.52 | [ ] |

Brun, T.A. | 2002 | Am. J. Phys. | 158 | 7.90 | [ ] |

Bender, C.M. | 2013 | Am. J. Phys. | 149 | 16.56 | [ ] |

Boatman, E.M. | 2005 | J. Chem. Educ. | 149 | 8.76 | [ ] |

Singh, C. | 2001 | Am. J. Phys. | 148 | 7.05 | [ ] |

Case, W.B. | 2008 | Am. J. Phys. | 137 | 9.79 | [ ] |

Laloë, F. | 2001 | Am. J. Phys. | 129 | 6.14 | [ ] |

Corresponding Author | Publication Year | Journal | LCS | GCS | Reference |
---|---|---|---|---|---|

Singh, C. | 2001 | Am. J. Phys. | 38 | 148 | [ ] |

Müller, R. | 2002 | Am. J. Phys. | 31 | 101 | [ ] |

Singh, C. | 2008 | Am. J. Phys. | 28 | 104 | [ ] |

Galvez, E.J. | 2005 | Am. J. Phys. | 27 | 63 | [ ] |

Kohnle, A. | 2014 | Eur. J. Phys. | 25 | 46 | [ ] |

Wittmann, M.C. | 2002 | Am. J. Phys. | 24 | 88 | [ ] |

Dehlinger, D. | 2002 | Am. J. Phys. | 22 | 85 | [ ] |

Zollman, D.A. | 2002 | Am. J. Phys. | 23 | 96 | [ ] |

Singh, C. | 2008 | Am. J. Phys. | 22 | 85 | [ ] |

Cataloglu, E. | 2002 | Am. J. Phys. | 21 | 78 | [ ] |

Cluster | Researchers and Countries | Exemplary Publication(s) |
---|---|---|

Brown | Perkins, Wieman, McKagan (USA) | [ ] |

Blue | Krijtenburg-Lewerissa, Pol, Brinkman, van Joolingen (The Netherlands) | [ ] |

Orange | Emigh, Passante, Shaffer (USA) | [ ] |

Light red | Belloni, Doncheski, Robinett (USA) | [ , , ] |

Dark Purple | Singh, Marshman, Zhu, Sayer (USA) | [ , ] |

Yellow | di Uccio, Colantonio, Galano, Marzoli, Trani, Testa (Italy) | [ , ] |

Green | Bøe, Henriksen, Bungum, Angell (Norway) | [ , ] |

Turquoise | Malgieri, Onorato, De Ambrosis (Italy) | [ ] |

Red | Baily, Finkelstein, Pollock (USA), Kohnle (UK) | [ , ] |

Light purple | Dür (Austria), Heusler (Germany) | [ , ] |

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## Share and Cite

Bitzenbauer, P. Quantum Physics Education Research over the Last Two Decades: A Bibliometric Analysis. Educ. Sci. 2021 , 11 , 699. https://doi.org/10.3390/educsci11110699

Bitzenbauer P. Quantum Physics Education Research over the Last Two Decades: A Bibliometric Analysis. Education Sciences . 2021; 11(11):699. https://doi.org/10.3390/educsci11110699

Bitzenbauer, Philipp. 2021. "Quantum Physics Education Research over the Last Two Decades: A Bibliometric Analysis" Education Sciences 11, no. 11: 699. https://doi.org/10.3390/educsci11110699

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## Researchers demonstrate how to build 'time-traveling' quantum sensors

by Nick Oakes, Washington University in St. Louis

The idea of time travel has dazzled sci-fi enthusiasts for years. Science tells us that traveling to the future is technically feasible, at least if you're willing to go near the speed of light, but going back in time is a no-go. But what if scientists could leverage the advantages of quantum physics to uncover data about complex systems that happened in the past?

New research indicates that this premise may not be that far-fetched. In a paper published June 27, 2024, in Physical Review Letters , Kater Murch, the Charles M. Hohenberg Professor of Physics and Director of the Center for Quantum Leaps at Washington University in St. Louis, and colleagues Nicole Yunger Halpern at NIST and David Arvidsson-Shukur at the University of Cambridge demonstrate a new type of quantum sensor that leverages quantum entanglement to make time -traveling detectors.

Murch describes this concept as analogous to being able to send a telescope back in time to capture a shooting star that you saw out of the corner of your eye. In the everyday world, this idea is a non-starter. But in the mysterious and enigmatic land of quantum physics, there may be a way to circumvent the rules. This is thanks to a property of entangled quantum sensors that Murch refers to as "hindsight."

The process begins with entanglement of two quantum particles in a quantum singlet state—in other words, two qubits with opposite spin—so that no matter what direction you consider, the spins point in opposing directions. From there, one of the qubits—the "probe," as Murch calls it—is subjected to a magnetic field that causes it to rotate.

The next step is where the proverbial magic happens. When the ancillary qubit (the one not used as the probe in the experiment) is measured, the properties of entanglement effectively send its quantum state (i.e. spin) "back in time" to the other qubit in the pair. This takes us back to the second step in the process, where the magnetic field rotated the "probe qubit," and it is where the real advantage of hindsight comes in.

