Freie Universität Berlin Dahlem Center for Complex Quantum Systems Complex quantum systems from a signal processing perspective Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften am Fachbereich Physik der Freien Universität Berlin vorgelegt von Adrian Steffens Berlin, 2017 Supervisor Erstgutachter Prof. Dr. Jens Eisert, Freie Universität Berlin Second reviewer Zweitgutachter Prof. Dr. Christoph Karrasch, Freie Universität Berlin Date of defense Tag der Disputation 11.03.2019 Meinen Eltern Juliane und Sigurd Steffens IV CONTENTS Summary of publications ........................................... 1 1 Introduction ................................................. 2 2 Compressed sensing quantum tomography ............................. 6 2.1 Quantum state tomography ................................... 6 2.2 Compressed sensing ........................................ 9 2.3 Experimental application and model selection ........................ 15 3 Quantum field tomography ....................................... 30 3.1 Tensor network states ....................................... 30 3.2 Continuous matrix product states for quantum tomography . 34 3.3 Tomography of ultracold Bose gases .............................. 70 4 Quantum algorithms ........................................... 79 5 Conclusion and outlook ......................................... 97 Acknowledgments ............................................... 100 References ..................................................... 101 Appendix A Coauthored publications ......................................... 117 A.1 Quantum transport experiments ................................ 117 A.2 Quantum singular value decomposition . 133 A.3 Compressive density functional theory ............................ 141 B Miscellaneous ................................................ 147 B.1 Abstract ................................................ 147 B.2 The publications and the author’s contributions . 149 B.3 List of figures ............................................ 151 V VI SUMMARY OF PUBLICATIONS The present dissertation is cumulative according to the doctorate rules of the department ofphysics of the Free University of Berlin and is based on the following first-authored publications. [1] Experimentally exploring compressed sensing quantum tomography . 17 Adrian Steffens, Carlos A. Riofrío, Will McCutcheon, Ingo Roth, Bryn A. Bell, Alex McMillan, Mark S. Tame, John G. Rarity, and Jens Eisert, Quantum Science and Technology 2:025005, 2017. [2] Quantum field tomography . 38 Adrian Steffens, Carlos A. Riofrío, Robert Hübener, and Jens Eisert, New Journal of Physics 16:123010, 2014. [3] Towards experimental quantum-field tomography with ultracold atoms . 73 Adrian Steffens, Mathis Friesdorf, Tim Langen, Bernhard Rauer, Thomas Schweigler, Robert Hübener, Jörg Schmiedmayer, Carlos A. Riofrı́o, and Jens Eisert, Nature Communications 6:7663, 2015. [4] An efficient quantum algorithm for spectral estimation . 83 Adrian Steffens, Patrick Rebentrost, Iman Marvian, Jens Eisert, and Seth Lloyd New Journal of Physics 19:033005, 2017. Closely related to these are the following coauthored publications, which are provided in the ap- pendix. [5] Continuous matrix product state tomography of quantum transport experiments . 120 Géraldine Haack, Adrian Steffens, Jens Eisert, and Robert Hübener, New Journal of Physics 17:113024, 2015. [6] Quantum singular value decomposition of non-sparse low-rank matrices . 136 Patrick Rebentrost, Adrian Steffens, and Seth Lloyd, ArXiv e-prints 1607.05404, 2016. Ref. [3] was developed in collaboration with the Schmiedmayer group at the Vienna University of Technology, Austria, Ref. [1] in collaboration with the Quantum Engineering Technology Labs at the University of Bristol, UK, and Refs. [4, 6] were established during the author’s research visit at the Research Laboratory of Electronics at the Massachusetts Institute of Technology, Cambridge, ma, USA. Not yet published is project [7], see p. 141, which resulted from the author’s work as part of the theory group of the Fritz Haber Institute of the Max Planck Society, Berlin. 1 1 INTRODUCTION Since its inception at the beginning of the 19th century, quantum mechanics has been fundamental in understanding the physical principles of nature, explaining phenomena from black body radia- tion and the photoelectric effect to superconductivity. This has made it possible to use quantum effects in technical applications such as solar cells or lasers. With the technical progress of the past years, the ability to actively control and manipulate complex quantum systems has come into reach, leading to a second quantum revolution [8] with entirely new technological prospects: Quantum com- puters are expected to substantially impact high-end computing, quantum cryptography protocols may provide intrinsically secure data transmission, and quantum metrology could allow for ultra- precise sensors. Commercial interest is growing, and a multi-billion Euro market is predicted to emerge.