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Non-Gaussian Noise Spectroscopy with Superconducting Qubits Youngkyu Sung

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Non-Gaussian Noise Spectroscopy with Superconducting Qubits Youngkyu Sung Non-Gaussian Noise Spectroscopy with Superconducting Qubits by Youngkyu Sung B.A., Seoul National University (2016) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2018 ○c Massachusetts Institute of Technology 2018. All rights reserved. Author................................................................ Department of Electrical Engineering and Computer Science August 31, 2018 Certified by. William D. Oliver Professor of the Practice, Physics Department Thesis Supervisor 2nd Certified by . Simon Gustavsson Principal Research Scientist, Research Laboratory Of Electronics Thesis Supervisor Accepted by . Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Non-Gaussian Noise Spectroscopy with Superconducting Qubits by Youngkyu Sung Submitted to the Department of Electrical Engineering and Computer Science on August 31, 2018, in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering Abstract Most quantum control and quantum error-correction protocols assume that the noise causing decoherence is described by Gaussian statistics. However, the Gaussianity assumption breaks down when the quantum system is strongly coupled to a sparse environment or has a non-linear response to external degrees of freedom. Here, we experimentally validate an open-loop quantum control protocol [2] that reconstructs the higher-order spectrum of a non-Gaussian dephasing process using a superconduct- ing qubit as a noise spectrometer. This experimental demonstration of non-Gaussian noise spectroscopy protocol represents a major step towards the goal of demonstrat- ing a complete noise spectral characterization of quantum devices. [1] E. Paladino et al. Rev. Mod. Phys. 86, 361 (2014). [2] L. M. Norris, G. A. Paz-Silva, L. Viola, Phys. Rev. Lett. 116, 150503 (2016) Thesis Supervisor: William D. Oliver Title: Professor of the Practice, Physics Department Thesis Supervisor: Simon Gustavsson Title: Principal Research Scientist, Research Laboratory Of Electronics 3 4 Acknowledgments I would like to thank my advisor, Professor Oliver, who gave me the opportunity to work on this exciting project and encouraged me to pursue my own research interests. For the past two years, I am deeply inspired by his extraordinarily broad knowledge in fields of science and engineering. His kindness and generosity with time areof great help to me and make me persevere with any experimental difficulty. I am always grateful to Dr. Gustavsson for his teaching about how to perform amazing experiments. Numerous discussions with him over the course of two years in the lab and his office gave me a tremendous amount of guidance in my academic journey. I can’t forget Professor Orlando’s lectures in solid state physics, and lectures on circuit quantization in our lab. Not only an amazing lecturer and theorist, he is also a great supportive mentor; conversations with him always relieve and inspire me. This work would not have been possible without our collaborators; Dr. Beaudoin, Dr. Norris and Professor Viola in Dartmouth college. It was truly an enjoyable experience to work with amazing theorists who are always ready to discuss and provide deep insights into physics obscured by puzzling experimental data. Our Skype calls usually go longer than two hours, which sounds quiet long, but it was actually a lot of fun and has made a lot of progress; we made things clear, set up a plan to figure out problems we had, and finally resolved the problems! I hope to continue our collaboration and explore new physics and cool techniques together! Dr. Yan was a postdoc in the group pioneering this project. From late nights in the lab to phone calls while he is driving, he has always been my mentor who taught me countless things, not only details about the experiment, but also important lessons to become a better scientist. Not only a mentor, he is one of the best friends who played and watched basketball games together until late night. This work would not have been possible without his research experience and kind help. Amy and Bharath are both cool lab-mates who always make me laugh and also outstanding teachers who are always ready to help. I feel particularly fortunate to work in a research group with these vibrant spirits! Also, it is such a pleasure to 5 have a friend who has the same tastes in food, Jack. He is also smart and talented in experiments, which makes me learn many things from him. Tim and Ben are great grads as well, who are doing amazing work and making a lot of progress, which motivate me a