Under usual circumstances for this kind of experiment, where the rotation of a spin is used to measure the size of a magnetic field, there is a one-in-three chance that the measurement will fail. This is because when the magnetic field interacts with the qubit along the x-, y-, or z-axis, if it is parallel or antiparallel to the direction of spin, the results will be nullified—there will be no rotation to measure.

Under normal conditions, when the magnetic field is unknown, scientists would have to guess along which direction to prepare the spin, leading to the one-third possibility of failure. The beauty of hindsight is that it allows experimenters to set the best direction for the spin—in hindsight—through time travel.

Einstein once referred to quantum entanglement as "spooky action at a distance." Perhaps the spookiest part about entanglement is that we can consider entangled particle pairs as being the very same particle, going both forward and backwards in time.

That gives quantum scientists creative new ways to build better sensors—in particular ones that you can effectively send backwards in time. There are a number of potential applications for these kinds of sensors, from detecting astronomical phenomena to the aforementioned advantage gained in studying magnetic fields, and more will surely come into focus as the concept is developed further.

Journal information: Physical Review Letters , arXiv

Provided by Washington University in St. Louis

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- Published: 02 July 2024

## Unravelling quantum dynamics using flow equations

- S. J. Thomson ORCID: orcid.org/0000-0001-9065-9842 1 nAff3 &
- J. Eisert ORCID: orcid.org/0000-0003-3033-1292 1 , 2

Nature Physics ( 2024 ) Cite this article

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- Computational science
- Condensed-matter physics
- Information theory and computation
- Statistical physics, thermodynamics and nonlinear dynamics
- Theoretical physics

The study of many-body quantum dynamics in strongly correlated systems is extremely challenging. To date, few numerical methods exist that are capable of simulating the non-equilibrium dynamics of two-dimensional quantum systems, which is partly due to complexity theoretic obstructions. In this work, we present a technique able to overcome this obstacle, by combining continuous unitary flow techniques with the newly developed method of scrambling transforms. We overcome the assumption that approximately diagonalizing the Hamiltonian cannot lead to reliable predictions for relatively long times. Rather, we show that the method achieves good accuracy in both localized and delocalized phases and makes reliable predictions for a number of quantities including infinite-temperature autocorrelation functions. We complement our findings with rigorous incremental bounds on the truncation error. Our approach shows that, in practice, the exploration of intermediate-scale time evolution may be more feasible than is commonly assumed, challenging near-term quantum simulators.

## Similar content being viewed by others

## Quantum algorithms for quantum dynamics

## Dynamics of quantum information

## Experimental extraction of the quantum effective action for a non-equilibrium many-body system

Taming the exponential complexity of many-body quantum systems remains one of the biggest challenges in modern physics. Exact numerical simulations provide the gold standard in accuracy. However, the computational cost quickly becomes prohibitive above a few tens of particles, and even rapid developments in computing power cannot outpace the exponential scaling of the complexity of fully solving a many-body quantum system. Although there are efficient methods for estimating the ground states of various quantum systems captured by local Hamiltonians, complexity becomes even more of an obstacle for time evolution. The time evolution of a given quantum state under the action of a local Hamiltonian is BQP complete in the worst case complexity. For this reason, one cannot hope to find universal classical methods that can accurately and efficiently simulate this evolution for all time and all local Hamiltonians 1 . Although the ultimate goal may be the development of flexible and reliable quantum simulators 2 , 3 , 4 able to directly realize many models of interest, in the near term we must continue to rely upon classical computers to simulate quantum matter.

To that end, many highly effective numerical techniques have been developed for studying many-body quantum systems subject to controlled and clear approximations. Leading the charge are tensor network methods 5 , 6 , which are instances of variational methods that build on tensor networks, particularly matrix product states in one dimension and projected entangled pair states in two dimensions. These methods work well for ground states and the short-time evolution of generic non-integrable systems, but are limited in how they can capture dynamics, a state of affairs sometimes dubbed the ‘entanglement barrier’. One may encounter indefinite oscillations and can, hence, capture long-time dynamics only for some specific and fine-tuned instances of weak ergodicity breaking 7 . Disorder will also assist in lessening the burden of long-time dynamics with tensor network methods 8 , 9 . That said, the core limitation stems from the generation of entanglement, as highly entangled systems require large bond dimensions, giving rise to computationally intractable situations, particularly in two dimensions. Quantum Monte Carlo techniques 10 are also widely used, including for non-equilibrium dynamics 11 , 12 . However, they suffer from the well-known sign problem and stability issues. Dynamical mean-field theory can also capture quantum dynamics 13 , 14 , but again stability matters arise. Recently, machine learning has been used to capture many-body dynamics, which constitutes a strikingly interesting approach 15 , 16 , 17 , but here, questions about the predictive power and the explanatory value emerge. These obstacles all reflect the computational hardness of the task and highlight the need for thinking about tools for many-body dynamics that are entirely different altogether.

