Deep Quantum Graph Dreaming: Deciphering Neural Network Insights into Quantum Experiments

Tareq Jaouni, Sören Arlt, Carlos Ruiz-Gonzalez, Ebrahim Karimi, Xuemei Gu, Mario Krenn

Despite their promise to facilitate new scientific discoveries, the opaqueness of neural networks presents a challenge in interpreting the logic behind their findings. Here, we use a eXplainable-AI (XAI) technique called inception or deep dreaming, which has been invented in machine learning for computer vision. We use this techniques to explore what neural networks learn about quantum optics experiments. Our story begins by training a deep neural networks on the properties of quantum systems. Once trained, we "invert" the neural network – effectively asking how it imagines a quantum system with a specific property, and how it would continuously modify the quantum system to change a property. We find that the network can shift the initial distribution of properties of the quantum system, and we can conceptualize the learned strategies of the neural network. Interestingly, we find that, in the first layers, the neural network identifies simple properties, while in the deeper ones, it can identify complex quantum structures and even quantum entanglement. This is in reminiscence of long-understood properties known in computer vision, which we now identify in a complex natural science task. Our approach could be useful in a more interpretable way to develop new advanced AI-based scientific discovery techniques in quantum physics.

Roadmap on structured waves

Konstantin Y Bliokh, Ebrahim Karimi, Miles J Padgett, Miguel A Alonso, Mark R Dennis, Angela Dudley, Andrew Forbes, Sina Zahedpour, Scott W Hancock, et al.

Experimental Solutions to the High-Dimensional Mean King's Problem

Tareq Jaouni, Xiaoqin Gao, Sören Arlt, Mario Krenn, Ebrahim Karimi

In 1987, Vaidman, Aharanov, and Albert put forward a puzzle called the Mean<br>King's Problem (MKP) that can be solved only by harnessing quantum<br>entanglement. Prime-powered solutions to the problem have been shown to exist,<br>but they have not yet been experimentally realized for any dimension beyond<br>two. We propose a general first-of-its-kind experimental scheme for solving the<br>MKP in prime dimensions ($D$). Our search is guided by the digital discovery<br>framework PyTheus, which finds highly interpretable graph-based representations<br>of quantum optical experimental setups; using it, we find specific solutions<br>and generalize to higher dimensions through human insight. As proof of<br>principle, we present a detailed investigation of our solution for the three-,<br>five-, and seven-dimensional cases. We obtain maximum success probabilities of<br>$72.8 \%$, $45.8\%$, and $34.8 \%$, respectively. We, therefore, posit that our<br>computer-inspired scheme yields solutions that exceed the classical probability<br>($1/D$) twofold, demonstrating its promise for experimental implementation.<br>

Recent advances in the Self-Referencing Embedding Strings (SELFIES) library

Alston Lo, Robert Pollice, AkshatKumar Nigam, Andrew D. White, Mario Krenn, Alán Aspuru-Guzik

String-based molecular representations play a crucial role in cheminformatics applications, and with the growing success of deep learning in chemistry, have been readily adopted into machine learning pipelines. However, traditional<br>string-based representations such as SMILES are often prone to syntactic and semantic errors when produced by generative models. To address these problems, a novel representation, SELF-referencIng Embedded Strings (SELFIES), was proposed that is inherently 100% robust, alongside an accompanying open-source implementation. Since then, we have generalized SELFIES to support a wider range of molecules and semantic constraints and streamlined its underlying grammar. We have implemented this updated representation in subsequent versions<br>of \selfieslib, where we have also made major advances with respect to design, efficiency, and supported features. Hence, we present the current status of \selfieslib (version 2.1.1) in this manuscript.

Quantum interference between distant creation processes

Johannes Pseiner, Manuel Erhard, Mario Krenn

The search for macroscopic quantum phenomena is a fundamental pursuit in<br>quantum mechanics. It allows us to test the limits quantum physics and provides new avenues for exploring the interplay between quantum mechanics and relativity. In this work, we introduce a novel approach to generate macroscopic quantum systems by demonstrating that the creation process of a quantum system can span a macroscopic distance. Specifically, we generate photon pairs in a coherent superposition of two origins separated by up to 70 meters. This new<br>approach not only provides an exciting opportunity for foundational experiments<br>in quantum physics, but also has practical applications for high-precision measurements of distributed properties such as pressure and humidity of air or gases.

Multiphoton non-local quantum interference controlled by an undetected photon

Kaiyi Qian, Kai Wang, Leizhen Chen, Hou Zhaohua, Mario Krenn, Shining Zhu, Xiao-Song Ma

The interference of quanta lies at the heart of quantum physics. The multipartite generalization<br>of single-quanta interference creates entanglement, the coherent superposition of states shared by several quanta. Entanglement allows non-local correlations between many quanta and hence is a key resource for quantum information technology. Entanglement is typically considered to be essential for creating non-local correlations, manifested by multipartite interference. Here, we show that this is not the case and demonstrate multiphoton non-local quantum interference without entanglement of any intrinsic properties of the photons. We harness the superposition of the physical origin of a four-photon product state, which leads to constructive and destructive interference of the photons’ mere existence. With the intrinsic indistinguishability in the generation process of photons, we realize four-photon frustrated quantum interference. We furthermore establish non-local control of multipartite quantum interference, in which we tune the phase of one undetected photon and observe the interference of the other three photons. Our work paves the way for fundamental studies of non-locality and potential applications in quantum technologies.

AI-discovery of a new charging protocol in a micromaser quantum battery

Carla Rodríguez, Dario Rosa, Jan Ollé

We propose a general computational framework for optimizing model-dependent<br>parameters in quantum batteries (QB). We apply this method to two different<br>charging scenarios in the micromaser QB and we discover a new charging protocol<br>for stabilizing the battery in upper-laying Hilbert space chambers in a<br>controlled and automatic way. This protocol is found to be stable and robust,<br>and it leads to an improved charging efficiency in micromaser QBs. Moreover,<br>our optimization framework is highly versatile and efficient, holding great<br>promise for the advancement of QB technologies at all scales.<br>

On-chip quantum interference between the origins of a multi-photon state

Lan-Tian Feng, Ming Zhang, Di Liu, Yu-Jie Cheng, Guo-Ping Guo, Dao-Xin Dai, Guang-Can Guo, M. Krenn, Xi-Feng Ren

Quantum mechanically, multiple particles can jointly be in a coherent superposition of two or more different states at the same time. This property is called quantum entanglement, and gives rise to characteristic nonlocal interference and stays at the heart of quantum information process. Here, rather than interference of different intrinsic properties of particles, we experimentally demonstrated coherent superposition of two different birthplaces of a four-photon state. The quantum state is created in four probabilistic photon-pair sources, two combinations of which can create photon quadruplets. Coherent elimination and revival of distributed 4-photons can be fully controlled by tuning a phase. The stringent coherence requirements are met by using a silicon-based integrated photonic chip that contains four spiral waveguides for producing photon pairs via spontaneous four-wave mixing. The experiment gives rise to peculiar nonlocal phenomena without any obvious involvement of entanglement. Besides several potential applications that exploit the new on-chip technology, it opens up the possibility for fundamental studies on nonlocality with spatially separated locations.

Artificial Intelligence and Machine Learning for Quantum Technologies

Mario Krenn, Jonas Landgraf, Thomas Fösel, Florian Marquardt

In recent years the dramatic progress in machine learning has begun to impact many areas of science and technology significantly. In the present perspective article, we explore how quantum technologies are benefiting from this revolution. We showcase in illustrative examples how scientists in the past few years have started to use machine learning and more broadly methods of artificial intelligence to analyze quantum measurements, estimate the parameters of quantum devices, discover new quantum experimental setups, protocols, and feed- back strategies, and generally improve aspects of quantum computing, quantum communication, and quantum simulation. We highlight open challenges and future possibilities and conclude with some speculative visions for the next decade.

