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University of Southern California Dissertations and Theses
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Modeling and engineering noise in superconducting qubits
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Modeling and engineering noise in superconducting qubits
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Modeling and Engineering Noise in Superconducting Qubits by Haimeng Zhang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2023 A c kno wledgmen ts The Quantum Error Correction course taught by Professor Daniel Lidar was where this journey started for me. When I took the class in 2018, I had little background in quantum information. However, the course was taught pedagogically, and the theory derivation was crystal clear. Daniel had a rather old-school way of teaching with a board and a marker. The course was smooth, like a well-narrated movie, with a guaranteed voilà moment at the end. I also had a chance to give a lecture on the stabilizer code. I want to thank all my classmates for sitting through my very first lecture. I also picked up my first project as the course’s final project, where we wanted to demonstrate an error-corrected logical qubit via channel-optimized error correction code. The goal we set at the time ended up being too ambitious to finish as a course project. Still, I learned Python from my great teammate Hannes Leipold and control theory from our great mentor, Dr. Robert Kosut. I also fostered precious friendships with Bibek Pokharel and Namit Anand through the study group after class. After the quantum error correction course, I decided to switch to the field of quantum computing, so I approached Daniel. Because of my prior experience in experiments, Daniel suggested he could co-advise me with Professor Eli Levenson-Falk, who then started a new lab in superconducting circuits. I talked to Eli, and he took me in. Working in two groups was more challenging than I thought, yet it was also more than rewarding. I especially want to thank my advisor Prof. Eli Levenson-Falk for teaching me how to be an experimentalist. When the lab started, and we were testing our first recipe for transmon qubits, he taught me step by step in the lab how to perform the development step, how to cleave a chip, and so on. He tries his best to spend a decent amount of time with students in the lab every day, doing all sorts of work, big or small, measurement, debugging, or ii simply fixing or cleaning things. To this day, I am still amazed at his ability to look at a measurement setup that we are trying to debug for hours, starting at it for 5 minutes, and then pointing out a problem we overlooked. He used to emphasize an experimentalist’s paranoia and pushed me to pay the most attention to details. Anyone who worked with Eli side by side in the lab can also speak to his fast unit conversion by heart, for example, from noise temperature to power unit to frequency unit, which comes from both his experience and intuition. He is a true experimentalist by heart. He also cares about his students deeply. He always makes time to look at my writing or slides or does practice talks, sometimes on short notice. Eli, thank you for your constant support in my research and career. I would also like to thank my advisor, Professor Daniel Lidar. He is the scientist that I will always aspire to be. Daniel is always clear and concise in his arguments. He pays extreme attention to details. He is always active and engaging at meetings and asks great questions. I often find it hard not to be self-conscious about the scientific results I am showing. Daniel’s as-a-matter-of-fact attitude has always helped me to get straight to the point and find a solution that makes sense. Whenever I need advice, he provides a listening ear and wisdom. James Farmer is the person I turn to for fridge operation when there is an emergency or want to have someone bounce off ideas on how to take a measurement. He knows the fridge and the lab inside and out. He is always willing to sit down and spend time helping someone in the lab. James is one we all look up to as a role model for his knowledge, problem-solving skills, and work ethic. I learned how to do soldering and make a twisted DC wire properly from James. James, thank you for the many Fridays closing the fridge. And I miss the days climbing and biking with you and Darian in Long Beach. I learned most of the device physics in cQED from Vinay. Vinay knows a great deal about how to model superconducting qubits on a circuit level. He is a great teacher and always has a contagious passion to talk about his research. Since we share the same office, Vinay has been there for me on several most stressful days late at night and has always made me feel better with a warm curry dish from T&J’s or a treat of sweets. I am fortunate to have iii his friendship. In the last year of my Ph.D., I have been working on the dissipator project with a team of young students full of ideas and energy, Vivek Maurya, Jocelyn Liu, Clark Miyamoto, Andre Luo, and Daria Kowsari. Jocelyn, thank you for asking all the insightful questions. You have pushed me to delve deeper. Vivek, thank you for giving me so much support and trust in the direction we are going. Our record of one cycle of cooldown and measurement per week was only made possible because of your commitment to keeping us on schedule. Clark, thank you for making the coding and qubit designing part of this project joyful and artsy. Andre and Daria, thank you for taking up the heavy lifting of fabrication when I take a step back to work on my thesis, and you are the true heroes of ensuring we have devices to measure. I take pride in our small team and working closely together. I am sure we will be prouder when we get our results out! Research here has been a collaborative effort. It was with Evangelos that we made the first 3D cavity to house the lab’s first transmon. Huo Chen’s expertise in open-system simulation provided a whole toolbox for efficient and flexible open-system modeling. Darian’s thorough work in perturbation analysis and numerical simulation laid the foundation for the dissipator project. I also had multiple discussions with Humberto, Pat, and Victor on bootstrapping, tensor network, and Fourier analysis, which deepened my understanding of those mathematical and statistical tools. I am fortunate to be in an open system where everyone is willing to share their expertise, and through interactions, I can learn and grow. For that, I thank you all. One of the things that I am most grateful for during my grad school is getting to know Bibek Pokharel. Over the years, I have seen how he pushed himself for excellence with discipline and consistency. My first publication in this field is with Bibek. He is a great person to collaborate with, who gets down to execution and keeps the project on the timeline. During the time when we were roommates, I also got to know more about his genuine curiosity about board subjects. Bibek is always enthusiastic about sharing his knowledge with people iv around him. Luckily, as his roommate, through him, I got to be introduced to Emacs (which Bibek went out of the way to set up for me on my laptop), literate programming and stayed up to date with the latest productivity enhancement software. We also had long conversations on cultures and social political matters. He has always listened to my opinions, often immature or untested, without judgment, and some of the views and books he shared with me are educational. He can lighten up the room with his cheerful tone and friendly jokes, even when I am only two minutes away from a presentation that I am nervous about. It was through Namit Anand that I learned how to make strong espresso from a Moka pot. Back when Bibek, Namit, and I had a study group for the Quantum Error Correction course, Namit taught me how to do tensor products. He still takes out his time after a long workday to give me a whole lecture on quantum many-body localization from scratch. Thank you for showing me the beauty of physics, sharing your philosophy of life with me, and lending me a helping hand when I most needed it. You and Bibek made Los Angeles like home to me, and I found peace and inspiration every time I talked to you. Keep your coffee strong, my friend. I was born in a small town, Deyang. It is inland China, surrounded by mountains. My father has a sailor’s dream. He gave me a name with ocean and dream in its characters. Mom and Dad, thank you for encouraging me to pursue my dreams. Thank you for teaching me curiosity and bravery; I will take them with me as I go see the world. v T able of Con ten ts Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1 Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Transmon qubits and their readout . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Noise in superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Chapter 2: Theory background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Lindblad formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Noise mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Beyond the Markovian approximation . . . . . . . . . . . . . . . . . . . . . 16 3 Chapter 3: Predicting non-Markovian dynamics . . . . . . . . . . . . . . . . . . . 21 3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4 Chapter 4: Theory of bath engineering via cavity cooling . . . . . . . . . . . . . . 53 vi 4.1 Parametric driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Cooling mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5 Chapter 5: Experiment for on-demand dissipation . . . . . . . . . . . . . . . . . . 70 5.1 Device design considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Device tune-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3 Logical qubit T 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Optimal drive parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6 Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1 Recent progress in noise characterization and modeling . . . . . . . . . . . . 96 6.2 Scalability of hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A Appendix A: Damping basis construction . . . . . . . . . . . . . . . . . . . . . . . 121 B Appendix B: Analytical solution of the PMME . . . . . . . . . . . . . . . . . . . 125 C Appendix C: Complete positivity of the PMME . . . . . . . . . . . . . . . . . . . 127 D Appendix D: PMME solution with the specific Lindbladian and kernels . . . . . . 130 E Appendix E: Best-fit parameters of the nested PMME models . . . . . . . . . . . 