In this work, we develop a radically different approach to time evolution in closed quantum systems. Combining the established method of continuous unitary transforms (CUTs), which are also known as ‘flow equations’ 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , with the newly developed method of scrambling transforms (sketched in Fig. 1 ), we present a flexible and powerful approach to diagonalizing large Hamiltonians and computing time evolution to very long times. The key ingredient in our work is the use of scrambling transforms to improve the convergence properties of CUT-based methods, as this significantly improves their accuracy and validity. We demonstrate the potential of this technique by computing the dynamics of disordered quantum systems in one and two dimensions. The limitation is very different compared to tensor network approaches. Here, it is not entanglement that provides the limitation but the accuracy of the approximate transform used. Heuristically, we find that in practice, this restriction is less severe than overcoming the entanglement barrier.

a , The conventional CUT process, which uses a single unitary transform U to diagonalize a Hamiltonian H , smoothly transforming it from the initial basis ( l = 0) to the diagonal basis ( l → ∞ ). b , The scrambling transform S first induces ‘effective disorder’, even in completely clean systems, which allows established CUT techniques to then take over and efficiently diagonalize the scrambled Hamiltonian S † H S with a second unitary transform U .

We will focus on a generic system of interacting fermions, captured as

where : . . . : represents normal ordering with respect to the vacuum, and \(| {{{\mathcal{L}}}}| =:L\) is the system size. We make no assumptions as to the form of the couplings nor the dimensionality of the system. The complexity of the calculation is set by the total number of lattice sites L , not by their geometry or the size of the local Hilbert spaces. A two-dimensional (2D) or three-dimensional system can be unfolded onto a one-dimensional (1D) system with long-range hopping, as sketched in Fig. 2 , which does not pose a problem for CUT-based techniques.

Illustration of how a 2D lattice can be mapped onto a 1D chain with correlated long-range hopping, which can be easily handled with CUT-based techniques.

Flow equation methods diagonalize the Hamiltonian by successively applying infinitesimal unitary transforms \(\mathrm{d}U(l)=\exp (-\eta (l)\,{{{\rm{d}}}}l)=\) \(1-\eta(l){\rm{d}}l\) , where η ( l ) is the generator and l represents a fictitious ‘flow time’ such that l = 0 is the initial Hamiltonian. The parameterized Hamiltonian H ( l ) ≔ U † ( l ) H U ( l ) becomes diagonal in the limit l → ∞ , where the full unitary transform \(U(l)=\) is a time-ordered integral over flow time l . The diagonalization procedure can be recast as solving the equation of motion d H /d l = [ η ( l ), H ( l )] (refs. 23 , 24 ). We store H (2) as a matrix with O ( L 2 ) entries and H (4) as a tensor of order four with O ( L 4 ) real entries, and we employ a similar procedure for the generator η ( l ) =: η (2) ( l ) + η (4) ( l ). This allows the relevant commutators to be computed efficiently as the sum of all one-point contractions of pairs of matrices or tensors 26 , at a cost polynomial in system size. In all of the following, we truncate at fourth order, \({{{\mathcal{O}}}}({L}^{4})\) . The main consequence of fermionic statistics is the minus signs, which arise when computing the contractions. The method can be applied to bosons with minor changes.

A common choice of generator is η ( l ) ≔ [ H 0 , V ( l )], where H 0 ( l ) and V ( l ) are, respectively, the diagonal and off-diagonal parts of the Hamiltonian. In the following, we use the symbol V for off-diagonal elements. This is often known as the Wegner generator 23 , 24 . The diagonalization can be seen because the squared \(\parallel V(l){\parallel }_{2}^{2}\) is non-increasing in the fictitious time l as \({{{\rm{d}}}}\parallel V(l){\parallel }_{2}^{2}/{{{\rm{d}}}}l=-2\parallel \eta (l){\parallel }_{2}^{2}\le 0\) (see, for example, ref. 27 ). Convergence relies upon the model in question having a clear separation of energy scales in the initial basis. Models where this is not true (such as homogeneous systems and disordered systems with many near-degeneracies) cannot be fully diagonalized by this generator, as they act like unstable fixed points. Perturbing the Hamiltonian away from this fixed point can allow the flow to begin. However, small perturbations can result in long convergence times, whereas large perturbations improve convergence but risk changing the underlying physics. Here, we resolve this by introducing scrambling transforms, which are targeted unitary transforms aimed at lifting degeneracies, which the Wegner procedure alone is unable to resolve. As they are unitary, they cannot change the underlying physics. They simply act to ‘prepare’ the Hamiltonian in a basis more amenable to being diagonalized by the conventional Wegner flow (Fig. 1 ). The (infinitesimal) scrambling transform takes the form \(\mathrm{d}S(l)=\exp (-\lambda (l)\,{{{\rm{d}}}}l)\) , with a generator λ ( l ) given by

with \(\delta h=\varepsilon | {H}_{ii}^{\,(2)}(l)-{H}_{jj}^{\,(2)}(l)|\) , where ε > 0 is the threshold parameter, which controls how easily the scrambling transform triggers. For ε = 0, this reduces to the Toda–Mielke generator 27 , 28 . Here, we use ε = 0.5. The full scrambling transform S ( l ) can be written as a time-ordered integral over d S ( l ). It is employed at the beginning of the flow and during the diagonalization procedure if degeneracies are encountered (see Supplementary Fig. 1 for details).