Digital Discovery of 100 diverse Quantum Experiments with PyTheus

Carlos Ruiz-Gonzalez, Sören Arlt, Jan Petermann, Sharareh Sayyad, Tareq Jaouni, Ebrahim Karimi, Nora Tischler, Xuemei Gu, Mario Krenn

Photons are the physical system of choice for performing experimental tests of the foundations of quantum mechanics. Furthermore, photonic quantum technology is a main player in the second quantum revolution, promising the development of better sensors, secure communications, and quantum-enhanced computation. These endeavors require generating specific quantum states or efficiently performing quantum tasks. The design of the corresponding optical experiments was historically powered by human creativity but is recently being automated with advanced computer algorithms and artificial intelligence. While several computer-designed experiments have been experimentally realized, this approach has not yet been widely adopted by the broader photonic quantum optics community. The main roadblocks consist of most systems being closed-source, inefficient, or targeted to very specific use-cases that are difficult to generalize. Here, we overcome these problems with a highly-efficient, open-source digital discovery framework PyTheus, which can employ a wide range of experimental devices from modern quantum labs to solve various tasks. This includes the discovery of highly entangled quantum states, quantum measurement schemes, quantum communication protocols, multi-particle quantum gates, as well as the optimization of continuous and discrete properties of quantum experiments or quantum states. PyTheus produces interpretable designs for complex experimental problems which human researchers can often readily conceptualize. PyTheus is an example of a powerful framework that can lead to scientific discoveries -- one of the core goals of artificial intelligence in science. We hope it will help accelerate the development of quantum optics and provide new ideas in quantum hardware and technology.

Digital Discovery of a Scientific Concept at the Core of Experimental Quantum Optics

Sören Arlt, Carlos Ruiz-Gonzalez, Mario Krenn

Entanglement is a crucial resource for quantum technologies ranging from quantum communication to quantum-enhanced measurements and computation. Finding experimental setups for these tasks is a conceptual challenge for human scientists due to the counterintuitive behavior of multiparticle interference and the enormously large combinatorial search space. Recently, new possibilities have been opened by artificial discovery where artificial intelligence proposes experimental setups for the creation and manipulation of high-dimensional multi-particle entanglement. While digitally discovered experiments go beyond what has been conceived by human experts, a crucial goal is to understand the underlying concepts which enable these new useful experimental blueprints. Here, we present Halo (Hyperedge Assembly by Linear Optics), a new form of multiphoton quantum interference with surprising properties. Halos were used by our digital discovery framework to solve previously open questions. We -- the human part of this collaboration -- were then able to conceptualize the idea behind the computer discovery and describe them in terms of effective probabilistic multi-photon emitters. We then demonstrate its usefulness as a core of new experiments for highly entangled states, communication in quantum networks, and photonic quantum gates. Our manuscript has two conclusions. First, we introduce and explain the physics of a new practically useful multi-photon interference phenomenon that can readily be realized in advanced setups such as integrated photonic circuits. Second, our manuscript demonstrates how artificial intelligence can act as a source of inspiration for the scientific discoveries of new actionable concepts in physics.

SELFIES and the future of molecular string representations

Mario Krenn, Qianxiang Ai, Senja Barthel, Nessa Carson, Angelo Frei, Nathan C. Frey, Pascal Friederich, Théophile Gaudin, Alberto Alexander Gayle, et al.

Artificial intelligence (AI) and machine learning (ML) are expanding in popularity for broad applications to challenging tasks in chemistry and materials science. Examples include the prediction of properties, the discovery of new reaction pathways, or the design of new molecules. The machine needs to read and write fluently in a chemical language for each of these tasks. Strings are a common tool to represent molecular graphs, and the most popular molecular string representation, SMILES, has powered cheminformatics since the late 1980s. However, in the context of AI and ML in chemistry, SMILES has several shortcomings -- most pertinently, most combinations of symbols lead to invalid results with no valid chemical interpretation. To overcome this issue, a new language for molecules was introduced in 2020 that guarantees 100\% robustness: SELFIES (SELF-referencIng Embedded Strings). SELFIES has since simplified and enabled numerous new applications in chemistry. In this manuscript, we look to the future and discuss molecular string representations, along with their respective opportunities and challenges. We propose 16 concrete Future Projects for robust molecular representations. These involve the extension toward new chemical domains, exciting questions at the interface of AI and robust languages and interpretability for both humans and machines. We hope that these proposals will inspire several follow-up works exploiting the full potential of molecular string representations for the future of AI in chemistry and materials science.

Design of quantum optical experiments with logic artificial intelligence

Alba Cervera-Lierta, Mario Krenn, Alán Aspuru-Guzik

Logic Artificial Intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and solved by checking their satisfiability (SAT). In contrast to machine learning approaches where the results can be approximations or local minima, Logic AI delivers formal and mathematically exact solutions to those problems. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI significantly improves the resolution of this problem, paving the path to developing more formal-based approaches in the context of quantum physics experiments.

On scientific understanding with artificial intelligence

Mario Krenn, Robert Pollice, Si Yue Guo, Matteo Aldeghi, Alba Cervera-Lierta, Pascal Friederich, Gabriel dos Passos Gomes, Florian Häse, Adrian Jinich, et al.

An oracle that correctly predicts the outcome of every particle physics experiment, the products of every possible chemical reaction or the function of every protein would revolutionize science and technology. However, scientists would not be entirely satisfied because they would want to comprehend how the oracle made these predictions. This is scientific understanding, one of the main aims of science. With the increase in the available computational power and advances in artificial intelligence, a natural question arises: how can advanced computational systems, and specifically artificial intelligence, contribute to new scientific understanding or gain it autonomously? Trying to answer this question, we adopted a definition of ‘scientific understanding’ from the philosophy of science that enabled us to overview the scattered literature on the topic and, combined with dozens of anecdotes from scientists, map out three dimensions of computer-assisted scientific understanding. For each dimension, we review the existing state of the art and discuss future developments. We hope that this Perspective will inspire and focus research directions in this multidisciplinary emerging field.<br><br><br><br>

Predicting the Future of AI with AI: High-quality link prediction in an exponentially growing knowledge network

Mario Krenn, Lorenzo Buffoni, Bruno Coutinho, Sagi Eppel, Jacob Gates Foster, Andrew Gritsevskiy, Harlin Lee, Yichao Lu, Joao P. Moutinho, et al.

A tool that could suggest new personalized research directions and ideas by taking insights from the scientific literature could significantly accelerate the progress of science. A field that might benefit from such an approach is artificial intelligence (AI) research, where the number of scientific publications has been growing exponentially over the last years, making it challenging for human researchers to keep track of the progress. Here, we use AI techniques to predict the future research directions of AI itself. We develop a new graph-based benchmark based on real-world data -- the Science4Cast benchmark, which aims to predict the future state of an evolving semantic network of AI. For that, we use more than 100,000 research papers and build up a knowledge network with more than 64,000 concept nodes. We then present ten diverse methods to tackle this task, ranging from pure statistical to pure learning methods. Surprisingly, the most powerful methods use a<br>carefully curated set of network features, rather than an end-to-end AI approach. It indicates a great potential that can be unleashed for purely ML approaches without human knowledge. Ultimately, better predictions of new future research directions will be a crucial component of more advanced research suggestion tools.

Curiosity in exploring chemical spaces: Intrinsic rewards for deep molecular reinforcement learning

Luca A. Thiede, Mario Krenn, AkshatKumar Nigam, Alán Aspuru-Guzik

Computer-aided design of molecules has the potential to disrupt the field of drug and material discovery. Machine learning, and deep learning, in particular, have been topics where the field has been developing at a rapid pace. Reinforcement learning is a particularly promising approach since it allows for molecular design without prior knowledge. However, the search space is vast and efficient exploration is desirable when using reinforcement learning agents. In this study, we propose an algorithm to aid efficient exploration. The algorithm is inspired by a concept known in the literature as curiosity. We show on three benchmarks that a curious agent finds better performing molecules. This indicates an exciting new research direction for reinforcement learning agents that can explore the chemical space out of their own motivation. This has the potential to eventually lead to unexpected new molecules that no human has thought about so far.

Quantum indistinguishability by path identity and with undetected photons

Armin Hochrainer, Mayukh Lahiri, Manuel Erhard, Mario Krenn, Anton Zeilinger

Two processes of photon-pair creation can be arranged such that the paths of the emitted photons are identical. The path information is thereby not erased but rather never born in the first place due to this path identity. In addition to its implications for fundamental physics, this concept has recently led to a series of impactful discoveries in the fields of imaging, spectroscopy, and quantum information science. Here the idea of path identity is presented and a comprehensive review of recent developments is provided. Specifically, the concept of path identity is introduced based on three defining experimental ideas from the early 1990s. The three experiments have in common that they contain two photon-pair sources. The paths of one or both photons from the different sources overlap such that no measurement can recognize from which source they originate. A wide range of noteworthy quantum interference effects (at the single- or two-photon level), such as induced coherence, destructive interference of photon pairs, and entanglement generation, are subsequently described. Progress in the exploration of these ideas has stagnated and has gained momentum again only in the last few years. The focus of the review is the new development in the last few years that modified and generalized the ideas from the early 1990s. These developments are overviewed and explained under the same conceptual umbrella, which will help the community develop new applications and realize the foundational implications of this sleeping beauty.