135 F Appendix F: Fabrication Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 G Appendix G: Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 vii List of T ables 3.1.1 Qubit calibration information of the ibmq_athens processor on the dates of data collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 The set of initial statesP used in our experiments, corresponding to the states shown in Fig. 3.1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Trace-norm distances corresponding to the results shown in Fig. 3.2.3. . . . . 36 3.2.2 Degree of non-Markovianity for three different initial states of the spectator qubits (column 1) as computed from the experimental data (column 2) and the two non-Markovian models (columns 3 and 4). . . . . . . . . . . . . . . . 40 3.2.3 A heuristic interpretation of AIC differences ∆ i reported in Fig. 3.2.3. The larger ∆ i is, the less plausible it is that the modelM i is the best model. . . 49 5.2.1 Fitted parameters for device Device08_07A and Diss08_09C . . . . . . . . . 81 E.0.1Best-fit parameters of the models M 0 ,M 1 andM 2 constructed using the fit- ting datasets in Fig. 3.2.2, Fig. 3.2.5, Fig. 3.2.6 and Fig. 3.2.7. The values in the parentheses correspond to the 2σ error bars (95% confidence intervals) es- timated using the bootstrap method. Parameters with best-fit values smaller than 1e-5 are expressed as zero as their effects on the dynamical predictions are negligible and well below the measurement precision of the tomography experiment. For the same reason, we report the non-zero best-fit parameters with two significant figures and their confidence intervals with one significant figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 viii List of Figures 2.1.1 (a) The exact dynamics, (b) the measurement interpretation of the exact dy- namics, (c) the measurement interpretation of the LME, (d) the measurement interpretation of the PMME. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.1 The nested candidate models we use to describe the qubit free evolution: the Lindblad modelM 0 , parameterized by a Hamiltonian term and the Lindbla- dian, and the PMME modelsM 1 ,M 2 with their additional kernel parameters. 26 3.1.2 The initial states used for the fitting data set (yellow) and the testing data set (blue). The initial states are chosen such that the statesjψ i (0)i, i = 0,1,2,3 form a tetrahedron on the Bloch sphere, where jψ 1 (0)i = j1i is the excited state. The statejψ 4 (0)i is a fixed, randomly chosen state on the Bloch sphere. The same set of five states are used in all our experiments. . . . . . . . . . . 29 3.2.1 Analysis protocol for PMME tomography. The tomography data sets for qubit free evolution with different initial states are divided into a fitting set (in our case with a single initial statejψ 0 (0)i) and a testing set (in our case with initial states jψ 1,2,3,4 (0)i). The fitting set is used to fit the dynamical modelsfM i g using a classical optimizer based on the MLE method. . . . . . 33 ix 3.2.2 PMME tomography protocol applied to single-qubit free evolution, with the spectator qubits all in their ground state. (a0-c0) The free-evolution tomogra- phy data ˆ v exp (t) of the fitting data set with the qubit initialized in jψ 0 (0)i and the best-fit models for the Lindbladian model M 0 (orange lines), the PMME model with kernel type 1,M 1 (blue lines), and the PMME model with kernel type 2,M 2 (red lines). The shaded regions denote the 95% confidence region of the model predictions. (a1-c4) The free-evolution tomography data in the testing data set with the qubit initialized infjψ i (0)ig 4 i=1 , and the prediction from the best-fit models from the fitting data set in (a0-c0). (d0-d4) The empirical purity Tr[ρ 2 ] of the qubit and that predicted by the best-fit models. (e0-e4) The distance between the tomographically constructed state and the state predicted by the best-fit models. All models perform equally well at predicting the dynamics of the excited state jψ 1 i = j1i (M 0 is obscured by M 1 and M 2 ), while the PMME models M 1 and M 2 predict the dynamics offjψ i (0)ig 4 i=2 better than the Lindbladian model. . . . . . . . . . . . . . . 34 3.2.3 The predictive ability of different models for the fitting data set (left) and the testing data set (right) for ibmq_athens data. For both datasets, we show the trace-norm distance between the tomographically constructed state fˆ ρ exp g and the model predicted state fρ prd g. The box plot shows the 5th, 50th (median), and 95th percentiles for this distance over t j and respective initial states. The open circles denote extremal outliers. The PMME models M 1 andM 2 describe both the fitting data set and the testing data set better than the Lindblad model M 0 (see Table 3.2.1). We also report the AIC of the models on the fitting data (purple squares, right axis), again with better performance by the non-Markovian models; note that all AIC values are negative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 x 3.2.4 Non-Markovianity of qubit free-evolution dynamics for spectator qubits in the ground state (a,b), the excited state (c,d) and thej+i state (e,f). (a,c,e) The trace-norm distanceD(ρ 1 (t),ρ 2 (t)) predicted by the best-fit models (solid lines) and experimentally measured by performing free-evolution tomography with a pair of initial states ρ 1 (0) =j+ih+j and ρ 2 (0) =jihj (grey circles). (b,d,f) The derivative σ(t), defined in Eq. ( 2.35), predicted by the best-fit models (solid lines), and approximated experimentally using forward differ- encing based on the tomography data in (a,c,e) (grey circles). . . . . . . . . 39 3.2.5 The tomography data set and the corresponding model predictions used to calculate the degree of non-Markovianity in Fig. 3.2.4 (a,b) when the spectator qubits are initialized in the ground state. (a0-c0) The fitting data set with the qubit initialized injψ 0 (0)i that is used to find the best-fit Lindblad model M 0 (orange lines), the PMME model with the type 1 kernel M 1 , and the PMME model with the type 2 kernel M 2 . (a1-c2) The tomography data set with qubit initialized injψ +x i,jψ −x i and the prediction from the best-fit models from the fitting data set in (a0-c0). The data sets are used to evaluate the degree of non-Markovianity in Fig. 3.2.4 (a,b). (a3-c4) The tomography data set with the qubit initialized in jψ +y i, jψ −y i, and the prediction from the best-fit models from the fitting data set in (a0-c0). The data sets are used to evaluate the degree of non-Markovianity in Fig. 3.2.10 (a,b). (d0-d4) The distance between the tomographically constructed state and the state predicted by the best-fit models. . . . . . . . . . . . . . . . . . . . . . . . . . 41 xi 3.2.6 The tomography data set and the corresponding model predictions used to calculate the degree of non-Markovianity in Fig. 3.2.4 (c,d) when the spectator qubits are initialized in the excited state. (a0-c0) The fitting data set with the qubit initialized injψ 0 (0)i that is used to find the best-fit Lindblad model M 0 (orange lines), the PMME model with the type 1 kernel M 1 , and the PMME model with the type 2 kernel M 2 . (a1-c2) The tomography data set with qubit initialized injψ +x i,jψ −x i and the prediction from the best-fit models from the fitting data set in (a0-c0). The data sets are used to evaluate the degree of non-Markovianity in Fig. 3.2.4 (c,d). (a3-c4) The tomography data set with the qubit initialized in jψ +y i, jψ −y i, and the prediction from the best-fit models from the fitting data set in (a0-c0). The data sets are used to evaluate the degree of non-Markovianity in Fig. 3.2.10 (c,d). (d0-d4) The distance between the tomographically constructed state and the state predicted by the best-fit models. . . . . . . . . . . . . . . . . . . . . . . . . . 42 xii 3.2.7 The tomography data set and the corresponding model predictions used to calculate the degree of non-Markovianity in Fig. 3.2.4 (e,f) when the spectator qubits are initialized in the j+i state. (a0-c0) The fitting data set with the qubit initialized in jψ 0 (0)i that is used to find the best-fit Lindblad model M 0 (orange lines), the PMME model with the type 1 kernel M 1 , and the PMME model with the type 2 kernel M 2 . (a1-c2) The tomography data set with qubit initialized injψ +x i,jψ −x i and the prediction from the best-fit models from the fitting data set in (a0-c0). The data sets are used to evaluate the degree of non-Markovianity in Fig. 3.2.4 (e,f). (a3-c4) The tomography data set with the qubit initialized in jψ +y i, jψ −y i, and the prediction from the best-fit models from the fitting data set in (a0-c0). The data sets are used to evaluate the degree of non-Markovianity in Fig. 3.2.10 (e,f). (d0-d4) The distance between the tomographically constructed state and the state predicted by the best-fit models. . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.8 Summary of the results of running the PMME tomography protocol on the IBMQE processor ibmq_athens, showing how well the model candidates de- scribe the fitting data set and the testing data set in Fig. 3.2.5 for the spectator qubits in the ground state (a), for those in Fig. 3.2.6 for the spectator qubits in the excited state (b) and for those in Fig. 3.2.7 for the spectator qubits in the j+i state (c). The box plots show the trace norm distance (see text) and the median is reported as the middle value of fD j g. The lower line of the box corresponds to the lower quartile of the data (25th percentile, Q1), and the upper line of the box corresponds to the upper quartile of the data (75th percentile, Q3). Let IQR denote the interquartile range: IQR = Q3-Q1. The outliers, plotted in circles, are the data outside the range (Q1-1.5*IQR, Q3+1.5*IQR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 xiii 3.2.9 The kernels in the constructed modelsM 1 andM 2 using the fitting datasets in Fig. 3.2.2 (a), Fig. 3.2.5 (b), Fig. 3.2.6 (c), and Fig. 3.2.7 (d). The shaded regions denote the 95% confidence region of the kernel function due to the uncertainty in the best-fit kernel parameters in Table E.0.1. . . . . . . . . . 46 3.2.10 Non-Markovianity of qubit free-evolution dynamics for spectator qubits in the ground state (a,b), the excited state (c,d) and thej+i state (e,f). (a,c,e) The trace-norm distanceD(ρ 1 (t),ρ 2 (t)) predicted by the best-fit models (solid lines) and experimentally measured by performing free-evolution tomography with a pair of initial states ρ 1 (0) = j+iih+ij and ρ 2 (0) = jiihij (grey circles). (b,d,f) The derivative σ(t), defined in Eq. ( 2.35), predicted by the best-fit models (solid lines), and approximated experimentally using forward differencing based on the tomography data in (a,c,d) (grey circles). . . . . . 