The scrambling transform used here is quadratic and does not induce any new higher-order terms. However, the action of the Wegner generator will typically lead to the generation of new terms containing six or more fermionic operators, like the way that such terms arise in renormalization group procedures. The central approximation of the CUT technique is that the Hamiltonian must be truncated and terms above a certain order neglected. We shall present rigorous error bounds later; for the moment, we emphasize that in cases where the method is insufficiently exact, higher-order terms can be systematically included until the desired precision is reached, at a cost polynomial in the system size.

We will investigate initial local Hamiltonians of the form \(H=\sum_{i\in {{{\mathcal{L}}}}}\) \([{h}_{i}:{n}_{i}:+J(:{c}_{i}^{{\dagger} }{c}_{i+1}:+\;\mathrm{H.c.})+{\Delta }_{0}:{n}_{i}{n}_{i+1}:]\) , using open boundary conditions, with J = 1 and Δ 0 = 0.1. In one dimension, this Hamiltonian maps onto the XXZ chain by a Jordan–Wigner transform. We diagonalize these Hamiltonians in both one and two dimensions, for two different choices of h i : random disorder ( h i ∈ [− d , d ]) and quasi-periodic potentials (QP). For the latter, in one spatial dimension, \({h}_{i}:= d\cos\) \((2\pi i/\phi +\theta )\) , with \(\phi := (1+\sqrt{5})/2\) and θ a (real) randomly chosen phase that plays the role of a ‘disorder realization’. In two dimensions, \({h}_{i}:= d(\cos(2\pi {i}_{x}/\phi +\theta )+\cos (2\pi {i}_{y}/{\phi }_{2}+{\theta }_{2}))\) , where ( i x , i y ) represent the coordinates of lattice site i , \({\phi }_{2}=1+\sqrt{2}\) and θ 2 is another random phase. For simplicity, we refer to d as the ‘disorder strength’ in both cases. This model (and its spin chain equivalent) has been extensively studied in the context of many-body localization in the presence of both random and QP potentials, particularly in one dimension 29 , 30 , 31 , 32 but also in two dimensions 33 . The end point is an (approximately) diagonal Hamiltonian:

where \({{{\mathcal{R}}}}\) represents neglected higher-order terms, typically of order \(O({\Delta }_{0}^{2})\) and higher. The interaction coefficients decay exponentially with distance in strongly (quasi)disordered systems, Δ i j ∝ e − ∣ i − j ∣ / ξ (refs. 26 , 33 , 34 , 35 , 36 , 37 ), where the \({\tilde{n}}_{i}\) operators are known as local integrals of motion. We emphasize, however, that the form of equation ( 3 ) does not assume localization or the existence of local integrals of motion. This construction is equally valid whether the unitary transform is quasilocal (as for many-body localization) or entirely non-local.

Once the Hamiltonian has been diagonalized, it is possible to obtain a closed-form solution (within a given truncation scheme) to the Heisenberg equation of motion for any operator O expressed in the diagonal basis. The operator must first be transformed according to the flow equation \({{{\rm{d}}}}O/{{{\rm{d}}}}l=[\overline{\eta }(l\,),O(l\,)]\) , where \(\overline{\eta }(l\,)\) collectively denotes both the scrambling and Wegner generators. This transformed operator also contains valuable information about the locality of the unitary transform and can be used to extract both a localization length and a measure of the ‘complexity’ of the diagonalization procedure, which can be linked to the existence of Lieb–Robinson bounds in flow time 38 . Specifically, the transformed creation operator takes the form \({c}_{i}^{{\dagger} }={\sum }_{j}\,{A}_j^{(i)}{\tilde{c}}_j^{{\dagger} }+{\sum }_{j,k,q}{B}_{jkq}^{(i)}{\tilde{c}}_{j}^{{\dagger} }{\tilde{c}}_{k}^{{\dagger} }{\tilde{c}}_{q}\) , and higher-order terms are neglected. A measure of the complexity of the transformed operators is given by the fraction of non-zero terms in this operator expansion. Intuitively, we would expect \({c}_{i}^{{\dagger}}\) to remain sparse in a localized phase but not in a delocalized phase. Supporting analysis is shown in Supplementary Information Section 3. In practice, we choose a cutoff value ϵ = 10 −6 below which we consider terms to be zero. The complexity is defined as

where ( A ∪ B ) represents the set of all coefficients A i and B i j k in the operator expansion of \({c}_{i}^{{\dagger} }\) . We also define \(\overline{\chi (\epsilon )}=| \{x\in (A\cup B)| {x}^{2} > {\epsilon }^{2}\}|\) . The results in Fig. 3 demonstrate a qualitative difference between one and two spatial dimensions. In one dimension, we find a phase where \(\overline{\chi (\epsilon )}\) tends to a constant and χ ( ϵ ) ∝ (1/ L ) 3 for large system sizes, indicating a ‘low complexity’ situation at strong disorder, as well as a higher complexity phase at small values of d where \(\overline{\chi (\epsilon )}\) increases rapidly with system size, suggestive of thermalization. In two dimensions, we find that \(\overline{\chi (\epsilon )}\) always increases, although for a quasi-periodic potential at large values of d , it increases sufficiently slowly that the normalized complexity χ ( ϵ ) still vanishes. By contrast, for small values of d in two dimensions, the complexity χ ( ϵ ) remains much larger than zero for all system sizes studied here. This suggests a slow crossover from a high complexity phase (consistent with the expectation of thermalization at small values of d ) to a low complexity phase with anomalous thermalization properties. This notion of complexity is reminiscent of circuit complexity 39 , 40 .