Learning Interpretable Representations of Entanglement in Quantum Optics Experiments using Deep Generative Models

Daniel Flam-Shepherd, Tony Wu, Xuemei Gu, Alba Cervera-Lierta, M. Krenn, Alan Aspuru-Guzik

Quantum physics experiments produce interesting phenomena such as interference or entanglement, which is a core property of numerous future quantum technologies. The complex relationship between a quantum experiment's structure and its entanglement properties is essential to fundamental research in quantum optics but is difficult to intuitively understand. We present the first deep generative model of quantum optics experiments where a variational autoencoder (QOVAE) is trained on a dataset of experimental setups. In a series of computational experiments, we investigate the learned representation of the QOVAE and its internal understanding of the quantum optics world. We demonstrate that the QOVAE learns an intrepretable representation of quantum optics experiments and the relationship between experiment structure and entanglement. We show the QOVAE is able to generate novel experiments for highly entangled quantum states with specific distributions that match its training data. Importantly, we are able to fully interpret how the QOVAE structures its latent space, finding curious patterns that we can entirely explain in terms of quantum physics. The results demonstrate how we can successfully use and understand the internal representations of deep generative models in a complex scientific domain. The QOVAE and the insights from our investigations can be immediately applied to other physical systems throughout fundamental scientific research.

Experimental high-dimensional Greenberger-Horne-Zeilinger entanglement with superconducting transmon qutrits

Alba Cervera-Lierta, Mario Krenn, Alan Aspuru-Guzik, Alexey Galda

Multipartite entanglement is one of the core concepts in quantum information science with broad applications that span from condensed matter physics to quantum physics foundations tests. Although its most studied and tested forms encompass two-dimensional systems, current quantum platforms technically allow the manipulation of additional quantum levels. We report the first experimental demonstration of a high-dimensional multipartite entangled state in a superconducting quantum processor. We generate the three-qutrit Greenberger-Horne-Zeilinger state by designing the necessary pulses to perform high-dimensional quantum operations. We obtain the fidelity of 76 ±1%, proving the generation of a genuine three-partite and three-dimensional entangled state.<br>To this date, only photonic devices have been able to create and manipulate these high-dimensional states. Our work demonstrates that another platform, superconducting systems, is ready to exploit<br>high-dimensional physics phenomena and that a programmable quantum device accessed on the<br>cloud can be used to design and execute experiments beyond binary quantum computation.

Quantum Optical Experiments Modeled by Long Short-Term Memory

Thomas Adler, Manuel Erhard, M. Krenn, Johannes Brandstetter, Johannes Kofler, Sepp Hochreiter

We demonstrate how machine learning is able to model experiments in quantum physics. Quantum entanglement is a cornerstone for upcoming quantum technologies such as quantum computation and quantum cryptography. Of particular interest are complex quantum states with more than two particles and a large number of entangled quantum levels. Given such a multiparticle high-dimensional quantum state, it is usually impossible to reconstruct an experimental setup that produces<br>it. To search for interesting experiments, one thus has to randomly create millions of setups on a computer and calculate the respective output states. In this work, we show that machine learning models can provide significant improvement over random search. We demonstrate that a long short-term memory (LSTM) neural network can successfully learn to model quantum experiments by correctly predicting output state characteristics for given setups without the necessity of computing the states themselves. This approach not only allows for faster search but is also an essential step towards automated design of multiparticle high-dimensional quantum experiments using generative machine learning models.

Design of quantum optical experiments with logic artificial intelligence

Alba Cervera-Lierta, M. Krenn, Alan Aspuru-Guzik

Logic artificial intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and be solved by checking their satisfiability (SAT). Recently, SAT solvers have become a sophisticated and powerful computational tool capable, among other things, of solving long-standing mathematical conjectures. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logic-based algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the state-of-the-art algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI improves significantly the resolution of this problem, paving the path to develop more formal-based approaches in the context of quantum physics experiments.

Conceptual Understanding through Efficient Automated Design of Quantum Optical Experiments

Mario Krenn, Jakob S. Kottmann, Nora Tischler, Alan Aspuru-Guzik

Artificial intelligence (AI) is a potentially disruptive tool for physics and science in general. One crucial question is how this technology can contribute at a conceptual level to help acquire new scientific understanding. Scientists have used AI techniques to rediscover previously known concepts. So far, no examples of that kind have been reported that are applied to open problems for getting new scientific concepts and ideas. Here, we present THESEUS, an algorithm that can provide new conceptual understanding, and we demonstrate its applications in the field of experimental quantum optics. To do so, we make four crucial contributions. (i) We introduce a graph-based representation of quantum optical experiments that can be interpreted and used algorithmically. (ii) We develop an automated design approach for new quantum experiments, which is orders of magnitude faster than the best previous algorithms at concrete design tasks for experimental configuration. (iii) We solve several crucial open questions in experimental quantum optics which involve practical blueprints of resource states in photonic quantum technology and quantum states and transformations that allow for new foundational quantum experiments. Finally, and most importantly, (iv) the interpretable representation and enormous speed-up allow us to produce solutions that a human scientist can interpret and gain new scientific concepts from outright. We anticipate that THESEUS will become an essential tool in quantum optics for developing new experiments and photonic hardware. It can further be generalized to answer open questions and provide new concepts in a large number of other quantum physical questions beyond quantum optical experiments. THESEUS is a demonstration of explainable AI (XAI) in physics that shows how AI algorithms can contribute to science on a conceptual level.

Deep molecular dreaming: inverse machine learning for de-novo molecular design and interpretability with surjective representations

Cynthia Shen, Mario Krenn, Sagi Eppel, Alán Aspuru-Guzik

Computer-based de-novo design of functional molecules is one of the most prominent challenges in cheminformatics today. As a result, generative and evolutionary inverse designs from the field of artificial intelligence have emerged at a rapid pace, with aims to optimize molecules for a particular chemical property. These models 'indirectly' explore the chemical space; by learning latent spaces, policies, and distributions, or by applying mutations on populations of molecules. However, the recent development of the SELFIES (Krenn 2020 Mach. Learn.: Sci. Technol. 1 045024) string representation of molecules, a surjective alternative to SMILES, have made possible other potential techniques. Based on SELFIES, we therefore propose PASITHEA, a direct gradient-based molecule optimization that applies inceptionism (Mordvintsev 2015) techniques from computer vision. PASITHEA exploits the use of gradients by directly reversing the learning process of a neural network, which is trained to predict real-valued chemical properties. Effectively, this forms an inverse regression model, which is capable of generating molecular variants optimized for a certain property. Although our results are preliminary, we observe a shift in distribution of a chosen property during inverse-training, a clear indication of PASITHEA's viability. A striking property of inceptionism is that we can directly probe the model's understanding of the chemical space on which it is trained. We expect that extending PASITHEA to larger datasets, molecules and more complex properties will lead to advances in the design of new functional molecules as well as the interpretation and explanation of machine learning models.

Quantum computer-aided design of quantum optics hardware

Jakob S. Kottmann, Mario Krenn, Thi Ha Kyaw, Sumner Alperin-Lea, Alan Aspuru-Guzik

The parameters of a quantum system grow exponentially with the number of involved quantum particles. Hence, the associated memory requirement to store or manipulate the underlying wavefunction goes well beyond the limit of the best classical computers for quantum systems composed of a few dozen particles, leading to serious challenges in their numerical simulation. This implies that the verification and design of new quantum devices and experiments are fundamentally limited to small system size. It is not clear how the full potential of large quantum systems can be exploited. Here, we present the concept of quantum computer designed quantum hardware and apply it to the field of quantum optics. Specifically, we map complex experimental hardware for high-dimensional, many-body entangled photons into a gate-based quantum circuit. We show explicitly how digital quantum simulation of Boson sampling experiments can be realized. We then illustrate how to design quantum-optical setups for complex entangled photonic systems, such as high-dimensional Greenberger-Horne-Zeilinger states and their derivatives. Since photonic hardware is already on the edge of quantum supremacy and the development of gate-based quantum computers is rapidly advancing, our approach promises to be a useful tool for the future of quantum device design.