47 4.1.1 Illustration of a readout cavity capacitively coupled to a frequency-tunable “dissipator” with coupling strength g. The cavity is modeled as a quantum harmonic oscillator. The dissipator is modeled as a qubit whose frequency is parametrically modulated with H p . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1.2 Energy levels of the coupled cavity-dissipator system: the parametric drive turns the first-order sideband transitions and enables energy swapping in the subspace that preserves one-photon excitation, shown by the blue arrows. The excitation then quickly dissipates due to the large dissipator decay rate. 56 4.1.3 Setup for cavity cooling with a filter mode. The filter mode is introduced to modify the dissipator’s coupling to the lossy bath, hence determining the dissipator decay rate κ. In the cavity-dissipator-filter system, by choosing the appropriate drive frequency, the first-order sideband transition can either turn on (a) the cavity-dissipator transition or (b) the cavity-filter transition. 61 xiv 4.2.1 Decay diagrams for (a) weak and (b) strong coupling regimes in the rotat- ing frame of the parametric drive. The target mode state has its intrinsic temperatureT target due to the equilibrium of the relaxation processjei!jgi with rate γ − and the excitation process jgi ! jei with rate γ + . (a) When jg n j/κ d 1, theje,0i state can decay back to the ground statejg,0i through a two-step processje0i!jg1i!jg0i which is limited by the slower process of the two. (b) When jg n j/κ d 1, the swap rate is strong enough to give rise to a coherent oscillation betweenje,0i andjg,1i states fast enough to be viewed as an equally weighted mixture of the two. The decay rate from this mixed state is effectively the average of γ − and κ d . . . . . . . . . . . . . . . 65 4.2.2 Schematic representation of a quantum absorption refrigerator. (a) a three- level quantum refrigerator where each transition is coupled to a different bath. ω c corresponds to the target model frequency to coolω r orω q ,ω w corresponds to the drive(pump) frequency ω p , and ω h corresponds to the dissipator fre- quencyω d , adopted from Ref. [1]. (b) A quantum heat pumpH S that couples a work reservoir with temperature T w , a hot reservoir with temperature T h , and a cold reservoir with temperature T c . The heat and work current are indicated by J c , J h and J h . In the steady state J h +J c +P = 0, adopted from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.1 Circuit model of the device. (a) a notch-type readout resonator (orange) ca- pacitively coupled to the logical qubit (purple) and the flux-tunable dissipator (blue). The readout cavity is on resonance with the notch-type Purcell fil- ter (green), effectively shorting out the 50-Ohm output environment at the readout frequency. (b) An updated design based on the circuit in (a) with the Purcell filter (green) acts as a bandpass filter at the dissipator operating frequency. The dissipator (blue) is lossy when tuned in-band with the Purcell filter due to stronger coupling to the 50-Ohm output environment. . . . . . . 72 xv 5.1.2 SEM image of the device using design in circuit model Fig. 5.1.1(b). . . . . . 74 5.2.1 The Resonator response as a function of flux bias in single-tone spectroscopy measurement(a) ω max d < ω r , no avoided crossing pattern (b) ω max d ≳ ω r , the dissipator minimum frequency is close to resonator at the avoided crossing, and the dissipator is punched out as a result (c) ω max d > ω r , the dissipator tuned through the resonator and the avoided crossing pattern is observed. . 79 5.2.2 Avoided crossing pattern in single-tone spectroscopy measurement for device Diss08_09C. The data to fit the model in Eq. ( 5.7a) and extract the relevant parameters as summarized in Table 5.2.1. The white dashed line shows the fit from the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2.3 Cavity magnitude response as a function of readout power. (a) FFL drive off: cavity response exhibit ”punch out” shift. (b) FFL drive on: cavity line width broadens, and the dip becomes shallower, suggesting a reduced cavity internal Q as photons are pumped out of the cavity by the parametric drive. 82 5.2.4 FFL calibration: (a) readout cavity response as a function of the coil bias I coil , (b) readout cavity response as a function of the FFL bias I FFL , (c) the extracted cavity resonance frequencies at each coil biasfω i r (I i coil )g in orange and at each FFL bias ∆I (i) FFL in blue, (d) the conversion relation of the flux control between the coil and the FFL. . . . . . . . . . . . . . . . . . . . . . 84 5.2.5 FFL spectroscopy as a function of dissipator flux bias. The dissipator fre- quency is tuned down as the flux bias changes from 90 µA to 175 µA. We see feature 1 at around 2.8 GHz at the cavity-filter detuning frequency. This feature gets broader as the dissipator is tuned closer. We see feature 2 start- ing at 3.8 GHz when flux bias is 90 µA, corresponding to cavity-dissipator detuning frequency, and this feature moves down in frequency as flux bias increases. We do not know what transition corresponds to feature 3, but it gets broader as the dissipator is tuned down. . . . . . . . . . . . . . . . . . . 86 xvi 5.2.6 Ringdown time measurement: (a) pulse sequence to measure cavity ringdown time, (b) ringdown time with and without the cavity-reset drive. Device: Diss08_09C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3.1 Pulse sequence to measure logical qubit T 1 with the parametric drive on. Drive frequency at cavity-dissipator detuning. Device: diss08_07A. . . . . . 88 5.3.2 Logical qubit T 1 with the cavity-reset drive on. (a) Logical qubit T 1 with weak cavity-reset drive (orange) and strong cavity-reset drive (blue); drive frequency at cavity-dissipator detuning, (b) logical qubit T 1 degrades as the cavity-reset drive becomes stronger, (c). the cavity-reset drive also effectively makes the logical qubit steady state hotter, possibly because it is driving higher-level transitions in the logical qubit. Device: diss08_07A. . . . . . . . 89 5.4.1 Ringdown time as a function of drive frequency and amplitude at different dissipator flux biases: (a) flux bias = 140 µA, (b) flux bias = 155 µA, (c) flux bias = 165 µA. Device: diss08_09C. . . . . . . . . . . . . . . . . . . . . . . 90 5.4.2 Ringdown time as a function of drive amplitude for different drive frequencies: (a) flux bias = 140 µA, (b) flux bias = 155 µA, (c) flux bias = 165 µA. Device: diss08_07A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.1 (a) The pulse sequence to test the efficacy of the cavity-reset pulse on logical qubit coherence preservation by removing residue readout photons after a readout. (b) The pulse sequence to test the efficacy of the cavity-cooling drive on logical qubit coherence preservation by removing thermal photons from the readout cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5.2 The qubit-dissipator transition for qubit reset . . . . . . . . . . . . . . . . . 93 5.5.3 (a) The pulse sequence to measure logical qubit T 1 and test the efficacy of qubit-reset drive at the dissipator-qubit detuning frequency. (b) The pulse sequence to test the efficacy of the qubit-reset pulse on preparing the qubit ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 xvii 5.5.4 The pulse sequences to measure the initial ground state population (a) and the excited state population (b) to estimate the qubit temperature with or without the qubit-cooling drive. . . . . . . . . . . . . . . . . . . . . . . . . . 95 G.0.1Measurement setup with VNA . . . . . . . . . . . . . . . . . . . . . . . . . . 141 G.0.2Measurement setup with Quantum Machine OPX controller . . . . . . . . . 142 xviii Abstract A central challenge in controlling and programming quantum processors is to overcome noise that arises from the system’s unwanted interaction with the environment. From the system’s perspective, the open system effects alter the desired system evolution, leading to decoherence and dissipation. From the environment’s perspective, an engineered controllable Hamiltonian modifies the effective environment that the qubit undergoes, and the open system effects can be used as a resource. In the first part of the thesis, I discuss how to model noise in a superconducting qubit. I focus on non-Markovian noise, that is, temporally correlated noise, and it is challenging to mitigate. I show how to construct a simple phenomenological dynamical model known as the post-Markovian master equation to accurately capture and predict non-Markovian noise in superconducting qubits. The model also allows the extraction of information about cross-talk from near-neighboring qubits and measures of non-Markovianity. The system’s coupling to the environmental degree of freedom can also be used to our advantage if we can control it. In the second part of the thesis, I discuss how to engineer a system’s coupling to a cold source of dissipation to remove entropy from a system with unwanted excitations. This is useful to speed up readout cavity reset, qubit reset, or preserve qubit coherence by eliminating unwanted thermal residue photons in the readout cavity. I show the design and operation of an active, on-demand source of dissipation via parametric coupling. Our experiment demonstrates it can be used to reset the readout resonator at a rate greater than 20 MHz through a quantum refrigeration process. This thesis addresses the interplay between system dynamics and environment fluctuation for a small system size with the outlook that noise modeling and engineering techniques discussed can be extended to larger system sizes. xix Chapter 1: In tro duction The field of quantum computing has been growing rapidly over the years, enabled by advances in quantum hardware. The goal of the field is to solve computational tasks beyond the capabilities of modern supercomputers. To achieve this goal, attempts are made to scale up the hardware to access large problem sizes to compete with classical computers. In terms of hardware, there are many candidates for a quantum computer: ion traps, cold atoms, spin qubits, etc. Among them, superconducting qubits [3] is a leading candidate, the system this thesis focuses on. In recent years, there has been tremendous progress in hardware and software development in the field. From the hardware perspective, many new qubit designs have been proposed, and the fabrication process has been optimized. On the software level, the field has developed systematical tools for qubit calibration, robust pulse control, and efficient gate compilation. However, superconducting qubits today still face significant challenges as the size of quantum computers scales up. One of the main challenges is that superconducting qubits are susceptible to noise; this greatly hinders the performance of quantum computers in the current NISQ era. The main topics of this thesis are noise in superconducting qubits, how to model them, and how to engineer them for quantum information processing tasks. The first question we ask is: how to arrive at an accurate description of the noise dynamics for superconducting qubits? Specially, we will focus on the non-Markovian aspect of the noise process, that is, temporally correlated noise that standard Markovian models cannot describe. Dissipation acting on a qubit is mostly thought of as a source of decoherence, but it can also be used as a resource. The second question we ask is: how to engineer and control the dissipation process of a qubit on demand? This will allow us to control the system’s coupling to the environment tunably. The example we demonstrate here is to use active, on-demand dissipation to quickly remove excitations from systems, which is useful to reset qubits and readout cavities during quantum computation promptly. In this work, the physical building block we use to encode quantum information is a transmon qubit. There are abundant physics and engineering challenges, even at the single qubit level. First, I will present the basic background of the hardware used in these projects. 