a – d , Results are averaged over disorder realizations, with N s ∈ [20, 1,024] samples depending on system size ( Supplementary Information ). Error bars show the standard deviation. a , b , Results in one dimension ( L = 8, 10, 12, 16, 24, 36, 48, 64, 100) for random ( a ) and QP potentials ( b ). c , d , The same in two dimensions ( L 2 = 9, 16, 25, 36, 49, 64, 100) for random ( c ) and QP potentials ( d ). Dashed black lines close to the origin in a and b are fits with the form χ ∝ 1/ L 3 , which suggests localization and is valid for large systems and strong disorder. Insets show the unnormalized complexity (that is, the numerator of equation ( 4 )), which tends to a constant in strongly disordered 1D chains but grows in two dimensions even for strong disorder.

Previous works that used CUT methods to compute non-equilibrium dynamics 33 , 36 , 41 employed a computationally costly inversion of the unitary transform to obtain time-evolved operators in the original basis. Here, we circumvent this limitation and directly obtain the infinite-temperature autocorrelation function. This highly non-trivial quantity fully characterizes the transport properties of the system. The thermal expectation value of any arbitrary operator O is given by \(\langle O\rangle =\operatorname{Tr}[\exp (-\beta H)O]/\operatorname{Tr}[\exp (-\beta H)]\) , where β = 1/ T is the inverse temperature (in units of k B = 1). In the limit T → ∞ , the expectation value becomes a uniform average over eigenstates, which in the diagonal basis are trivial product states. We approximate this average for large systems by randomly sampling \({{{{\mathcal{N}}}}}_{s}\in [50,256]\) half-filled eigenstates. Specifically, we compute

To minimize boundary effects, we choose i to be in the centre of the system. This is a highly demanding quantity that can be extremely challenging to compute with other methods but can be obtained very efficiently with the flow equation approach. Results for system size L = 100 ( L 2 = 10 × 10 for two dimensions) are shown in Fig. 4 . The results remain reasonable far beyond the naive expectation that the accuracy should break down beyond timescales \(tJ \approx1/{\Delta }_{0}^{2}\) , corresponding to the typical inverse magnitude of the terms cut off by the truncation, as verified by comparison with exact diagonalization (Supplementary Fig. 7 ). Figure 5 shows results for system sizes up to L = 100 ( L 2 = 10 × 10 in two dimensions), along with a linear fit indicating the L → ∞ behaviour. Strikingly, very little dependence on system size is observed in one dimension, although in two dimensions there is a slow trend towards decreasing values of C ( t ) as the system size increases, except for strong QP potentials. This is consistent with the expectation that many-body localization may be ultimately unstable in two dimensions. For weak random disorder in two dimensions, the linear fit for large values of L breaks down, suggesting that our results probably overestimate the long-time value of C ( t ) as L → ∞ . Additionally, at any finite order of truncation, there may still exist higher-order processes that could contribute to thermalization on very long timescales. Nonetheless, for a given truncation scheme, we can make precise statements about the validity of this technique.

a – d , Infinite-temperature correlation functions: a , 1D, random, L = 100; b , 1D, QP, L = 100; c , 2D, random, L 2 = 10 × 10; d , 2D, QP, L 2 = 10 × 10. Grey dot-dashed vertical lines indicate the approximate timescale beyond which accuracy cannot be guaranteed. However, the results typically remain reasonable until much longer timescales. Black dashed lines show the long-time average computed directly without explicit time evolution, which is valid at strong (quasi)disorder only. The results are averaged over N s ∈ [50, 128] disorder realizations. Error bars indicate the variance over disorder realizations. For both D = 1 and D = 2, the QP potential exhibits most much robust localization at large values of d / J , but by contrast also exhibits more complete thermalization at low values of d / J due to the underlying single-particle phase transition at d / J = 2.

a – d , Results for various system sizes, disorder types and strengths, and dimensionalities: a , 1D, random; b , 1D, QP; c , 2D, random; d , 2D, QP. Error bars indicate the variance over disorder realizations. The black squares indicate the results obtained from exact diagonalization, shown for small system sizes only. System sizes are L = 8, 10, 12, 16, 24, 36, 48, 64 and 100 in one dimension, and L 2 = 9, 16, 25, 36, 49, 64 and 100 in two dimensions. e , The result of linearly extrapolating to L → ∞ . Error bars indicate the uncertainty (variance) in the fits (shown as dashed lines in a – d ) to extract the scaling behaviour. Lines are guides to the eye.