Scientific intuition inspired by machine learning-generated hypotheses

Pascal Friederich, Mario Krenn, Isaac Tamblyn, Alan Aspuru-Guzik

Machine learning with application to questions in the physical sciences has become a widely used tool, successfully applied to classification, regression and optimization tasks in many areas. Research focus mostly lies in improving the accuracy of the machine learning models in numerical predictions, while scientific understanding is still almost exclusively generated by human researchers analysing numerical results and drawing conclusions. In this work, we shift the focus on the insights and the knowledge obtained by the machine learning models themselves. In particular, we study how it can be extracted and used to inspire human scientists to increase their intuitions and understanding of natural systems. We apply gradient boosting in decision trees to extract human-interpretable insights from big data sets from chemistry and physics. In chemistry, we not only rediscover widely know rules of thumb but also find new interesting motifs that tell us how to control solubility and energy levels of organic molecules. At the same time, in quantum physics, we gain new understanding on experiments for quantum entanglement. The ability to go beyond numerics and to enter the realm of scientific insight and hypothesis generation opens the door to use machine learning to accelerate the discovery of conceptual understanding in some of the most challenging domains of science.

Beyond generative models: superfast traversal, optimization, novelty, exploration and discovery (STONED) algorithm for molecules using SELFIES

AkshatKumar Nigam, Robert Pollice, Mario Krenn, Gabriel dos Passos Gomes, Alan Aspuru-Guzik

Inverse design allows the generation of molecules with desirable physical quantities using property optimization. Deep generative models have recently been applied to tackle inverse design, as they possess the ability to optimize molecular properties directly through structure modification using gradients. While the ability to carry out direct property optimizations is promising, the use of generative deep learning models to solve practical problems requires large amounts of data and is very time-consuming. In this work, we propose STONED - a simple and efficient algorithm to perform interpolation and exploration in the chemical space, comparable to deep generative models. STONED bypasses the need for large amounts of data and training times by using string modifications in the SELFIES molecular representation. First, we achieve non-trivial performance on typical benchmarks for generative models without any training. Additionally, we demonstrate applications in high-throughput virtual screening for the design of drugs, photovoltaics, and the construction of chemical paths, allowing for both property and structure-based interpolation in the chemical space. Overall, we anticipate our results to be a stepping stone for developing more sophisticated inverse design models and benchmarking tools, ultimately helping generative models achieve wider adoption.

Data-Driven Strategies for Accelerated Materials Design

Robert Pollice, Gabriel dos Passos Gomes, Matteo Aldeghi, Riley J. Hickman, M. Krenn, Cyrille Lavigne, Michael Lindner-D’Addario, AkshatKumar Nigam, Cher Tian Ser, et al.

The ongoing revolution of the natural sciences by the advent of machine learning and artificial intelligence sparked significant interest in the material science community in recent years. The intrinsically high dimensionality of the space of realizable materials makes traditional approaches ineffective for large-scale explorations. Modern data science and machine learning tools developed for increasingly complicated problems are an attractive alternative. An imminent climate catastrophe calls for a clean energy transformation by overhauling current technologies within only several years of possible action available. (...)

Compact Greenberger—Horne—Zeilinger state generation via frequency combs and graph theory

Xuemei Gu, Mario Krenn

Self-referencing embedded strings (SELFIES): A 100% robust molecular string representation

Mario Krenn, Florian Häse, AkshatKumar Nigam, Pascal Friederich, Alan Aspuru-Guzik

The discovery of novel materials and functional molecules can help to solve some of society's most urgent challenges, ranging from efficient energy harvesting and storage to uncovering novel pharmaceutical drug candidates. Traditionally matter engineering–generally denoted as inverse design–was based massively on human intuition and high-throughput virtual screening. The last few years have seen the emergence of significant interest in computer-inspired designs based on evolutionary or deep learning methods. The major challenge here is that the standard strings molecular representation SMILES shows substantial weaknesses in that task because large fractions of strings do not correspond to valid molecules. Here, we solve this problem at a fundamental level and introduce SELFIES (SELF-referencIng Embedded Strings), a string-based representation of molecules which is 100% robust. Every SELFIES string corresponds to a valid molecule, and SELFIES can represent every molecule. SELFIES can be directly applied in arbitrary machine learning models without the adaptation of the models; each of the generated molecule candidates is valid. In our experiments, the model's internal memory stores two orders of magnitude more diverse molecules than a similar test with SMILES. Furthermore, as all molecules are valid, it allows for explanation and interpretation of the internal working of the generative models.

Path identity as a source of high-dimensional entanglement

Jaroslav Kysela, Manuel Erhard, Armin Hochrainer, Mario Krenn, Anton Zeilinger

We present an experimental demonstration of a general entanglement-generation framework, where the form of the entangled state is independent of the physical process used to produce the particles. It is the indistinguishability of multiple generation processes and the geometry of the setup that give rise to the entanglement. Such a framework, termed entanglement by path identity, exhibits a high degree of customizability. We employ one class of such geometries to build a modular source of photon pairs that are high-dimensionally entangled in their orbital angular momentum. We demonstrate the creation of three-dimensionally entangled states and show how to incrementally increase the dimensionality of entanglement. The generated states retain their quality even in higher dimensions. In addition, the design of our source allows for its generalization to various degrees of freedom and even for the implementation in integrated compact devices. The concept of entanglement by path identity itself is a general scheme and allows for construction of sources producing also customized states of multiple photons. We therefore expect that future quantum technologies and fundamental tests of nature in higher dimensions will benefit from this approach.

Computer-inspired quantum experiments

Mario Krenn, Manuel Erhard, Anton Zeilinger

The design of new devices and experiments has historically relied on the intuition of human experts. Now, design inspirations from computers are increasingly augmenting the capability of scientists. We briefly overview different fields of physics that rely on computer-inspired designs using a variety of computational approaches based on topological optimization, evolutionary strategies, deep learning, reinforcement learning or automated reasoning. Then we focus specifically on quantum physics. When designing new quantum experiments, there are two challenges: quantum phenomena are unintuitive, and the number of possible configurations of quantum experiments explodes exponentially. These challenges can be overcome by using computer-designed quantum experiments. We focus on the most mature and practical approaches to find new complex quantum experiments, which have subsequently been realized in the lab. These methods rely on a highly efficient topological search, which can inspire new scientific ideas. We review several extensions and alternatives based on various optimization and machine learning techniques. Finally, we discuss what can be learned from the different approaches and outline several future directions.

Computer-Inspired Concept for High-Dimensional Multipartite Quantum Gates

Xiaoqin Gao, Manuel Erhard, Anton Zeilinger, Mario Krenn

An open question in quantum optics is how to manipulate and control complex quantum states in an experimentally feasible way. Here we present concepts for transformations of high-dimensional multiphotonic quantum systems. The proposals rely on two new ideas: (i) a novel high-dimensional quantum nondemolition measurement, (ii) the encoding and decoding of the entire quantum transformation in an ancillary state for sharing the necessary quantum information between the involved parties. Many solutions can readily be performed in laboratories around the world and thereby we identify important pathways for experimental research in the near future. The concepts have been found using the computer algorithm MELVIN for designing computer-inspired quantum experiments. As opposed to the field of machine learning, here the human learns new scientific concepts by interpreting and analyzing the results presented by the machine. This demonstrates that computer algorithms can inspire new ideas in science, which has a widely unexplored potential that goes far beyond experimental quantum information science.

Advances in high-dimensional quantum entanglement

Manuel Erhard, Mario Krenn, Anton Zeilinger

Since its discovery, quantum entanglement has challenged some of the best established views of the world: locality and reality. Quantum technologies promise to revolutionize computation, communication, metrology and imaging. Here we review conceptual and experimental advances in complex entangled systems involving many multilevel quantum particles. We provide an overview of the latest technological developments in the generation and manipulation of high-dimensionally entangled photonic systems encoded in various discrete degrees of freedom such as path, transverse spatial modes or time-frequency bins. This overview should help to transfer various physical principles for the generation and manipulation from one degree of freedom to another and thus inspire new technical developments. We also show how purely academic questions and curiosity led to new technological applications. Fundamental research provides the necessary knowledge for upcoming technologies, such as a prospective quantum internet or the quantum teleportation of all information stored in a quantum system. Finally, we discuss some important problems in the area of high-dimensional entanglement and give a brief outlook on possible future developments.<br> The study of higher-dimensional quantum states has seen numerous conceptual and technological developments. This review discusses various techniques for the generation and processing of qudits, which are stored in the momentum, path, time-/frequency-bins, or the orbital angular momentum of photons.