1.1 Transmon qubits and their readout Transmons are a type of superconducting qubits widely adopted in today’s quantum com- puter architecture [4, 5]. Their design is simple and fabrication robust. They serve as the main building block for the experiment conducted in this thesis. This section gives a broad overview of their key properties to introduce the basic parameters needed to describe their behavior and establish the transmon qubit’s representation as a circuit, an anharmonic oscillator, and a qubit. A transmon is a capacitively shunted Cooper-pair box. The circuit Hamiltonian of a transmon qubit is: H = 4E C (ˆ nn g ) 2 E J cos ˆ ϕ (1.1) where ˆ n denotes the number of excess Cooper pairs on the island, ˆ ϕ is the 2π-periodic operator of the phase difference across the Josephson junction and n g is the offset charge. The operators satisfy the commutation relationship [ ˆ ϕ,ˆ n] = i. The potential landscape and the nature of the qubit encoded states are determined by the relative strength of the energies associated with various circuit elements, including the Josephson energyE J and the capacitive charge energyE C . The characteristics of a transmon is that it has a largeE J /E C , 2 typically around 50–100, which makes transmon qubits less susceptible to charge noise. The transmon can be modeled as a weakly anharmonic oscillator with anharmonicity around a few percent of its resonance frequency. The transmon qubit can be frequency tunable by replacing the single Josephson junction with a loop interpreted by two identical junctions, forming a superconducting quantum interference device (SQUID). Due to the interference of the two arms of the SQUID loop, the effective critical current of the two parallel junctions can be decreased by applying external magnetic flux threading the loop, hence the effective Josephson energy E ′ J is tunable via the external flux as: E ′ J (φ e ) = 2E J jcos(φ e )j (1.2) where φ e =πΦ ext /Φ 0 and Φ 0 =h/2e is the magnetic flux quantum. The state of the superconducting qubit is read out by coupling it with a readout resonator with frequencyω r . Taking the rotating-wave approximation, the joint qubit-resonator system is described by the Jaynes Cummings Hamiltonian H JC = 1 2 ℏω q ˆ σ z +ℏω r ˆ a † ˆ a+1/2 +ℏg ˆ aˆ σ + +ˆ a † ˆ σ − (1.3) In the dispersive regime, where the qubit-resonator coupling strengthg is much smaller than the detuning ∆ = ω q ω r , the two systems have no direct energy exchange. Instead, the resonator frequency acquires a frequency shift dependent on the qubit state. H int,disp = ℏg 2 ∆ ˆ a † ˆ aˆ σ z (1.4) The dispersive shift of a qubit χ is g 2 /∆, for a transmon, the dispersive shift is modified to g 2 ∆ α α+∆ , where α denotes the qubit anharmonicity. In a physical system, measurement takes a finite time, during which the readout resonator rings up and down. During the ring-up time, the resonator is populated with photons. 3 Photons then interact with the qubit. The photons coming out of the resonator encodes the information of the qubit state. After the readout, one must wait for photons to leave the cavity to perform another logical operation. An on-demand dissipation process can be used to speed up this process. 1.2 Noise in superconducting qubits A central challenge in controlling and programming quantum processors is to overcome noise. Noise in superconducting qubits refers to the unwanted dissipation and decoherence pro- cess of qubits. A consequence of noise is that qubit states have a finite lifetime, which limits the number of operations that can be faithfully performed before qubits lose their quantum nature and become classical bits. The noise is detrimental to the performance of quantum algorithms. When scaling up the hardware, as the quantum circuit width and depth increase, circuit fidelity is observed to decrease exponentially [ 6]. This makes the NISQ era’s quantum computers less powerful than an ideal quantum computer promises. In recent years, tremendous efforts have been focused on understanding the noise mechanism in superconducting qubits and developing error mitigation techniques. Some key examples in error mitigation techniques are dynamical decoupling [7], zero-noise extrapolation [8], probabilistic error cancellation [9], and measurement error mitigation [10]. The solution to noise, ultimately, relies on quantum error correction and fault tolerance. The intrinsic noise on the device level needs to be low enough to achieve that. The error threshold will depend on the type of error correction code used, the device topology, the gate set, and the noise model of the device. The surface code for 2D nearest neighbor architectures has a relatively high threshold error rate of approximately 1% [11, 12]. The Steane and Bacon–Shor codes, when implemented on two-dimensional lattices with nearest-neighbor coupling, are found to have thresholds of about 2 10 −5 per operation [13, 14]. To make superconducting qubits practical for fault tolerance, understanding the noise mechanism at a single device level is still essential to unleash the full potential of quantum computing. 4 1.3 Thesis overview Chapter 2 begins with a general setup for the theory of open quantum systems and then reviews the Lindblad formalism, emphasizing the implication of its Markovian approxima- tion. It then overviews some main noise mechanisms in superconducting qubits and how they are described under this formalism. The exact dynamical description of noise from first principles is usually hard to obtain or numerically hard to solve, so we seek simple but ac- curate phenomenological noise models. To describe a more general noise process beyond the Markovian approximation, I introduce the post-Markovian master equation (PMME) and give a sketch of deriving its analytical solution, which serves as the theoretical foundation for Chapter 3. Chapter 3 describes how to use what we call the “PMME tomography” to construct a model to describe the free evolution of a single superconducting qubit on an IBM quantum processor. Instead of solving a master equation, we address its inverse problem: given a time series of quantum states obtained from a tomography experiment, what is the simplest model to describe it? I further show how the constructed model is tested by using it to predict the non-Markovian dynamics. Chapter 4 introduces the theory that describes dissipation under parametric driving in superconducting qubits. I start by presenting a classical picture of driven dissipation using LC circuits, which provides intuition. I then explain how driven dissipation can be viewed as a quantum refrigerator, which serves as the key mechanism to demonstrate reset for a readout cavity or a qubit. I then introduce the quantum treatment in the closed system, where analytical expressions can be derived with approximation. I then present the setup of the full quantum open-system treatment for the numerical simulation of the experiment. Chapter 5 describes the experimental realization of on-demand dissipation for qubit and cavity reset. I begin with the design consideration of the device and then present two iterations of the designs. The chapter continues by describing the device fabrication process 5 and measurement apparatus. I then explain the operation and characterization of the device. Next, I show measurements on the performance of using the cooling protocol to reset a readout cavity, with further investigation on the optimal control parameters of the parametric drive. Lastly, I discuss future experiments to demonstrate qubit reset. Finally, in Chapter 6, I present some ideas for future experiments and an outlook on the scalability of the noise modeling and engineering methods discussed in this thesis. 1.4 Summary of key results For noise modeling, we consider the free evolution of a single superconducting qubit, and show that a simple phenomenological dynamical model based on PMME accurately cap- tures and predicts non-Markovian noise in a superconducting qubit system. The PMME is constructed using experimentally measured state dynamics of an IBM Quantum Experience cloud-based quantum processor, and the model thus constructed successfully predicts the non-Markovian dynamics observed in later experiments. The model also allows the extrac- tion of information about cross-talk and measures of non-Markovianity. The results demon- strate definitively that the PMME model predicts subsequent dynamics of the processor better than the standard Markovian master equation. Our result confirms a non-Markovian model is necessary to capture correlated noise in real superconducting devices. For noise engineering, we demonstrate active, on-demand dissipation and use it to reset a readout cavity. The cavity can be reset at a rate greater than 20 MHz when the parametric drive is on, which is sped up by 20 times compared with passive waiting. We also show that this dissipation can be turned off readily, leaving a regular readout operation unaffected. 