To do so, we develop an incremental bound on the error in the unitary transform. If at each flow time step we discard all newly generated terms above fourth order, we obtain

where d H ( l ) = [ η (2) ( l ), H (2) ( l )] + [ η (2) ( l ), H (4) ( l )] + [ η (4) ( l ), H (2) ( l )] represents the terms of the flow that are kept and \(A(l)=[{\eta }^{(4)}(l),{H}^{\,(4)}(l)]+{{{\mathcal{T}}}}\) represents the truncation error, where the higher-order terms \({{{\mathcal{T}}}}\) are assumed to be negligible in what follows. The norm of the truncation error A ( l ) at each infinitesimal time step is upper bounded by

using the submultiplicativity of the ∥ . ∥ 2 norm 42 . The total truncation error in flow time can be written as an integral of equation ( 7 ):

over l . Typical values of V (2) ( l ) decay exponentially in flow time, that is, \({[{V}^{\,(2)}(l)]}_{ij}\propto \exp (-{({h}_{i}-{h}_{j})}^{2}l){[{V}^{\,(2)}(0)]}_{ij}\) . Assuming random disorder drawn from a box distribution of width [− d , d ] such that the mean value of this squared energy difference is 2 d 2 /3 and that the largest parts of the interaction tensor remain proportional to the initial interaction strength (as new terms induced by the flow should always be smaller than the initial interactions), the error can be approximated as

For weak (quasi)disorder, the disorder bandwidth d is replaced by the effective bandwidth \(\tilde{d}\ge d\) induced by the scrambling transform (Supplementary Fig. 2 ). A numerical analogue can be computed by replacing the Hilbert–Schmidt (or Frobenius) norms in equation ( 7 ) with tensor Frobenius norms; the typical truncation error at each flow time step is well below 1% (Supplementary Fig. 6 ).

The above analysis indicates that energy differences below \(O(\,J{\Delta }_{0}^{2}/{d}^{\,2})\) cannot be reliably resolved, implying that the method will break down on timescales of the order \(t\propto {d}^{\,2}/(\,J{\Delta }_{0}^{2})\) when oscillations at corresponding frequencies \(\omega \approx O(\,J{\Delta }_{0}^{2}/{d}^{\,2})\) become relevant for the dynamics. The accuracy of the method can be systematically improved by incorporating additional higher-order terms into the truncated Hamiltonian, which allows accurate simulations of quantum dynamics to even longer times (proportional to \(1/{\Delta }_{0}^{3}\) at the next order of approximation) with a computational cost that remains polynomial in the system size. Future developments in massively parallel implementations of the tensor flow equation method 26 used in this work, as well as advances in computer hardware, will facilitate the extension of this method to larger system sizes, longer timescales, stronger interactions and additional physical systems (including both driven 37 , 43 and dissipative 44 systems, which have been previously studied with CUT-based techniques). Scrambling transforms may be of interest in a variety of other contexts, as they are essentially a way of transforming a highly entangled system into a simpler representation that is easier to simulate.

We end the discussion by briefly comparing our findings with those of tensor network methods. Standard tensor network methods are challenged in time evolution by the exponentially growing bond dimension that is requ ired to accommodate states faithfully in time. Some steps have already been taken to allow tensor networks to access longer times 45 , 46 , 47 , 48 , for example, with folding techniques 47 or adaptive mode transformations 46 . The ideas introduced here show that one can go further than that, as we find that the fermionic mode transformations do not have to be linear. There are good reasons to believe that this is a favourable way to reach long simulation times. In fact, once the Hamiltonian is diagonalized, in principle, all times are available and the accumulating errors can be upper bounded. First connections between flow equation and tensor network methods have been made 49 , in anticipation of the time-dependent variational principle based on a differential geometric picture 50 . It is conceivable that the ideas introduced here can be further merged with tensor network techniques, as one could think of final Hamiltonians that are not treated as fully diagonal ones. There is also the intriguing possibility of combining scrambling transforms and other CUT-based techniques with tensor network approaches, such as entanglement-based CUTs 51 , which may allow tensor network methods to break through the entanglement barrier.

In this work, we have introduced a flow-based method equipped with scrambling transforms that can simulate interacting fermionic quantum many-body systems to good accuracy for intermediate and long times. Such a classical development can also be seen as a challenge to dynamical quantum simulators 2 , 4 , which aim to probe non-equilibrium properties of quantum matter beyond the reach of classical computers. These are exciting avenues for future progress.

## Computing and integrating the flow equation

All commutators computed in this work follow the scheme of ref. 26 , in which the representation of the Hamiltonian in terms of a quadratic component (stored in memory as a matrix) and quartic component (stored as a tensor) allow the commutators to be recast in terms of matrix/tensor contractions, which are highly optimized linear algebra operations that can be performed efficiently on modern computing hardware. A complete description is contained in Supplementary Information Section 2 . We used vacuum normal ordering such that higher-order terms in the running Hamiltonian have no feedback onto lower-order terms. The incorporation of additional non-perturbative corrections due to different choices of normal ordering has previously been done for the time-independent scenario 52 , but has been left for future work in the non-equilibrium setting. This would require specifying a particular reference state with respect to which the corrections are computed. Calculations for all system sizes with more than a total of 16 lattice sites were performed on graphics processing units (GPUs; specifically, NVIDIA RTX A5000 GPUs with 24 Gb RAM and NVIDIA RTX 2080Ti GPUs with 12 Gb RAM) using single-precision arithmetic.