Phenomenology of complex structured light in turbulent air

Xuemei Gu, Lijun Chen, Mario Krenn

The study of light propagation has been a cornerstone of progress in physics and technology. Recently, advances in control and shaping of light have created significant interest in the propagation of complex structures of light - particularly under realistic terrestrial conditions. While theoretical understanding of this research question has significantly grown over the last two decades, outdoor experiments with complex light structures are rare, and comparisons with theory have been nearly lacking. Such situations show a significant gap between theoretical models of atmospheric light behaviour and current experimental effort. Here, in an attempt to reduce this gap, we describe an interesting result of atmospheric models that are feasible for empirical observation. We analyze in detail light propagation in different spatial bases and present results of the theory that the influence of atmospheric turbulence is basis-dependent. Concretely, light propagating as eigenstate in one complete basis is more strongly influenced by atmosphere than light propagating in a different, complete basis. We obtain these results by exploiting a family of the continuously adjustable, complete basis of spatial modes-the Ince-Gauss modes. Our concrete numerical results will hopefully inspire experimental efforts and bring the theoretical and empirical study of complex light patterns in realistic scenarios closer together. Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License.

The sounds of science—a symphony for many instruments and voices

Gerianne Alexander, Roland E Allen, Anthony Atala, Warwick P Bowen, Alan A Coley, John B Goodenough, Mikhail I Katsnelson, Eugene V Koonin, Mario Krenn, et al.

Quantum experiments and hypergraphs: Multiphoton sources for quantum interference, quantum computation, and quantum entanglement

Xuemei Gu, Lijun Chen, Mario Krenn

We introduce the concept of hypergraphs to describe quantum optical experiments with probabilistic multiphoton sources. Every hyperedge represents a correlated photon source, and every vertex stands for an optical output path. Such a general graph description provides new insights for producing complex high-dimensional multiphoton quantum entangled states, which go beyond limitations imposed by pair creation via spontaneous parametric down-conversion. Furthermore, the properties of hypergraphs can be investigated experimentally. For example, the NP-complete problem of deciding whether a hypergraph has a perfect matching can be answered by experimentally detecting multiphoton events in quantum experiments. By introducing complex weights in hypergraphs, we show a general many-particle quantum interference and manipulating entanglement in a pictorial way. Our work paves the path for the development of multiphoton high-dimensional state generation and might inspire new applications of quantum computations using hypergraph mappings.

Predicting research trends with semantic and neural networks with an application in quantum physics

Mario Krenn, Anton Zeilinger

The vast and growing number of publications in all disciplines of science cannot be comprehended by a single human researcher. As a consequence, researchers have to specialize in narrow subdisciplines, which makes it challenging to uncover scientific connections beyond the own field of research. Thus, access to structured knowledge from a large corpus of publications could help push the frontiers of science. Here, we demonstrate a method to build a semantic network from published scientific literature, which we call SEMNET. We use SEMNET to predict future trends in research and to inspire personalized and surprising seeds of ideas in science. We apply it in the discipline of quantum physics, which has seen an unprecedented growth of activity in recent years. In SEMNET, scientific knowledge is represented as an evolving network using the content of 750,000 scientific papers published since 1919. The nodes of the network correspond to physical concepts, and links between two nodes are drawn when two concepts are concurrently studied in research articles. We identify influential and prize-winning research topics from the past inside SEMNET, thus confirming that it stores useful semantic knowledge. We train a neural network using states of SEMNET of the past to predict future developments in quantum physics and confirm high-quality predictions using historic data. Using network theoretical tools, we can suggest personalized, out-of-the-box ideas by identifying pairs of concepts, which have unique and extremal semantic network properties. Finally, we consider possible future developments and implications of our findings.

Augmenting Genetic Algorithms with Deep Neural Networks for Exploring the Chemical Space

AkshatKumar Nigam, Pascal Friederich, Mario Krenn, Alán Aspuru-Guzik

Challenges in natural sciences can often be phrased as optimization problems. Machine learning techniques have recently been applied to solve such problems. One example in chemistry is the design of tailor-made organic materials and molecules, which requires efficient methods to explore the chemical space. We present a genetic algorithm (GA) that is enhanced with a neural network (DNN) based discriminator model to improve the diversity of generated molecules and at the same time steer the GA. We show that our algorithm outperforms other generative models in optimization tasks. We furthermore present a way to increase interpretability of genetic algorithms, which helped us to derive design principles.

Physics Insights from Neural Networks

Mario Krenn

Researchers probe a machine-learning model as it solves physics problems in order to understand how such models "think."

Quantenteleportation in höheren Dimensionen

Manuel Erhard, Mario Krenn

Vor rund zwei Jahrzehnten gelang erstmals die Quantenteleportation, also die Übertragung von Quanteninformation mit Hilfe der Verschränkung. Unserer Gruppe am Institute for Quantum Optics and Quantum Information (IQOQI) Vienna ist es nun in Kooperation mit einem chinesischen Team gelungen, dreidimensionale Zustände von Photonen zu übertragen.

Quantum Teleportation in High Dimensions

Yi-Han Luo, Han-Sen Zhong, Manuel Erhard, Xi-Lin Wang, Li-Chao Peng, Mario Krenn, Xiao Jiang, Li Li, Nai-Le Liu, et al.

Quantum teleportation allows a "disembodied" transmission of unknown quantum states between distant quantum systems. Yet, all teleportation experiments to date were limited to a two-dimensional subspace of quantized multiple levels of the quantum systems. Here, we propose a scheme for teleportation of arbitrarily high-dimensional photonic quantum states and demonstrate an example of teleporting a qutrit. Measurements over a complete set of 12 qutrit states in mutually unbiased bases yield a teleportation fidelity of 0.75(1), which is well above both the optimal single-copy qutrit state-estimation limit of 1/2 and maximal qubit-qutrit overlap of 2/3, thus confirming a genuine and nonclassical three-dimensional teleportation. Our work will enable advanced quantum technologies in high dimensions, since teleportation plays a central role in quantum repeaters and quantum networks.

Questions on the Structure of Perfect Matchings Inspired by Quantum Physics

Mario Krenn, Xuemei Gu, Daniel Soltesz

We state a number of related questions on the structure of perfect matchings. Those questions are inspired by and directly connected to Quantum Physics. In particular, they concern the constructability of general quantum states using modern photonic technology. For that we introduce a new concept, denoted as inherited vertex coloring. It is a vertex coloring for every perfect matching. The colors are inherited from the color of the incident edge for each perfect matching.<br>First, we formulate the concepts and questions in pure graph-theoretical language, and finally we explain the physical context of every mathematical object that we use. Importantly, every progress towards answering these questions can directly be translated into new understanding in quantum physics.

Quantum experiments and graphs. III. High-dimensional and multiparticle entanglement

Xuemei Gu, Lijun Chen, Anton Zeilinger, Mario Krenn

Quantum entanglement plays an important role in quantum information processes, such as quantum computation and quantum communication. Experiments in laboratories are unquestionably crucial to increase our understanding of quantum systems and inspire new insights into future applications. However, there are no general recipes for the creation of arbitrary quantum states with many particles entangled in high dimensions. Here we exploit a recent connection between quantum experiments and graph theory and answer this question for a plethora of classes of entangled states. We find experimental setups for Greenberger-Horne-Zeilinger states, W states, general Dicke states, and asymmetrically high-dimensional multipartite entangled states. This result sheds light on the producibility of arbitrary quantum states using photonic technology with probabilistic pair sources and allows us to understand the underlying technological and fundamental properties of entanglement.

Quantum experiments and graphs II: Quantum interference, computation, and state generation

Xuemei Gu, Manuel Erhard, Anton Zeilinger, Mario Krenn

We present an approach to describe state-of-the-art photonic quantum experiments using graph theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that introducing complex weights in graphs naturally leads to quantum interference. This viewpoint immediately leads to many interesting results, some of which we present here. First, we identify an experimental unexplored multiphoton interference phenomenon. Second, we find that computing the results of such experiments is #P-hard, which means it is a classically intractable problem dealing with the computation of a matrix function Permanent and its generalization Hafnian. Third, we explain how a recent no-go result applies generally to linear optical quantum experiments, thus revealing important insights into quantum state generation with current photonic technology. Fourth, we show how to describe quantum protocols such as entanglement swapping in a graphical way. The uncovered bridge between quantum experiments and graph theory offers another perspective on a widely used technology and immediately raises many follow-up questions.

Arbitrary d-dimensional Pauli X gates of a flying qudit

Xiaoqin Gao, Mario Krenn, Jaroslav Kysela, Anton Zeilinger

High-dimensional degrees of freedom of photons can encode more quantum information than their two-dimensional counterparts. While the increased information capacity has advantages in quantum applications (such as quantum communication), controlling and manipulating these systems has been challenging. Here we show a method to perform deterministic arbitrary high-dimensional Pauli X gates for single photons carrying orbital angular momentum. The X gate consists of a cyclic permutation of qudit basis vectors and, together with the Z gate, forms the basis for performing arbitrary transformations. The proposed experimental setups only use two basic optical elements such as mode sorters and mode shifters and thus could be implemented in any system where these experimental tools are available. Furthermore the number of involved interferometers scales logarithmically with the dimension, which is important for practical implementation.