6 Chapter 2: Theory bac kground This chapter overviews some key theoretical frameworks and models to describe noise in superconducting qubits. The chapter starts with a general setup in open quantum systems. I then present the Lindblad master equation (LME). The LME describes the dynamics of an open quantum system where the qubits undergo decoherence and dissipation through interaction with their environment. The LME serves as an approximation to the actual noise process of a quantum processor. Different noise processes can be characterized by their bath correlation functions or the noise spectral densities, and they enter the LME through the Lindblad rates. I will review examples of common noise sources pertinent to today’s superconducting qubits, mechanisms, and noise spectra. We will see that deriving a microscopic description of noise from first principles is often difficult or impossible. This motivates us to develop phenomenological models. One such example is the post-Markovian master equation (PMME), which goes beyond the Lindblad formalism to account for the temporally correlated noise. 2.1 General setting The theory of open quantum systems describes the dynamics of quantum systems that interact and exchange information with their surrounding environment. In this setting, we consider two composite systems: the quantum system of interest (S), and the external environment that it interacts with, usually referred to as the “bath” (B). The system S we consider here is superconducting qubits, and together with the bathB comprise the lab, or even the universe, and can be thought of as a closed system. The composite system undergoes a joint unitary evolution generated by the total Hamiltonian H(t): H(t) =H S (t)+H B +H SB (t) (2.1) where H S is the system Hamiltonian, H B is the bath Hamiltonian and H SB is the system- bath interaction Hamiltonian. The initial joint state is ρ SB (0). Then, by Schrödinger’s equation, ρ SB (t) =U(t)ρ SB (0)U † (t) (2.2) As the system evolution is what we are interested in, we take a partial trace over the bath: ρ(t) = Tr B [ρ SB (t)] Φ[ρ(0)] (2.3) Here Φ denotes a map from the initial system state ρ(0) to its final state. For a quantum map to be physical, it needs to be completely positive and trace-preserving. Equivalently, this is to say map Φ has a Kraus operator sum representation, i.e., Eq. 2.3 can be rewritten as: ρ(t) = Φ[ρ(0)] = X k A † k (t)ρ(0)A k (t) (2.4) where A k ’s are Kraus operators, and satisfy P k A † k A k =I. Eq. 2.4 can be interpreted from a measurement perspective. Fig. 2.1.1a is a schematic to show the procedure to arrive at the reduced system state at the final time; this can be understood equivalently as a projective measurement of the bath at the final time, depicted schematically in Fig. 2.1.1b. In this measurement interpretation, imagine the bath acts as a probe coupled to the system at t = 0. They jointly evolve under the unitary U(t) generated by H, U(t) = T exp h i R t 0 H(t ′ )dt ′ i . To study the system state, a single pro- 8 Figure 2.1.1: (a) The exact dynamics, (b) the measuremen t in terpretation of the exact dynam- ics, (c) the measuremen t in terpretation of the LME, (d) the measuremen t in terpretation of the PMME. jective measurement is performed on the bath at time t with a complete set of projection operators jiihij, H B = Spanfjiig i . The measurement yields the result k and collapses the state to the corresponding eigenstatejki with probability p k = Tr S [hkjρ SB (t)jki]. The sys- tem state reduces to ρ k (t) = hkjρ SB (t)jki/p k =: A † k ρ(0)A k /p k . If we repeat this process for an ensemble with an identical initial state ρ SB (0), the average system state becomes ρ(t) = Σ k p k ρ k (t) = Tr B U(t)ρ SB (0)U † (t) , which is just Eq. 2.3. In the next section, we will see that measurement interpretation is useful for understanding the Markovian approxima- tion in the Lindblad master equation. 2.2 Lindblad formalism Open quantum system dynamics are often modeled using the the Gorini–Kossakowski– Sudarshan to Lindblad (GKSL) master equation [15, 16], also commonly known as the 9 Lindblad master equation: ˙ ρ = X α γ α L α ρ(t)L † α 1 2 L † α L α ,ρ(t) L D ρ, (2.5) whereL D , the Lindbladian, is the generator of the evolution: L D () = X α γ α L α L † α 1 2 L † α L α , , γ α 0 (2.6) L D generates decoherence and dissipation due to the system-bath interactions. The Lindblad operator L α describes various dissipation channels. For example, the dephasing process corresponds to a Lindblad operator of σ z , and an amplitude damping process corresponds to a Lindblad operator of j0ih1j. However, the LME is derived under the assumption of Markovianity. Loosely, this assumption amounts to an environment that is ‘memoryless’ and is only valid when the system is weakly coupled to a bath whose characteristic timescale is much shorter than that of the system dynamics [17, 18]. This can also be understood in the measurement interpretation as shown in Fig. 2.1.1c. For the Lindblad master equation Eq. 2.6, expanding Eq. 2.6 to first order in the short time interval τ yields ρ(t+τ) = " I(τ/2) X α L † α L α # ρ(t) " I(τ/2) X α L † α L α # +τ X α L α ρ(t)L † α (2.7) To the same order, we also have the normalization condition " I(τ/2) X α L † α L α #" I(τ/2) X α L † α L α # +τ X α L † α L α =I (2.8) 10 Thus, the Lindblad equation is cast in the Kraus operator sum as in Eq. 2.4, with the identification: A 0 =I(τ/2) X α L † α L α (2.9a) A α = p τL α , α 1 (2.9b) to the first order in τ, which is the coarse-graining timescale for which the Markovian ap- proximation is valid. 2.3 Noise mechanisms In this section, we will take a closer look at the specific forms of system-bath interactions in H SB for superconducting qubits: H SB = X α g α A α B α (2.10) which describes qubit’s degree of freedom A α coupled to a bath operator B α with coupling strength g α . An important property that characterizes the bath is its bath two-time corre- lation function: B αβ (t,t ′ ) B † α (t)B β (t ′ ) B (2.11) wherehi denotes the thermal averagehi B Tr[ρ B ]. Assuming different noise sources are uncorrelated, i.e., B αβ = 0, we will only look at the bath correlation function for the case α = β. Further, assuming the bath is stationary, the bath correlation is only a function of τ tt ′ : B αα (t,t ′ )B α =B α (tt ′ ) =B α (τ). (2.12) The Fourier transform of the bath correlation function encodes the frequency distribution 11 of the noise power. It is defined as S α (ω) = Z ∞ −∞ dτe iωτ B α (τ), (2.13) and is referred to as the bath spectral density (or noise power spectral density, PSD). When the noise can be described by a classical fluctuating parameter, the PSD is symmetric in frequency. For quantum noise, the autocorrelation function is complex-valued. The time ordering of the operator matters, and the PSD need not be symmetric in frequency. The effect of noise on superconducting qubits, broadly speaking, can be characterized by two rates: the longitudinal relaxation rates 1/T 1 and the transverse dephasing rates 1/T 2 . When the system-bath interaction in Eq. 2.10 is transverse to the qubit, e.g., A α is of the typeσ x or (a+a † ), then the noise at the qubit frequency can cause state transitions between qubit eigenstates. Formally, the T 1 related decay rate can be estimated by Fermi’s golden rule: 1 T 1 = 1 ℏ X α * g ∂ ˆ H S ∂B α e + 2 S α (ω q ) (2.14) where∂ ˆ H S /∂B α is the qubit’s susceptibility to the fluctuation due to bath operator B α and ω q is the qubit frequency. If the system-bath interaction in Eq. 2.10 is longitudinal with respect to the qubit, e.g., the system operator A α is of the type σ z or a † a, the noise causes fluctuation in qubit frequency and thereby introduce a stochastic phase evolution for a qubit superposition state. This causes the qubit state to lose phase information and is referred to as a pure dephasing process. TheT 2 combines the relaxation rates and dephasing rates into an overall decoherence rate: 1 T 2 = 1 2T 1 + 1 T φ (2.15) where 1/T φ is the pure dephasing rate. 12 The noise PSD can be characterized experimentally using several approaches, for example, via dynamical decoupling [19] and Rabi spectroscopy [20]. In the following, I review different sources of noise and their spectral properties. 1 Charge noise Charge noise refers to the charge fluctuation δn that arises from material defects or surface dielectric loss. Charge noise in superconducting qubits is modeled by an ensemble of fluctuating two-level systems (TLSs). A TLS is a low-energy excitation and is generally visualized as an ion or electron. Due to the charged nature of the TLS, it has a dipole moment that interacts with the electromagnetic fields. Depending on the coupling mechanism, they can exchange energy with a qubit or cause fluctuation in qubit frequency. An ensemble of TLSs can contribute to both qubit relaxation and the dephasing process. The charge noise spectrum has low-frequency noise and high-frequency components (Ohmic). For the low-frequency range, it is modeled using a 1/f noise spectrum S Q (ω) =A 2 Q 2π1 Hz ω γ Q (2.16) Typical values are A 2 Q = (10 −3 e) 2 /Hz at 1 Hz, and γ Q 1. At high frequencies, the noise power spectrum takes the form of Nyquist noise: S Q (ω) =A 2 Q [ω/(2π1 Hz)] (2.17) where the noise strengthA 2 Q at 1 Hz can assume a range of values depending on the level of dissipation in the system. A transmon qubit is less susceptible to charge noise by introducing a big shunting capacitanceC, resulting in a small capacitance energyE C =e 2 /2C. One can estimate the prefactor of the transmon’s susceptibility to charge fluctuation ∂H/∂n is 8E C ˆ n, and a detailed analysis [21] shows that the suppression of charge sensitivity is exponential 13 in the parameter p 8E J /E C . A large ratio between E J and E C makes transmon qubits less sensitive to charge noise at the price of a relatively low anharmonicity, typicallyE J /E C ≳ 50. 2 Magnetic flux noise Another common source of noise present in superconducting qubits is magnetic flux noise. This noise is thought to originate from collections of spins on the surface of a superconducting film, which can couple a noisy flux into a SQUID loop. In a flux-tunable transmon qubit, the magnetic field couples longitudinally to the qubit, and the fluctuation in an external magnetic field can cause frequency fluctuation resulting in pure dephasing. The noise spectrum of flux noise contains a significant low-frequency contribution, which is commonly referred to as 1/f noise, and its PSD is modeled as: S Φ (ω) =A 2 Φ 2π1 Hz ω γ Φ (2.18) There have been extensive efforts to characterize and understand 1/f flux noise in su- perconducting qubits [19]. Experimental characterization suggests γ Φ 0.8 1.0 and A 2 Φ (1µΦ 0 ) 2 /Hz. This 1/f spectrum has been shown to extend from less than millihertz to beyond gigahertz frequencies. The origin of flux noise is attributed to the stochastic spin flips on the surface of the superconducting metal. Their microscopic nature remains an open question [22]. 