lot. Conversations with our post-docs (Philip, Dan, Joel, Morten, Roni and Jochen) are always inspiring and intellectually pleasing. They have very solid backgrounds in various fields, so that I feel so thankful to learn a variety of things from them. I also would like to extend my thanks to Moritz, Uwe, Thorvald, Andreas, Niels Jacob and Antoine who have visited our group as visiting students and helped and taught me a lot. Additionally, I enjoyed the company of all my friends here in the MIT community. They made my passing years here an unforgettable memory. Finally, I must give my deep gratitude to my parents, and my brother, for their love and unconditional support. Their lovely messages and phone calls boost my energy more than any brand of energy drink or coffee. 6 Contents 1 Introduction 17 1.1 Quantum noise spectroscopy . 18 1.2 Non-Gaussian noise in superconducting qubits . 19 1.3 Overview of thesis . 20 2 Superconducting Qubits and Circuit QED 21 2.1 Superconducting Qubits . 21 2.1.1 Josephson junction as a non-linear inductor . 22 2.1.2 Cooper-pair box (CPB) . 23 2.1.3 Transmon . 24 2.1.4 Persistent-current flux qubit . 27 2.1.5 C-shunt flux qubit . 28 2.1.6 Quarton . 30 2.2 Cavity quantum electrodynamics . 34 2.2.1 Strong coupling regime . 34 2.2.2 Dispersive coupling regime . 36 2.3 Circuit QED . 36 2.3.1 Coupling a superconducting qubit to a coplanar waveguide res- onator . 36 2.3.2 Dispersive regime of circuit QED . 37 2.4 Coherence properties of superconducting qubits . 38 2.4.1 Coherence time and anharmonicity of a Quarton . 38 7 3 Quantum Noise Spectroscopy 41 3.1 A qubit as a noise probe . 41 3.1.1 Hamiltonian of open quantum system . 41 3.1.2 The relation between spin observables and bath-operator cu- mulants . 42 3.2 Quantum noise spectroscopy (QNS) protocols . 43 3.2.1 Bylander et al.’s protocol(2011): CPMG spectroscopy for Gaus- sian dephasing process . 44 3.2.2 Álvarez and Suter protocol (2011): Repetition of suitably de- signed pulse sequences . 46 3.2.3 Norris et al. protocol (2016): Generalization of the Álvarez and Suter protocol to Non-Gaussian dephasing processes . 47 3.2.4 Norris et al.’s protocol(2016) cont.: Reconstructing bispectrum 49 4 Experimental Setup and Details 57 4.1 Cryogenic setup . 57 4.2 Room temperature control . 60 4.3 Pulse control and modulation . 60 4.4 Materials and fabrication of superconducting qubits . 60 4.4.1 Growth and patterning of high-Q aluminum . 61 4.4.2 Patterning the qubit loop and Josephson junctions . 61 4.4.3 Dicing and packaging . 62 5 Reconstructing the Bispectrum of non-Gaussian Dephasing Pro- cesses in a Superconducting Qubit 63 5.1 Non-Gaussian dephasing in a superconducting qubit . 64 5.1.1 Non-linear energy dependence on external flux . 64 5.1.2 Squared Gaussian dephasing process . 65 5.2 Verifying the presence of non-Gaussianity . 67 5.3 Reconstruction of the bispectrum . 69 5.3.1 Review of the old approach (Section 3.2.4). 69 8 5.3.2 Reconstructing the bispectrum with new approach . 70 6 Conclusions and Future Work 77 6.1 Characterizing natural non-Gaussian dephasing processes . 77 6.2 Regularized maximum likelihood estimation for bispectrum reconstruc- tion . 78 6.3 Faster noise spectroscopy using active reset protocols . 79 A Moments and Cumulants 81 A.1 Moment and moment-generating function . 81 A.2 Cumulant and cumulant-generating function . 82 B Derivation of the Joint Hamiltonian in the Toggling Frame 83 C Timing digram of control pulse sequences 89 9 10 List of Figures 1-1 A schematic diagram showing the steps involved in quantum noise spectroscopy (QNS). 18 2-1 Circuit diagrams of cooper pair box(CPB) qubit and Transmon. 24 2-2 Charge dispersion. The energies of the lowest 5 levels of the charge qubit Hamiltonian in units of the charging energy EC. For low EJ=EC, the qubit is in the Cooper-pair box regime, and the energies are parabolic functions of the offset charge ng, with avoided crossings. As the ratio of EJ=EC is increased, the levels become exponentially flattened, as one enters the Transmon regime. 25 2-3 Potential function of the persistent-current flux qubit where 훼 = 0:75, f = 0:5. We can clearly see that the potential has a double-well profile, around the points where 휑p + 휑m ≈ 2푛휋, where n is an integer. 29 2-4 Near f = 0.5, the potential function assumes a double well profile. For f < 0.5, the left well has a lower energy (clockwise circulating cur- rent); for f > 0.5, the right well has a lower energy (counter-clockwise circulating current). At f = 0.5 the diabatic state energies are degener- ate. Tunneling energy mixes the states, forming superpositions of the circulating currents and the qubit has no magnetic polarization. Im- age courtesy of Prof.
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