Experimental Greenberger-Horne-Zeilinger entanglement beyond qubits

Manuel Erhard, Mehul Malik, Mario Krenn, Anton Zeilinger

Quantum entanglement is important for emerging quantum technologies such as quantum computation and secure quantum networks. To boost these technologies, a race is currently ongoing to increase the number of particles in multiparticle entangled states, such as Greenberger-Horne-Zeilinger (GHZ) states. An alternative route is to increase the number of entangled quantum levels. Here, we overcome present experimental and technological challenges to create a three-particle GHZ state entangled in three levels for every particle. The resulting qutrit-entangled states are able to carry more information than entangled states of qubits. Our method, inspired by the computer algorithm Melvin, relies on a new multi-port that coherently manipulates several photons simultaneously in higher dimensions. The realization required us to develop a new high-brightness four-photon source entangled in orbital angular momentum. Our results allow qualitatively new refutations of local-realistic world views. We also expect that they will open up pathways for a further boost to quantum technologies.

On small beams with large topological charge: II. Photons, electrons and gravitational waves

Mario Krenn, Anton Zeilinger

Beams of light with a large topological charge significantly change their spatial structure when they are focused strongly. Physically, it can be explained by an emerging electromagnetic field component in the direction of propagation, which is neglected in the simplified scalar wave picture in optics. Here we ask: is this a specific photonic behavior, or can similar phenomena also be predicted for other species of particles? We show that the same modification of the spatial structure exists for relativistic electrons as well as for focused gravitational waves. However, this is for different physical reasons: for electrons, which are described by the Dirac equation, the spatial structure changes due to a spin-orbit coupling in the relativistic regime. In gravitational waves described with linearized general relativity, the curvature of space-time between the transverse and propagation direction leads to the modification of the spatial structure. Thus, this universal phenomenon exists for both massive and massless elementary particles with spin 1 /2,1 and 2. It would be very interesting whether other types of particles such as composite systems (neutrons or C-60) or neutrinos show a similar behavior and how this phenomenon can be explained in a unified physical way.

Twisted photons: new quantum perspectives in high dimensions

Manuel Erhard, Robert Fickler, Mario Krenn, Anton Zeilinger

Twisted photons can be used as alphabets to encode information beyond one bit per single photon. This ability offers great potential for quantum information tasks, as well as for the investigation of fundamental questions. In this review article, we give a brief overview of the theoretical differences between qubits and higher dimensional systems, qudits, in different quantum information scenarios. We then describe recent experimental developments in this field over the past three years. Finally, we summarize some important experimental and theoretical questions that might be beneficial to understand better in the near future.

Gouy Phase Radial Mode Sorter for Light: Concepts and Experiments

Xuemei Gu, Mario Krenn, Manuel Erhard, Anton Zeilinger

We present an in principle lossless sorter for radial modes of light, using accumulated Gouy phases. The experimental setups have been found by a computer algorithm, and can be intuitively understood in a geometric way. Together with the ability to sort angular-momentum modes, we now have access to the complete two-dimensional transverse plane of light. The device can readily be used in multiplexing classical information. On a quantum level, it is an analog of the Stern-Gerlach experiment-significant for the discussion of fundamental concepts in quantum physics. As such, it can be applied in high-dimensional and multiphotonic quantum experiments.

Active learning machine learns to create new quantum experiments

Alexey A. Melnikov, Hendrik Poulsen Nautrup, Mario Krenn, Vedran Dunjko, Markus Tiersch, Anton Zeilinger, Hans J. Briegel

How useful can machine learning be in a quantum laboratory? Here we raise the question of the potential of intelligent machines in the context of scientific research. A major motivation for the present work is the unknown reachability of various entanglement classes in quantum experiments. We investigate this question by using the projective simulation model, a physics-oriented approach to artificial intelligence. In our approach, the projective simulation system is challenged to design complex photonic quantum experiments that produce high-dimensional entangled multiphoton states, which are of high interest in modern quantum experiments. The artificial intelligence system learns to create a variety of entangled states and improves the efficiency of their realization. In the process, the system autonomously (re)discovers experimental techniques which are only now becoming standard in modern quantum optical experiments-a trait which was not explicitly demanded from the system but emerged through the process of learning. Such features highlight the possibility that machines could have a significantly more creative role in future research.

Mit Lichtschrauben ans Quantenlimit

Robert Fickler, Mario Krenn, Anton Zeilinger

Licht erlaubt es, Quanteneffekte zwischen weit entfernten Systemen mit vielen Einstellmöglichkeiten zu präparieren. Dazu zählen auch „Lichtschrauben“ mit Bahndrehimpulsen hoher Quantenzahl. Mit speziellen Spiegeln wurden bereits Bahndrehimpulse mit Quantenzahlen bis zu 10 000 ħ erzeugt. Dies ist sogar bei einzelnen Photonen möglich. Damit ist eine Form der Quantenverschränkung nutzbar, welche die Bahndrehimpuls-Quantenzahl eines Photons mit der Polarisation eines Partnerphotons verschränkt. Bei hoher Drehimpulsquantenzahl lässt sich in einem solchen Paar entsprechend viel Quanteninformation speichern. Diese Verschränkung bleibt sogar bei der Übertragung über große Distanzen in Luft erhalten. Sie könnte damit die Bandbreite der Freiluft-Quanteninformationsübertragung erheblich steigern.

Generation of the complete four-dimensional Bell basis

Feiran Wang, Manuel Erhard, Amin Babazadeh, Mehul Malik, Mario Krenn, Anton Zeilinger

The Bell basis is a distinctive set of maximally entangled two-particle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding, and entanglement swapping. While the generation, manipulation, and measurement of two-level quantum states are well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a four-dimensional Hilbert space, comprising 16 orthogonal entangled Bell-like states encoded in the orbital angular momentum of photons. The states are created by the application of generalized high-dimensional Pauli gates on an initial entangled state. Our results pave the way for the application of high-dimensional quantum states in complex quantum protocols such as quantum dense coding. (c) 2017 Optical Society of America

Quantum Experiments and Graphs: Multiparty States as Coherent Superpositions of Perfect Matchings

Mario Krenn, Xuemei Gu, Anton Zeilinger

We show a surprising link between experimental setups to realize high-dimensional multipartite quantum states and graph theory. In these setups, the paths of photons are identified such that the photon-source information is never created. We find that each of these setups corresponds to an undirected graph, and every undirected graph corresponds to an experimental setup. Every term in the emerging quantum superposition corresponds to a perfect matching in the graph. Calculating the final quantum state is in the #P-complete complexity class, thus it cannot be done efficiently. To strengthen the link further, theorems from graph theory-such as Hall's marriage problem-are rephrased in the language of pair creation in quantum experiments. We show explicitly how this link allows one to answer questions about quantum experiments (such as which classes of entangled states can be created) with graph theoretical methods, and how to potentially simulate properties of graphs and networks with quantum experiments (such as critical exponents and phase transitions).

High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments

Amin Babazadeh, Manuel Erhard, Feiran Wang, Mehul Malik, Rahman Nouroozi, Mario Krenn, Anton Zeilinger

Transformations on quantum states form a basic building block of every quantum information system. From photonic polarization to two-level atoms, complete sets of quantum gates for a variety of qubit systems are well known. For multilevel quantum systems beyond qubits, the situation is more challenging. The orbital angular momentum modes of photons comprise one such high-dimensional system for which generation and measurement techniques are well studied. However, arbitrary transformations for such quantum states are not known. Here we experimentally demonstrate a four-dimensional generalization of the Pauli X gate and all of its integer powers on single photons carrying orbital angular momentum. Together with the well-known Z gate, this forms the first complete set of high-dimensional quantum gates implemented experimentally. The concept of the X gate is based on independent access to quantum states with different parities and can thus be generalized to other photonic degrees of freedom and potentially also to other quantum systems.

Quantum gate description for induced coherence without induced emission and its applications

Sahar Alipour, Mario Krenn, Anton Zeilinger

We introduce unitary quantum gates for photon pair creation in spontaneous parametric down-conversion nonlinear crystals (NLs) and for photon path alignment. These are the two key ingredients for the method of induced coherence without induced emission and many ensuing variations thereof. The difficulty in doing so stems from an apparent mixing of the mode picture (such as the polarization of photons) and the Fock picture (such as the existence of the photons). We illustrate utility of these gates by obtaining quantum circuits for the experimental setups of the frustrated generation of photon pairs, identification of a pointlike object with undetected photons, and creation of a Bell state. We also introduce an effective nonunitary description for the action of NLs in experiments where all the NLs are pumped coherently. As an example, by using this simplifying picture, we show how NLs can be used to create superposition of given quantum states in a modular fashion.