3 Photon shot noise The photon shot noise arises from photon number fluctuations in the readout resonator. In the dispersive readout regime, as in Eq. 1.4, the qubit interacts with the readout resonator through the term χσ z n, resulting in a qubit frequency shift that is linear in the average 14 photon number ¯ n in the resonator: ∆ Stark = 2ηχ¯ n (2.19) where η =κ 2 /(κ 2 +4χ 2 ), κ is the resonator decay rate, and χ is the dispersive shift. Since the photon shot noise is longitudinally coupled to the qubits, it contributes to qubit pure dephasing with a rate Γ ϕ =η 4χ 2 κ ¯ n (2.20) This dephasing can be understood as an unintentional measurement as photons leaving the cavity carry unrecorded qubit state information. The photon number fluctuation originates from the residual photons in the readout resonator, typically from thermal radiation from a higher temperature component near the qubit. The noise spectral density is of a Lorentzian type: S(ω) = 4χ 2 2η¯ nκ ω 2 +κ 2 (2.21) which exhibits white noise spectrum up to 3dB cutoff frequency ω κ set by the resonator decay rate κ. If the measurement is strong, and the number of photons in the resonator is larger than a critical value [23–25], residual photons in the cavity can also cause qubit state transition due to the breakdown of dispersive approximation, leading to readout error. It can also excite the qubit out of the computational subspace due to the breakdown of the rotating wave approximation [26, 27]. 15 2.4 Beyond the Markovian approximation As we have seen in the previous section, the actual noise process in superconducting qubits has different timescales, and the Markovian approximation is only a crude approximation. On the other hand, as it is often not feasible to derive noise models from first principles, we want to develop phenomenological models that can accurately capture the effect of noise. As a reminder, in the measurement interpretation of the open system dynamics, the exact dynamics corresponds to the joint evolution of coupled system and bath with a generalized measurement at the end (Fig. 2.1.1b), and the Lindblad equation corresponds to a series of measurements interrupting the joint evolution after each time interval τ (Fig. 2.1.1c). A natural way to go beyond the Markovian approximation is to relax the many-measurement process. The post-Markovian master equation [28] is derived based on a probabilistic single- shot measurement process as in Fig. 2.1.1(d). The PMME naturally interpolates between the exact dynamics (a completely positive map) and the Markovian Lindblad equation. The PMME is derived considering the following process: a probe is coupled to the system att = 0; they evolve jointly for timet ′ (0⩽t ′ <t). Att ′ , the system state isΛ(t ′ )ρ(0) where Λ(t ′ ) is a one-parameter map, at which moment the generalized measurement is performed on the bath. The system and bath continue their joint evolution fromt ′ tot, where the final measurement is applied. The final state at time t is ρ(t) = Λ(tt ′ )ρ(t ′ ). It is important to note that ρ(t ′ ) cannot be written as Λ(t ′ )ρ(0) since measurement selects ρ(t ′ ) as random. The time t ′ encodes bath memory effects and is a function of time scales characterizing the evolution. In the PMME, this is captured by a kernel functionk(tt ′ ,t) that assigns weights to different measurements at different times t ′ . For the full derivation of the PMME, please refer to Ref. [28] (see also [29] for an updated derivation). 16 The PMME takes the following form: ∂ t ρ =L Z t 0 k(τ)e Lτ ρ(tτ)dτ (2.22) =Lk(t)e Lt ρ(t), (2.23) Note that the PMME reduces to the standard Lindblad equation if we choose k(τ) = δ(ττ 0 ). One nice feature of the PMME is that it is analytically solvable via Laplace transform. s˜ ρ(s)ρ(0) = ˜ M(s)˜ ρ(s) (2.24) ˜ M(s) Lap k(t)Le Lt . (2.25) Assuming the invertibility of the superoperatorsI ˜ M(s), the solution can be expressed in Laplace domain as: ˜ ρ(s) = (sI ˜ M(s)) −1 ρ(0) (2.26) Moving back to the time domain, we have: ρ(t) =N(t)ρ(0), (2.27a) N(t) = Lap −1 [ ˜ N(s)], (2.27b) ˜ N(s) (sI ˜ M(s)) −1 (2.27c) Here, the Laplace transform, and its inverse on the superoperators are applied to their matrix elements. To make this explicit, we can introduce a good operator basis so that we can replace the superoperator L by its matrix representation L2 M (d 2 ,R). Assume L is diagonalizable and has distinct right and left eigenoperators, i.e., we can find operators R and L such thatLR =λR and L † L =λL † . The sets can be chosen to be mutually orthonormal 17 such that after renormalizationhL † i ,R j i = Tr[L i R j ] =δ ij and they define the damping basis ofL. The construction of the damping basis given a diagonalizable superoperatorL is given in Appendix A. Finding these left and right eigenvectors,fL i g andfR i g, makes it convenient to simplify N(t), and we can we solve for ρ(t) in the damping basis by expanding ρ(t) as: ρ(t) = X i µ i (t)R i (2.28) where the expansion coefficients are µ j (t) = X i µ i (t)Tr(L j R i ) = Tr[L j ρ(t)]. (2.29) Then one can solve for µ i (t) in the Laplace domain according to Appendix B and arrive at the solution below: µ i (t) =ξ i (t)µ i (0) (2.30) where: ξ i (t) = Lap −1 1 sλ i ˜ k(sλ i ) , (2.31a) µ i (0) = Tr[L i ρ(0)] (2.31b) The solution to the PMME can be viewed as a map: ρ(t) = X i µ i (t)R i = X i ξ i (t)µ i (0)R i = X i ξ i (t)Tr[L i ρ(0)]R i = Φ[ρ(0)] (2.32) 18 where Φ[X] X i ξ i (t)Tr[L i X]R i (2.33) Complete positivity is not always guaranteed in the PMME due to the freedom of choosing the kernel k(t). The Choi theorem [30] provides a way to construct a condition for PMME complete positivity. Constructing the Choi matrix χ Φ for the PMME map from Eq. 2.33, we arrive at the complete positivity condition on the PMME: χ Φ = X i ξ i (t)L T i R i 0 (2.34) From the information perspective, the presence of non-Markovianity indicates the backflow of information from bath to system. To quantify the degree of non-Markovianity, this thesis adopts the measure in Ref. [31]. We now define the positive semi-definite, trace-one linear operators acting on H as S(H). Letting ρ 1 (0) and ρ 2 (0) be a pair of initial qubit states ρ 1,2 2 S(H), the measure of non-Markovianity uses the rate of change of the trace-norm distanceD between the two states under some noise channel Φ t : σ(t,ρ 1 ,ρ 2 ) d dt D(Φ t (ρ 1 (0)),Φ t (ρ 2 (0))) (2.35) where the trace-norm distanceD between two states is defined as: D(ρ 1 ,ρ 2 ) := 1 2 kρ 1 ρ 2 k 1 = 1 2 Tr h p (ρ 1 ρ 2 ) † (ρ 1 ρ 2 ) i . (2.36) Under Markovian dynamics [Eq. (2.5)], the trace-norm distance between two quantum states is monotonically decreasing as a function of time, whereas non-Markovian dynamics violates this contractive property, i.e., there can be an increase in the trace-norm distance. In other words, non-Markovianity leads to a revival of distinguishability between two states at some 19 point during the evolution, and a process is non-Markovian if there exists any pair of initial states ρ 1 (0),ρ 2 (0) and a time t for which σ(t,ρ 1 ,ρ 2 )> 0 [31]. The measure for the degree of non-Markovianity of a quantum process is thus defined as: N(Φ) max ρ 1,2 ∈S(H) Z σ>0 dtσ(t,ρ 1 ,ρ 2 ). (2.37) Having introduced the PMME and how it can be used to model open quantum systems, I will next present experimental results using the PMME to model noise on IBM quantum processors. 20 Chapter 3: Predicting non-Mark o vian dynamics In this chapter, we ask, given a noisy superconducting qubit, what is the simplest model we can construct to describe and predict the effects of noise on the qubit system. One such choice is the LME, which is completely positive and is formally easily solvable. However, the LME is derived under the assumption of Markovianity. As we have seen in the previous chapter, although the Markovian assumption allows for significant simplifications, it is only an approximation and in reality it is often desirable to account for non-Markovian effects [ 32]. This is true in particular in the case of the dynamics of superconducting qubit sys- tems [33, 34]. For example, it has been observed on the IBM Quantum Experience (IBMQE) processors [35] that the fidelity of a gate operation is conditional on the gate operation that preceded it in a gate sequence [36]. This is an example of non-Markovian noise, which in- troduces temporally correlated errors. Non-Markovian effects may arise from, for example, spatially correlated noise that arises from nonlocal external pulse controls, coherent errors caused by residual Hamiltonian terms, or stochastic errors due to slow environmental fluc- tuations. Such correlations, as well as correlated errors on multiple qubits, have been shown to be a leading source of failure in achieving quantum error correction [37–39], and also in other near term quantum applications [40]. In other words, dealing with non-Markovianity will be vital to achieving fault-tolerant quantum computation [41–44]. Unfortunately, most device characterization and validation methods do not fully capture non-Markovian effects, as these methods either implicitly or explicitly make the Markovian approximation. For instance, the standard T 1 and T 2 measurements that quantify qubit lifetime assume exponential decay of the excited state population or the qubit coherence. Similarly, randomized benchmarking and gate set tomography [45] consider circuits of vary- ing length and assume that the fidelity of corresponding operations decay as circuits become longer. However, on real quantum processors, recent studies [46–48] have observed deviations of the qubit dynamics from the prediction of a purely Markovian treatment. In this work we focus on noise processes that govern the free (undriven) evolution of a su- perconducting quantum system. Here, the non-Markovian effects can be both coherent, e.g., due to unintentional crosstalk with neighboring qubits, or incoherent, e.g., due to coupling to magnetic impurities [49]. There has been extensive work on developing a set of master equations that are both easily solvable and account for non-Markovian effects, e.g., the Gaussian collapse model [ 50], quantum collisional models [51] and the time-convolutionless master equations [17]. Here, we choose to focus on the PMME, introduced in Sec. 2.4, which includes bath memory effects via a phenomenological memory kernel k(t). We