The flow equation \({{{\rm{d}}}}Hl/{{{\rm{d}}}}l=[\overline{\eta }(l),H(l)]\) was solved using a mixed fourth- and fifth-order Runge–Kutta integration method as implemented in the JAX library 53 , which applies an adaptive step-size algorithm. The maximum integration time was \({l}_{\max }=\text{1,000}\) , and the integration was stopped before then if the Hamiltonian was diagonalized to the target accuracy, which we chose to be when \(\max [| {V}^{\,(2)}| ] < 1{0}^{-6}\) and \(\max [| {V}^{\,(4)}| ] < 1{0}^{-3}\) . Results for longer integration times showed no significant increase in accuracy, despite incurring a significantly higher computational cost. This is because the running Hamiltonian H ( l ) approaches full diagonalization only asymptotically at large values of l , so using larger values of \({l}_{\max }\) leads to diminishing returns.

## Computing the dynamics

The transformed number operator was reconstructed from the transformed creation and annihilation operators for large fictitious time:

with the creation operator given by transformed creation and annihilation operators:

and the annihilation operator obtained by taking its Hermitian conjugate \({c}_{i}(l\to \infty )={({c}_{i}^{\dagger }(l\to \infty ))}^{\dagger}\) . Multiplying these together allowed us to reconstruct the number operator, including terms up to sixth-order in the fermionic creation/annihilation operators for the diagonal basis, \({\tilde{c}}{_{i}^{{\dagger} }}\) and \({\tilde{c}}_{i}\) . The number operator was then evolved in time in the diagonal basis according to the Heisenberg equation of motion, while neglecting newly generated higher-order terms, resulting in a closed-form solution. This step was performed on CPUs rather than GPUs due to memory limitations and is a prime candidate for future efficiency improvements. At long times, near-degenerate single-particle eigenvalues can still lead to divergent terms in this solution (consistent with the expectation that the simulation of a BQP-hard problem will eventually run into accuracy issues on a classical computer). However, these terms were strongly suppressed, arising only very rarely and at very long times. To prevent these rare scenarios from dominating the averaged data, we excluded disorder realizations when the maximum value of ∣ C ( t ) ∣ > 1.1. (Alternatively, we could have used the typical rather than mean value of C ( t ).) See Supplementary Information for full details of the calculation and for where divergent terms arise from. In one dimension, the divergent terms were rare enough to have essentially no effect. The long-time average was obtained directly by setting all off-diagonal terms in the transformed number operator to zero (as when time-evolved, they acquire oscillating phases that average to zero). For systems with greater than 36 lattice sites in total, we neglected the sixth-order contributions and kept only the quadratic and quartic terms when computing the dynamics. For the systems considered here, the sixth-order terms have a negligible effect, which can be seen from the qualitative agreement between small and larger systems.

## Rescaling the correlation function

As the norm of the number operator n i is not precisely conserved by the unitary transform, we rescaled the correlation function for each disorder realization according to the ansatz C ( t ) ↦ c 1 ( C ( t ) − c 2 ), where c 1 and c 2 are determined by minimizing the error with respect to the short-time dynamics of the non-interacting system (as many-body interactions are essentially irrelevant at very short times). This is computationally efficient, as we get the exact dynamics of the non-interacting system essentially for free in this formalism by just retaining the quadratic components of the Hamiltonian and relevant observables. The rescaling employed in this work is justified a posteriori by the clear agreement between the rescaled C ( t ) and the exact result, which were computed for system sizes small enough for the comparison to be practical. Further analysis may be found in Supplementary Information Section 9 . For small enough systems, an alternative would be to construct the operator as a matrix in the full Hilbert space and renormalize it by hand. However, this is not practical for systems as large as those considered here. We emphasize that the norm is preserved to high accuracy for sufficiently strong disorder, and the effects of this rescaling are most important for weakly disordered systems. This is independent of any error introduced in the eigenvalues and reflects the difficulty in simultaneously preserving the unitary evolution of both the Hamiltonian and the number operator within the same truncation scheme. The norm of the operator could, in principle, be exactly preserved by constructing the unitary transforms subject to additional constraints 24 . However, in practice this is challenging to implement. This underscores the need for further work in developing more flexible generators for the types of CUT developed here, perhaps in concert with machine learning approaches to design data-driven generators tailored for specific problems subject to specific hard-to-satisfy constraints.

## Data availability

Supporting data for this manuscript are available via Zenodo at https://doi.org/10.5281/zenodo.8144136 (ref. 54 ).

## Code availability

This work made use of the open source PyFlow library 43 , developed and maintained by S.J.T. based on ref. 26 . The script used to generate the main figures used in this work is available via Zenodo at https://doi.org/10.5281/zenodo.8144136 (ref. 54 ).

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## Acknowledgements

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101031489, Ergodicity Breaking in Quantum Matter (S.J.T.) and under the Quantum Flagship (PASQuanS2, Millenion) (J.E.). This project also received support from NVIDIA Corporation through the Academic Hardware Grant Program (S.J.T.), as well as the Federal Ministry of Education and Research, Germany (FermiQP, MuniQC-Atoms) (J.E.), the German Research Foundation (CRC 183) (J.E.) and the European Research Council (DebuQC) (J.E.). S.J.T. thanks J. Catton for providing the artwork used in Fig. 1 .