Orbital angular momentum of photons and the entanglement of Laguerre-Gaussian modes

Mario Krenn, Mehul Malik, Manuel Erhard, Anton Zeilinger

The identification of orbital angular momentum (OAM) as a fundamental property of a beam of light nearly 25 years ago has led to an extensive body of research around this topic. The possibility that single photons can carry OAM has made this degree of freedom an ideal candidate for the investigation of complex quantum phenomena and their applications. Research in this direction has ranged from experiments on complex forms of quantum entanglement to the interaction between light and quantum states of matter. Furthermore, the use of OAM in quantum information has generated a lot of excitement, as it allows for encoding large amounts of information on a single photon. Here, we explain the intuition that led to the first quantum experiment with OAM 15 years ago. We continue by reviewing some key experiments investigating fundamental questions on photonic OAMand the first steps to applying these properties in novel quantum protocols. At the end, we identify several interesting open questions that could form the subject of future investigations with OAM.<br> This article is part of the themed issue 'Optical orbital angular momentum'.

Entanglement by Path Identity

Mario Krenn, Armin Hochrainer, Mayukh Lahiri, Anton Zeilinger

Quantum entanglement is one of the most prominent features of quantum mechanics and forms the basis of quantum information technologies. Here we present a novel method for the creation of quantum entanglement in multipartite and high-dimensional systems. The two ingredients are (i) superposition of photon pairs with different origins and (ii) aligning photons such that their paths are identical. We explain the experimentally feasible creation of various classes of multiphoton entanglement encoded in polarization as well as in high-dimensional Hilbert spaces-starting only from nonentangled photon pairs. For two photons, arbitrary high-dimensional entanglement can be created. The idea of generating entanglement by path identity could also apply to quantum entities other than photons. We discovered the technique by analyzing the output of a computer algorithm. This shows that computer designed quantum experiments can be inspirations for new techniques.

Quantifying high dimensional entanglement with two mutually unbiased bases

Paul Erker, Mario Krenn, Marcus Huber

We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond being unbiased, thus enabling entanglement quantification with minimal assumptions. Furthermore, we show that it is feasible to experimentally implement our method with readily available equipment and even conservative estimates of physical parameters.

Quantum Communication with Photons

Mario Krenn, Mehul Malik, Thomas Scheidl, Rupert Ursin, Anton Zeilinger

The secure communication of information plays an ever increasing role in our society today. Classical methods of encryption inherently rely on the difficulty of solving a problem such as finding prime factors of large numbers and can, in principle, be cracked by a fast enough machine. The burgeoning field of quantum communication relies on the fundamental laws of physics to offer unconditional information security. Here we introduce the key concepts of quantum superposition and entanglement as well as the no-cloning theorem that form the basis of this field. Then, we review basic quantum communication schemes with single and entangled photons and discuss recent experimental progress in ground and space-based quantum communication. Finally, we discuss the emerging field of high-dimensional quantum communication, which promises increased data rates and higher levels of security than ever before. We discuss recent experiments that use the orbital angular momentum of photons for sharing large amounts of information in a secure fashion.

Twisted light transmission over 143 km

Mario Krenn, Johannes Handsteiner, Matthias Fink, Robert Fickler, Rupert Ursin, Mehul Malik, Anton Zeilinger

Spatial modes of light can potentially carry a vast amount of information, making them promising candidates for both classical and quantum communication. However, the distribution of such modes over large distances remains difficult. Intermodal coupling complicates their use with common fibers, whereas free-space transmission is thought to be strongly influenced by atmospheric turbulence. Here, we show the transmission of orbital angular momentum modes of light over a distance of 143 km between two Canary Islands, which is 50x greater than the maximum distance achieved previously. As a demonstration of the transmission quality, we use superpositions of these modes to encode a short message. At the receiver, an artificial neural network is used for distinguishing between the different twisted light superpositions. The algorithm is able to identify different mode superpositions with an accuracy of more than 80% up to the third mode order and decode the transmitted message with an error rate of 8.33%. Using our data, we estimate that the distribution of orbital angular momentum entanglement over more than 100 km of free space is feasible. Moreover, the quality of our free-space link can be further improved by the use of state-of-the-art adaptive optics systems.

Quantum optical rotatory dispersion

Nora Tischler, Mario Krenn, Robert Fickler, Xavier Vidal, Anton Zeilinger, Gabriel Molina-Terriza

The phenomenon of molecular optical activity manifests itself as the rotation of the plane of linear polarization when light passes through chiral media. Measurements of optical activity and its wavelength dependence, that is, optical rotatory dispersion, can reveal information about intricate properties of molecules, such as the three-dimensional arrangement of atoms comprising a molecule. Given a limited probe power, quantum metrology offers the possibility of outperforming classical measurements. This has particular appeal when samples may be damaged by high power, which is a potential concern for chiroptical studies. We present the first experiment in which multiwavelength polarization-entangled photon pairs are used to measure the optical activity and optical rotatory dispersion exhibited by a solution of chiral molecules. Our work paves the way for quantum-enhanced measurements of chirality, with potential applications in chemistry, biology, materials science, and the pharmaceutical industry. The scheme that we use for probing wavelength dependence not only allows one to surpass the information extracted per photon in a classical measurement but also can be used for more general differential measurements.

Cyclic transformation of orbital angular momentum modes

Florian Schlederer, Mario Krenn, Robert Fickler, Mehul Malik, Anton Zeilinger

The spatial modes of photons are one realization of a QuDit, a quantum system that is described in a D-dimensional Hilbert space. In order to perform quantum information tasks with QuDits, a general class of D-dimensional unitary transformations is needed. Among these, cyclic transformations are an important special case required in many high-dimensional quantum communication protocols. In this paper, we experimentally demonstrate a cyclic transformation in the high-dimensional space of photonic orbital angular momentum (OAM). Using simple linear optical components, we show a successful four-fold cyclic transformation of OAM modes. Interestingly, our experimental setup was found by a computer algorithm. In addition to the four-cyclic transformation, the algorithm also found extensions to higher-dimensional cycles in a hybrid space of OAM and polarization. Besides being useful for quantum cryptography with QuDits, cyclic transformations are key for the experimental production of high-dimensional maximally entangled Bell-states.

Multi-photon entanglement in high dimensions

Mehul Malik, Manuel Erhard, Marcus Huber, Mario Krenn, Robert Fickler, Anton Zeilinger

Forming the backbone of quantum technologies today, entanglement(1,2) has been demonstrated in physical systems as diverse as photons(3), ions(4) and superconducting circuits(5). Although steadily pushing the boundary of the number of particles entangled, these experiments have remained in a two-dimensional space for each particle. Here we show the experimental generation of the first multi-photon entangled state where both the number of particles and dimensions are greater than two. Two photons in our state reside in a three-dimensional space, whereas the third lives in two dimensions. This asymmetric entanglement structure(6) only appears in multiparticle entangled states with d > 2(6). Our method relies on combining two pairs of photons, high-dimensionally entangled in their orbital angular momentum(7). In addition, we show how this state enables a new type of 'layered' quantum communication protocol. Entangled states such as these serve as a manifestation of the complex dance of correlations that can exist within quantum mechanics.

Automated Search for new Quantum Experiments

Mario Krenn, Mehul Malik, Robert Fickler, Radek Lapkiewicz, Anton Zeilinger

Quantum mechanics predicts a number of, at first sight, counterintuitive phenomena. It therefore remains a question whether our intuition is the best way to find new experiments. Here, we report the development of the computer algorithm MELVIN which is able to find new experimental implementations for the creation and manipulation of complex quantum states. Indeed, the discovered experiments extensively use unfamiliar and asymmetric techniques which are challenging to understand intuitively. The results range from the first implementation of a high-dimensional Greenberger-Horne-Zeilinger state, to a vast variety of experiments for asymmetrically entangled quantum states-a feature that can only exist when both the number of involved parties and dimensions is larger than 2. Additionally, new types of high-dimensional transformations are found that perform cyclic operations. MELVIN autonomously learns from solutions for simpler systems, which significantly speeds up the discovery rate of more complex experiments. The ability to automate the design of a quantum experiment can be applied to many quantum systems and allows the physical realization of quantum states previously thought of only on paper.