choose the PMME for its conceptual and computational simplicity and because it has a closed form analytical solution in terms of a Laplace transform. It naturally interpolates between the exact dynamics (a completely positive map [52]) and the Markovian Lindblad equation, and at the same time, retains complete positivity with an appropriate choice of the form and parameters for k(t) [28, 53, 54], and remains analytically solvable. The PMME we consider in this chapter takes the following form: d dt ρ(t) =L 0 ρ(t)+L 1 Z t 0 dt ′ k(t ′ )exp[(L 0 +L 1 )t ′ ]ρ(tt ′ ). (3.1) Here ρ(t) is the reduced system state and L 0 and L 1 are Markovian (super-)generators in Lindblad form that describe the dissipative dynamics, whereL 0 can have additional Hermi- tian (i.e., unitary evolution generating) components. Non-Markovian effects in the evolution under the Lindblad superoperatorL 1 are introduced via a phenomenological memory kernel k(t) to assign weights to the previous “history” of the system state. We note that Eq. (3.1) 22 differs in an important way from the original PMME [ 28] in Eq. 2.22, in that the latter did not contain L 0 inside the integral. The reason behind this choice will become appar- ent below; in essence, it allows us more flexibility in partitioning the various terms in the Lindbladian. In our protocol, the PMME model is constructed by fitting to an ensemble of time- domain tomography measurements. Hence, we call our protocol PMME tomography. We demonstrate PMME tomography on an IBMQE device and then use the PMME model thus constructed to quantify the degree of non-Markovianity. Our procedure predicts the non-Markovian effects we observe in future measurements and can model the information backflow from the environment to the system on the device we have tested. Our protocol provides a robust estimation method for a continuous dynamical model beyond the com- monly assumed Markovian approximation, paving the way to more accurate modeling of noisy intermediate-scale quantum (NISQ) devices. In the following sections, we start by constructing the closed system model for the qubit evolution, introduce the non-Markovian open system model by proposing a Lindbladian and different memory kernel terms for Eq. ( 3.1), describe the data collection procedure of the state tomography experiments, and finally use the estimated qubit states sampled during the evolution to find the best-fit PMME model parameters. In Sec. 3.2 we apply the PMME model construction protocol to an IBMQE processor, discuss the descriptive power of the PMME models on the fitting data set, and discuss the models’ predictive power by using them to predict qubit dynamics in a previously unseen testing data set. We also quantify the degree of non-Markovianity based on the constructed PMME model. In Sec. 3.3 we contrast our method with previous work, and provide a discussion of the results and conclusions. 23 3.1 Methodology 1 Closed system model We consider a single qubit described by the effective Hamiltonian H = 1 2 ω z σ z . (3.2) which is written in the rotating frame of the qubit drive, whereω z accounts for the detuning between the qubit frequency and the drive frequency. In practice, the drive frequency is typically set to be the qubit frequency. The latter is determined via a calibration procedure, typically carried out on a single qubit with the rest of the neighboring, “spectator” qubits all in their ground states. A shift in qubit frequency can lead to a non-zero detuning between the qubit frequency and the drive, thus a non-zero ω z in the effective Hamiltonian. In addition, the sign and the magnitude of ω z can change depending on the initial state of the spectator qubits due to the presence of an always-on ZZ interaction which arises from unintended coupling between the qubit and its neighbors [48]. For these reasons, we include a Hamiltonian term in our model. 2 Open system models and their physical motivation Our task is to find a model that best describes a time series of state tomography observations and find the best-fit parameters of that model. In this case the model itself is represented by the functional form of the memory kernel k(t), while the model parameters are the parameters of the kernel function. We consider a sequence of models in order of increasing model complexity: the Lindblad modelM 0 and the PMME modelsM 1 andM 2 . In general, we denote the models byM i (θ) whereθ is a list ofp i free model parameters, withp i increasing monotonically with i. 24 We take as our simplest model (M 0 ) the Lindblad master equation dρ dt =L(ρ) =L 0 (ρ)+L 1 (ρ) (3.3a) =i[H,ρ]+ X k γ k V k ρV † k 1 2 n V † k V k ,ρ o . (3.3b) As is clear from Eq. (2.22), this is equivalent to a PMME with a delta-function kernel, k 0 (t) =δ(t). In the notation of Eq. (2.22), we choose the first generator as: L 0 =H+L GAD , (3.4) which has a Hermitian component H(ρ) = i[H,ρ] and a generalized amplitude-damping LindbladianL GAD with the Lindblad operatorsV k 2fσ + ,σ − g. We choose the second gener- ator as a pure dephasing Lindbladian: L 1 (ρ) =γ z (σ z ρσ z ρ) . (3.5) With this choice, which is motivated by our experiments with the IBMQE devices, the population decay (T 1 ) is essentially Markovian, as it is dominated by theL 0 ρ(t) term outside the integral in Eq. (2.22). The parameters of interest to us are the following: ω z , the amplitude of the static z-field due to the always-on ZZ coupling with the neighboring qubit(s); γ z , the pure dephasing rate; Γ s = γ + +γ − and Γ r = γ + /γ − , respectively the sum and ratio of the excitation rate and the relaxation rate. The Kubo–Martin–Schwinger (KMS) condition [17], which states that the rate of excitation in a system is exponentially suppressed relative to the rate of relaxation at the same frequency [γ(ω) = e −βω γ(ω), where β is the inverse temperature and ω> 0], implies that Γ r < 1. This Lindblad model has a total of p 0 = 4 parameters. To go beyond the Lindblad modelM 0 , we consider two PMME modelsM 1 andM 2 with 25 Figure 3.1.1: The nested candidate mo dels w e use to describ e the qubit free ev olution: the Lind- blad mo delM 0 , parameterized b y a Hamiltonian t erm and the Lindbladian, and the PMME mo delsM 1 ,M 2 with their additional k ernel parameters. extra parameters for the memory kernel. We consider a family of kernels whose Laplace transform can be written as rational functions, i.e., ˜ k 1 (s) = Lap[k(t)] =P(s)/Q(s) , (3.6) with polynomials P(s) =a m s m ++a 1 s+a 0 and Q(s) =b n s n ++b 1 s+b 0 of degree m andn, respectively. This include a large class of kernels which can be expressed as linear combinations of functions of the form t d e ct for complex c and integer d. For the PMME modelM 1 we choose the simplest kernel in this family: an exponentially decaying kernel k 1 (t) =a 0 exp(b 0 t) () ˜ k 1 (s) =a 0 /(s+b 0 ) , (3.7) where henceforth we impose the constraint k(0) = 1 (since the normalization can always be absorbed into L 1 ), which leads to a 0 = 1. This model thus has a total of p 1 = 5 free parameters: the Lindblad model parameters and b 0 . 26 The more complex PMME modelM 2 has two additional free parameters a 0 and b 1 : k 2 (t) = 8 > < > : e − b 1 2 t h 2a 0 −b 1 µ sinh( µ 2 t)+cosh( µ 2 t) i if B 0 e − b 1 2 t h 2a 0 −b 1 µ sin( µ 2 t)+cos( µ 2 t) i if B < 0 () ˜ k 2 (s) = (s+a 0 )/(s 2 +b 1 s+b 0 ) , (3.8) where B = b 2 1 4b 0 and µ = p jBj. This model has a total of p 2 = 7 free parameters (the Lindblad parameters and a 0 , b 0 , and b 1 ). The sign of B specifies whether the kernel is overdamped or underdamped. As illustrated in Fig. 3.1.1, the modelsM 0 ,M 1 andM 2 are a nested sequence of models of increasing complexity; e.g.,M 2 reduces toM 1 with the identifications a 0 = 0,b 0 = 0, and with a renaming of the parameter b 1 ! b 0 (since now the kernel function ˜ k(s) has a lower degree). The models fM i g predict a functional form of the dynamics depending on the model variables θ, so the predicted evolution of a state can be written as ρ prd (t) =f(tjθ) , (3.9) where θ = fω z ,γ z ,γ − ,γ + ,⃗ a, ⃗ bg is the list of model parameters. The kernel parameters are ⃗ a =fa 0 ,...,a m−1 g, and ⃗ b =fb 0 ,...,b n−1 g, some of which may be constrained, in addition to the positivity constraintfγ z ,γ − ,γ + g> 0 and the KMS constraint γ + /γ − < 1. The goal of this procedure is to specify the master equation that governs the dynamics of the system. We formulate this in terms of the inverse problem: given a discrete time series of measurement records of the state, we seek the dynamical modelM i (θ) that most closely matches the observations. 27 3 Quantum State Tomography To get complete qubit state information, we perform state tomography on a single qubit of the ibmq_athens processor, which is a 5-qubit processor consisting of superconducting transmon qubits. The main qubit we perform PMME tomography on is qubit 0 (Q0). The relevant device calibration details on the dates of data collection are provided in Table 3.1.1. Data set Fig. 3.2.2 Fig. 3.2.5, Fig. 3.2.6 Fig. 3.2.7 Date collected 6/25/2021 6/30/2021 7/1/2021 T 1 (µs) 72.6 70.8 75.2 T 2 (µs) 93.4 82.6 62.9 readout error [10 −2 ] 1.9 0.99 1.00 T able 3.1.1: Qubit calibration information of the ibmq_athens pro cessor on the dates of data collection. The data is collected with the main qubit state ρ S initialized in one of the five states in the preparation set P = fjψ i (0)ig i=4 i=0 ; and the rest of the processor’s spectator qubits initialized in the ground statej0i. The states inP are illustrated in Fig. 3.1.2 and specified in Table 3.1.2. After the qubit initialization, the main qubit undergoes free evolution for a variable time t j , and then state tomography is performed to construct the density matrix, augmented by measurement error mitigation (see Sec. 4). Specifically, the circuits of the tomography experiment contain the following steps: 1. State preparation: the qubit is initialized in the ground state, and a state-preparation gate is applied to initialize the qubit in one of the five fiducial states jψ i (0)i. 2. Evolution: the qubit undergoes free evolution, with a variable evolution time. This corresponds to applying a sequence of identity gates I and sweeping the number of gates. 28 Figure 3.1.2: The initial states used for the fitting data set (y ello w) and the testing data set (blue). The initial states are c hosen suc h that the statesjψ i (0)i , i = 0,1,2,3 form a tetrahe- dron on the Blo c h sphere, wherejψ 1 (0)i = j1i is the excited state. The stat ejψ 4 (0)i is a fixed, randomly c hosen state on the Blo c h sphere. The same set of fiv e states are used in all our exp eri- men ts. jψ 0 (0)i q 8 9 ,0, 1 3 jψ 1 (0)i (0,0,1) jψ 2 (0)i q 2 9 , q 2 3 , 1 3 jψ 3 (0)i q 2 9 , q 2 3 , 1 3 jψ 4 (0)i (0.50,0.75,0.41) T able 3.1.2: The set of initial statesP used in our exp erimen ts, corresp onding to the states sho wn in Fig. 3.1.2 . 