[9] Quantum systems, however, are very fragile and require thorough analysis if they are to be ef- fectively harnessed for quantum technologies. Accurately modeling quantum systems is necessary for explaining and predicting their behavior as building blocks of quantum devices. For a complete model describing a quantum system—a quantum state—the system has to be prepared multiple times and each time a certain quantum measurement has to be performed, resulting in data that can be processed to infer the model. This procedure is called quantum state tomography and will be a fundamental topic in the sections 2 and 3. With increasing size and complexity of the quantum system, this task soon becomes infeasible in the general case. This is due to an exponential increase of the number of modeling parameters and necessary measurements with the size of the quantum system, making it hard to find appropriate parameters (“curse of dimensionality”[10]). To cope with this problem, elaborate signal processing techniques have to be developed and employed. Signal processing comprises the transmission, manipulation, and representation of signals and their underlying information. It is fundamental to a wide range of fields such as audio, image, and video processing, telecommunication, or seismology. Signals can be continuous functions in time or space such as voltages, magnetic fields, sound waves, angles, forces, or temperatures. For process- ing, especially digital signal processing, such signals are typically discretized, i.e., sampled at discrete points. Higher sampling rates generally result in higher accuracy in describing the input and the output quantities of interest, but then require more computational resources, especially computa- tional time and memory. With finite resources, this calls for carefully weighing effort and quality of the output. In the context of complex quantum systems, the basic input signals consist of quantum mea- surements to estimate the parameters that constitute the corresponding quantum state. In general, it is desirable to restrict the parameter set in such a way that the properties of the quantum system are essentially preserved while keeping the processing effort manageable. Finding such “compressed models” is not trivial and requires a certain intrinsic compressibility of the parameter space. And yet it is not uncommon: Often in the history of physics it was vital to discard unnecessary properties 2 of a system to properly model particular parts of nature, such as observing the free fall in a vacuum without drag forces. In the case of complex quantum systems, it turns out that a large part of the entire parameter space is occupied by highly entangled quantum states, which are unlikely to ap- pear in large parts of nature. In many cases, the remaining states, the ones of actual interest, can be efficiently parametrized by use of tensor network states. They constitute a seminal class of quantum states with a plethora of groundbreaking applications, allowing the development of potent quan- tum state tomography protocols (see the publications [2] and [3] in section 3). For this, a crucial reconstruction step relies on a class of signal processing algorithms for spectral estimation, which provide frequency spectra for certain signals with high accuracy. Beyond quantum state tomogra- phy, in this thesis tensor network states are employed for succinctly describing physical processes such as quantum transport experiments (see publication [5] in appendix A). The established protocol fa- cilitates the study of the short-time dynamics of these quantum systems by providing statistics that are not directly accessible otherwise. If, on the other hand, one is interested in a very general and robust procedure with very few assumptions about the quantum state to be determined, which comes at the price of less total com- pressibility of the parameter space, one can make use of another famous signal processing paradigm: compressed sensing, which allows for the extraction of sparse solutions out of a higher-dimensional parameter space using only a comparably small number of measurements/samples (section 2). With this, signals can be recorded in a compressed way, i.e. substantially fewer input samples are required to recover the underlying model. Compressed sensing has vastly spread in recent years, making huge progress both in establishing provable theoretical recovery guarantees and practical applications in all kinds of fields ranging