On small beams with large topological charge

Mario Krenn, Nora Tischler, Anton Zeilinger

Light beams can carry a discrete, in principle unbounded amount of angular momentum. Examples of such beams, the Laguerre-Gauss modes, are frequently expressed as solutions of the paraxial wave equation. The paraxial wave equation is a small-angle approximation of the Helmholtz equation, and is commonly used in beam optics. There, the Laguerre-Gauss modes have well-defined orbital angular momentum (OAM). The paraxial solutions predict that beams with large OAM could be used to resolve arbitrarily small distances-a dubious situation. Here we show how to solve that situation by calculating the properties of beams free from the paraxial approximation. We find the surprising result that indeed one can resolve smaller distances with larger OAM, although with decreased visibility. If the visibility is kept constant (for instance at the Rayleigh criterion, the limit where two points are reasonably distinguishable), larger OAM does not provide an advantage. The drop in visibility is due to a field in the direction of propagation, which is neglected within the paraxial limit. Our findings have implications for imaging techniques and raise questions on the difference between photonic and matter waves, which we briefly discuss in the conclusion.

Physical meaning of the radial index of Laguerre-Gauss beams

William N. Plick, Mario Krenn

The Laguerre-Gauss modes are a class of fundamental and well-studied optical fields. These stable shape-invariant photons, exhibiting circular-cylindrical symmetry, are familiar from laser optics, micromechanical manipulation, quantum optics, communication, and foundational studies in both classical optics and quantum physics. They are characterized, chiefly, by two mode numbers: the azimuthal index indicating the orbital angular momentum of the beam, which itself has spawned a burgeoning and vibrant subfield, and the radial index, which up until recently has largely been ignored. In this paper we develop a differential operator formalism for dealing with the radial modes in both the position and momentum representations and, more importantly, give the meaning of this quantum number in terms of a well-defined physical parameter: the intrinsic hyperbolic momentum charge.

Twisted photon entanglement through turbulent air across Vienna

Mario Krenn, Johannes Handsteiner, Matthias Fink, Robert Fickler, Anton Zeilinger

Photons with a twisted phase front can carry a discrete, in principle, unbounded amount of orbital angular momentum (OAM). The large state space allows for complex types of entanglement, interesting both for quantum communication and for fundamental tests of quantum theory. However, the distribution of such entangled states over large distances was thought to be infeasible due to influence of atmospheric turbulence, indicating a serious limitation on their usefulness. Here we show that it is possible to distribute quantum entanglement encoded in OAM over a turbulent intracity link of 3 km. We confirm quantum entanglement of the first two higher-order levels (with OAM=+/- 1h and +/- 2h). They correspond to four additional quantum channels orthogonal to all that have been used in long-distance quantum experiments so far. Therefore, a promising application would be quantum communication with a large alphabet. We also demonstrate that our link allows access to up to 11 quantum channels of OAM. The restrictive factors toward higher numbers are technical limitations that can be circumvented with readily available technologies.

Communication with spatially modulated light through turbulent air across Vienna

Mario Krenn, Robert Fickler, Matthias Fink, Johannes Handsteiner, Mehul Malik, Thomas Scheidl, Rupert Ursin, Anton Zeilinger

Transverse spatial modes of light offer a large state- space with interesting physical properties. For exploiting these special modes in future long-distance experiments, the modes will have to be transmitted over turbulent free-space links. Numerous recent lab-scale experiments have found significant degradation in the mode quality after transmission through simulated turbulence and consecutive coherent detection. Here, we experimentally analyze the transmission of one prominent class of spatial modes-orbital-angular momentum (OAM) modes-through 3 km of strong turbulence over the city of Vienna. Instead of performing a coherent phase-dependent measurement, we employ an incoherent detection scheme, which relies on the unambiguous intensity patterns of the different spatial modes. We use a pattern recognition algorithm (an artificial neural network) to identify the characteristic mode patterns displayed on a screen at the receiver. We were able to distinguish between 16 different OAM mode superpositions with only a similar to 1.7% error rate and to use them to encode and transmit small grayscale images. Moreover, we found that the relative phase of the superposition modes is not affected by the atmosphere, establishing the feasibility for performing long-distance quantum experiments with the OAM of photons. Our detection method works for other classes of spatial modes with unambiguous intensity patterns as well, and can be further improved by modern techniques of pattern recognition.

Generation and confirmation of a (100 x 100)-dimensional entangled quantum system

Mario Krenn, Marcus Huber, Robert Fickler, Radek Lapkiewicz, Sven Ramelow, Anton Zeilinger

Entangled quantum systems have properties that have fundamentally overthrown the classical worldview. Increasing the complexity of entangled states by expanding their dimensionality allows the implementation of novel fundamental tests of nature, and moreover also enables genuinely newprotocols for quantum information processing. Here we present the creation of a (100 x 100)-dimensional entangled quantum system, using spatial modes of photons. For its verification we develop a novel nonlinear criterion which infers entanglement dimensionality of a global state by using only information about its subspace correlations. This allows very practical experimental implementation as well as highly efficient extraction of entanglement dimensionality information. Applications in quantum cryptography and other protocols are very promising.

Real-Time Imaging of Quantum Entanglement

Robert Fickler, Mario Krenn, Radek Lapkiewicz, Sven Ramelow, Anton Zeilinger

Quantum Entanglement is widely regarded as one of the most prominent features of quantum mechanics and quantum information science. Although, photonic entanglement is routinely studied in many experiments nowadays, its signature has been out of the grasp for real-time imaging. Here we show that modern technology, namely triggered intensified charge coupled device (ICCD) cameras are fast and sensitive enough to image in real-time the effect of the measurement of one photon on its entangled partner. To quantitatively verify the non-classicality of the measurements we determine the detected photon number and error margin from the registered intensity image within a certain region. Additionally, the use of the ICCD camera allows us to demonstrate the high flexibility of the setup in creating any desired spatial-mode entanglement, which suggests as well that visual imaging in quantum optics not only provides a better intuitive understanding of entanglement but will improve applications of quantum science.

Quantum orbital angular momentum of elliptically symmetric light

William N. Plick, Mario Krenn, Robert Fickler, Sven Ramelow, Anton Zeilinger

We present a quantum-mechanical analysis of the orbital angular momentum of a class of recently discovered elliptically symmetric stable light fields-the so-called Ince-Gauss modes. We study, in a fully quantum formalism, how the orbital angular momentum of these beams varies with their ellipticity, and we discover several compelling features, including nonmonotonic behavior, stable beams with real continuous (noninteger) orbital angular momenta, and orthogonal modes with the same orbital angular momenta. We explore, and explain in detail, the reasons for this behavior. These features may have applications in quantum key distribution, atom trapping, and quantum informatics in general-as the ellipticity opens up an alternative way of navigating the spatial photonic Hilbert space. DOI: 10.1103/PhysRevA.87.033806

Entangled singularity patterns of photons in Ince-Gauss modes

Mario Krenn, Robert Fickler, Marcus Huber, Radek Lapkiewicz, William Plick, Sven Ramelow, Anton Zeilinger

Photons with complex spatial mode structures open up possibilities for new fundamental high-dimensional quantum experiments and for novel quantum information tasks. Here we show entanglement of photons with complex vortex and singularity patterns called Ince-Gauss modes. In these modes, the position and number of singularities vary depending on the mode parameters. We verify two-dimensional and three-dimensional entanglement of Ince-Gauss modes. By measuring one photon and thereby defining its singularity pattern, we nonlocally steer the singularity structure of its entangled partner, while the initial singularity structure of the photons is undefined. In addition we measure an Ince-Gauss specific quantum-correlation function with possible use in future quantum communication protocols. DOI: 10.1103/PhysRevA.87.012326

Quantum Entanglement of High Angular Momenta

Robert Fickler, Radek Lapkiewicz, William N. Plick, Mario Krenn, Christoph Schaeff, Sven Ramelow, Anton Zeilinger

Single photons with helical phase structures may carry a quantized amount of orbital angular momentum (OAM), and their entanglement is important for quantum information science and fundamental tests of quantum theory. Because there is no theoretical upper limit on how many quanta of OAM a single photon can carry, it is possible to create entanglement between two particles with an arbitrarily high difference in quantum number. By transferring polarization entanglement to OAM with an interferometric scheme, we generate and verify entanglement between two photons differing by 600 in quantum number. The only restrictive factors toward higher numbers are current technical limitations. We also experimentally demonstrate that the entanglement of very high OAM can improve the sensitivity of angular resolution in remote sensing.