Open access funding provided by Freie Universität Berlin.

## Author information

S. J. Thomson

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S.J.T. conceived the project, developed the method, wrote the code and ran all simulations and analysis. S.J.T. and J.E. jointly wrote the final manuscript.

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Thomson, S.J., Eisert, J. Unravelling quantum dynamics using flow equations. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02549-2

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DOI : https://doi.org/10.1038/s41567-024-02549-2

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## Quantum Advantage

Quantum advantage: building 'time-traveling' quantum sensors.

In a paper published in Physical Review Letters, scientists at WashU and elsewhere demonstrate a new type of quantum sensor that leverages quantum entanglement to make time traveling detectors.

The idea of time travel has dazzled sci-fi enthusiasts for years. Science tells us that traveling to the future is technically feasible, at least if you’re willing to go near the speed of light, but going back in time is a no-go. But what if scientists could leverage the advantages of quantum physics to uncover data about complex systems that happened in the past? New research indicates that this premise may not be that far-fetched. In a new paper published June 27, 2024, in Physical Review Letters, Kater Murch , the Charles M. Hohenberg Professor of Physics and Director of the Center for Quantum Leaps at Washington University in St. Louis, and colleagues Nicole Yunger Halpern at NIST and David Arvidsson-Shukur at the University of Cambridge demonstrate a new type of quantum sensor that leverages quantum entanglement to make time- traveling detectors.

Murch describes this concept as analogous to being able to send a telescope back in time to capture a shooting star that you saw out of the corner of your eye. In the everyday world, this idea is a non-starter. But in the mysterious and enigmatic land of quantum physics, there may be a way to circumvent the rules. This is thanks to a property of entangled quantum sensors that Murch refers to as “hindsight.”

The process begins with entanglement of two quantum particles in a quantum singlet state—in other words, two qubits with opposite spin—so that no matter what direction you consider, the spins point in opposing directions. From there, one of the qubits—the “probe,” as Murch calls it—is subjected to a magnetic field that causes it to rotate.

The next step is where the proverbial magic happens. When the ancillary qubit (the one not used as the probe in the experiment) is measured, the properties of entanglement effectively send its quantum state (i.e. spin) “back in time” to the other qubit in the pair. This takes us back to the second step in the process, where the magnetic field rotated the “probe qubit”, and it is where the real advantage of hindsight comes in.

Under usual circumstances for this kind of experiment, where the rotation of a spin is used to measure the size of a magnetic field, there is a one-in-three chance that the measurement will fail. This is because when the magnetic field interacts with the qubit along the x-, y-, or z-axis, if it is parallel or antiparallel to the direction of spin, the results will be nullified–there will be no rotation to measure. Under normal conditions, when the magnetic field is unknown, scientists would have to guess along which direction to prepare the spin, leading to the one-third possibility of failure. The beauty of hindsight is that it allows experimenters to set the best direction for the spin—in hindsight—through time travel.

Einstein once referred to quantum entanglement as “spooky action at a distance.” Perhaps the spookiest part about entanglement is that we can consider entangled particle pairs as being the very same particle, going both forward and backwards in time. That gives quantum scientists creative new ways to build better sensors—in particular ones that you can effectively send backwards in time. There are a number of potential applications for these kinds of sensors, from detecting astronomical phenomena to the aforementioned advantage gained in studying magnetic fields, and more will surely come into focus as the concept is developed further.

## in the news:

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Help | Advanced Search

## Quantum Physics

Title: the xyz ruby code: making a case for a three-colored graphical calculus for quantum error correction in spacetime.

Abstract: Analyzing and developing new quantum error-correcting schemes is one of the most prominent tasks in quantum computing research. In such efforts, introducing time dynamics explicitly in both analysis and design of error-correcting protocols constitutes an important cornerstone. In this work, we present a graphical formalism based on tensor networks to capture the logical action and error-correcting capabilities of any Clifford circuit with Pauli measurements. We showcase the formalism on new Floquet codes derived from topological subsystem codes, which we call XYZ ruby codes. Based on the projective symmetries of the building blocks of the tensor network we develop a framework of Pauli flows. Pauli flows allow for a graphical understanding of all quantities entering an error correction analysis of a circuit, including different types of QEC experiments, such as memory and stability experiments. We lay out how to derive a well-defined decoding problem from the tensor network representation of a protocol and its Pauli flows alone, independent of any stabilizer code or fixed circuit. Importantly, this framework applies to all Clifford protocols and encompasses both measurement- and circuit-based approaches to fault tolerance. We apply our method to our new family of dynamical codes which are in the same topological phase as the 2+1d color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a Z3 automorphism to the logical Pauli group every three timesteps. We highlight some of its topological properties and comment on the anyon physics behind a planar layout. Lastly, we benchmark the performance of the XYZ ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of 0.18%, comparable with other Floquet codes and 2+1d color codes.

Comments: | 59 pages, 26 figures |

Subjects: | Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph) |

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View PDF HTML (experimental) Abstract: Analyzing and developing new quantum error-correcting schemes is one of the most prominent tasks in quantum computing research. In such efforts, introducing time dynamics explicitly in both analysis and design of error-correcting protocols constitutes an important cornerstone.

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