3. Measurement: one of the three single-qubit gates (I,M X , orM Y ) is applied before the measurement, corresponding to measurement in the Pauli z, x, and y-bases respec- tively. HereM X =H is a Hadamard gate andM Y =HS † whereS is a phase gate. We record the measurement outcomes 0 or 1. The steps above are repeated for all combinations of initial states, free evolution duration, and measurement basis, and each combination is repeated for N s = 8192 shots. Letting N jk denote the number of times out of N s shots that outcome 1 occurred at time t j in 29 measurement basis k, the state tomography raw data are the recorded outcome counts, fN jk jj = 1,...,n t ,k2fx,y,zgg (3.10) where n t is the total number of time points. We perform Bayesian measurement error mitigation on the raw data (see Sec. 4 for details) and the measurement mitigated data is then fed into a maximum likelihood estimation (MLE) routine. This routine determines the qubit state ˆ ρ(t j ) at time t j , represented by the experimentally-measured Bloch vector ⃗ v exp (t j ) as ˆ ρ(t j ) = 1 2 (I+⃗ v exp (t j )⃗ σ). The uncertainties associated with the tomographically constructed states, i.e., the standard deviations of the corresponding Bloch vector components, denoted as σ jk , k 2fx,y,zg, are estimated by Bayesian bootstrapping. The collected data sets are divided into the model fitting set fˆ ρg fit and the model testing setfˆ ρg test . The former contains a time series of qubit dynamics with a single initial statejψ 0 (0)i, the latter contains the other four time series with four different initial states fjψ i (0)ig i=4 i=1 . Finally, we use the constructed qubit states fˆ ρ(t j )g j=nt−1 j=0 from the fitting data set to construct the PMME description of the free-evolution channel. 4 Measurement error mitigation Measurement error mitigation is performed by using information from calibration experi- ments to remove any systematic bias in the measurement results [55]. The calibration exper- iments involve preparation of the computational basis statesjji, which are then used to learn the response matrixM. The entriesm kj = probability(preparejjijmeasure bitstring k) rep- resent conditional probabilities. Any subsequent experiment gives us the measured proba- bility vector ⃗ p(E) which is used to infer the true probability of vector ⃗ t(E) = f(⃗ p(E),M). The most commonly used MEM method – called response matrix inversion method – defines ⃗ t = M −1 ⃗ p. Crucially, M −1 is not stochastic, so ⃗ t(E) can have negative entries. Recently, a Bayesian solution to the non-stochasticity problem was proposed [56]. In this method, 30 inspired from similar unfolding methods in high-energy physics, we start with a prior truth spectrum ⃗ t 0 and update it using Bayes’ rule to get t n+1 i = X j M ji t n i P k M jk t n k p j . The prior t n i is updated using the response matrix M and gives the posterior t n+1 i , and the process proceeds for 100 iterations (in practice this was found to always be sufficient for convergence). After each tomography experiment, the probabilities of each measurement outcome are updated using measurement error mitigation. 5 Model fitting Given the observations from the state tomography experiment, our task is to find the best-fit model parameters θ, as parametrized in Sec. 2, of the dynamical model M i (θ). To tackle the problem, we perform a maximum likelihood estimation (MLE), a well-studied method of determining unknown parameters of a model from a set of data. The input data of this MLE procedure is a time series of the qubit evolution ft j ,ˆ ρ exp (t j )g j=Nt−1 j=0 , initialized in one of the states in the set P. The qubit state ˆ ρ exp at time t j is constructed from tomography experiment outcomes via a Qiskit state tomography MLE routine [57]. In the fitting procedure, we seek to minimize the distance between the observed state ˆ ρ exp (t j ) and the model predicted state ρ prd (t j ) for all the sampled time instances t j . We define the following standard objective function in the least squares form [58]: χ 2 (θ) = Nt−1 X j=0 X k=x,y,z [ˆ v exp jk (t j )v prd jk (t j ;θ)] 2 σ 2 jk , (3.11) where ⃗ v exp jk and ⃗ v prd jk denote the kth Bloch vector component of the qubit state at time t j constructed from the experimental observations and predicted by the model, respectively. The latter is obtained by solving the model M i (θ) for ρ prd (t). Since the PMME admits a 31 closed-form analytical solution, the evaluation of the objective function Eq. (3.11) is efficient to compute under the assumption that the noise associated with each data point follows a normal distribution. Minimizing the least-squares functionχ 2 (θ) for the model parametersθ is equivalent to maximizing the likelihood of observing the data set given that the underlying model is true. We would like the dynamical model to generate a CPTP map (though this is not strictly necessary, as non-CP maps are also valid physical models; see, e.g., Ref. [59]). For this reason, we restricted the Lindblad rates fγ z ,γ + ,γ − g to be positive. For the PMME models with different choices of kernel functions, we derive the condition that guarantees CPTP dynamics in App. C. We find that the conditions jξ 4 j < 1 and jξ 2 j = jξ 3 j < p (Γr+ξ 4 )(1+Γrξ 4 ) 1+Γr < 1 are necessary and sufficient for complete positivity, where ξ i (t) = Lap −1 h 1 s−λ 0 i −λ 1 i k(s−λ i ) i (the inverse Laplace transform), andλ 0 i ,λ 1 i , andλ i are the eigenvalues of the matrix representation ofL 0 ,L 1 , andL, respectively, in the Pauli basis (see Appendix D). The best-fit PMME parameters are found by solving the following minimization problem: minimize χ 2 ω z ,γ z ,γ + ,γ − ,⃗ a, ⃗ b (3.12) subject to γ z ,γ + ,γ − > 0, Γ r =γ + /γ − < 1, k(0) = 1 . After we fit the models by solving Eq. ( 3.12), we compute the trace-norm distance D(t j ) = 1 2 kˆ ρ exp (t j )ρ prd (t j )k 1 , (3.13) where kAk 1 = Tr[ p A † A] is the trace norm. D(t j ) quantifies the probability with which one can optimally discriminate the experimentally observed ˆ ρ(t j ) from the predicted state ρ(t j ) [60], and thus a smallD(t j ) indicates an accurate prediction. 32 fitting set testing set Lindblad model evaluate non-Markovianity model validation PMME models PMME models Figure 3.2.1: Analysis proto col for PMME tomograph y . The tomograph y data sets for qubit free ev olution with differen t initial states are divided in to a fitting set (in our case with a single initial statejψ 0 (0)i ) and a testing set (in our case with initial statesjψ 1,2,3,4 (0)i ). The fitting set is used to fi t the dynamical mo dels fM i g using a classical optimizer based on the MLE metho d. 3.2 Results We test our approach for tomographic PMME construction in three increasingly challenging settings, summarized in Fig. 3.2.1. We start with time-series state tomography data for a single initial state on a transmon qubit (the fitting data set on jψ 0 (0)i) and show that we can construct a sequence of nested PMME models to represent this evolution. We then compare the PMME models with the Markovian Lindblad counterpart and show that the PMME models provide a measurably better fit. Secondly, we test the PMME’s predicted evolution for quantum states that were not used to construct the PMME model. Again, we show that the PMME model faithfully predicts the evolution even for these new states without requiring full process tomography, while the Lindblad model does not. Lastly, we compute the degree of non-Markovianity in the evolution using the tomographically constructed PMME model and show that it correctly approximates the observed non-Markovianity in the qubit free-evolution dynamics, which the Lindblad model fails to do. 33 0.0 1.0 v x a0 |ψ 0 (0) fi a1 |ψ 1 (0) fi a2 |ψ 2 (0) fi a3 |ψ 3 (0) fi a4 |ψ 4 (0) fi -0.5 0.0 0.5 v y b0 b1 b2 b3 b4 -1.0 0.0 v z c0 c1 c2 c3 c4 0.6 0.8 1.0 Tr[ρ 2 ] d0 d1 d2 d3 d4 0 30 60 90 0.0 0.2 D(ˆ ρ exp ,ρ prd ) e0 0 30 60 90 e1 0 30 60 90 time (μs) e2 0 30 60 90 e3 0 30 60 90 e4 fitting data testing data tomography data M 0 M 1 M 2 Figure 3.2.2: PMME tomograph y proto col applied to single-qubit free ev olution, with the sp ec- tator qubits all in their ground state. (a0-c0) The free-ev olution tomograph y data ˆ v exp (t) of the fitting data set with the qubit initialized in jψ 0 (0)i and the b est-fit mo dels for the Lindbladian mo delM 0 (orange lines), the PMME mo del with k ernel t yp e 1,M 1 (blue lines), and the PMME mo del with k ernel t yp e 2,M 2 (red lines). The shaded regions denote the 95% confidence region of the mo del predictions. (a1-c4) The free-ev olution tomograph y data in the testing data set with the qubit initialized infjψ i (0)ig 4 i=1 , and the prediction from the b est-fit mo dels fr om the fitting data set in (a0-c0). (d0-d4) The empirical purit y Tr[ρ 2 ] of the qubit and that predicted b y the b est-fit mo dels. (e0-e4) The distance b et w een the tomographically constructed state and the state predicted b y the b est-fit mo dels. All mo dels p erform equally w ell at predicting the dynam- ics of the excited statejψ 1 i = j1i (M 0 is obscured b yM 1 andM 2 ), while the PMME mo dels M 1 andM 2 predict the dynamics offjψ i (0)ig 4 i=2 b etter than the Lindbladian mo del. 34 1 Fitting set First, we construct the dynamical modelsM 0 ,M 1 andM 2 using the time-series tomography data of a single initial state. Our results are displayed in the first column of Fig. 3.2.2, which shows the observed evolution⃗ v exp (grey circles with error bars) and the predicted evolution ⃗ v prd (solid lines) from the constructed dynamical models. The best-fit parameters of the constructed models M i are reported in Table E.0.1 in App. E. We plot the Bloch vector components in rows a, b, and c, the state purity Tr[ρ 2 ] in row d, and the trace-norm distance between the measured and the predicted state D(t j ) [Eq. (3.13)] in row e, and a smaller distance is more desirable. The PMME models M 1 and M 2 both faithfully capture the evolution of the system. To compare our models, we use the Akaike information criterion (AIC) [61], which is a model selection metric (see Sec. 4 for more details). AIC accounts for the goodness of fit of a model and penalizes models with more parameters. As shown by the purple squares in Fig. 3.2.3, M 1 and M 2 have much lower AIC than M 0 and going from M 1 to M 2 decreases the AIC slightly. Overall, M 1 and M 2 are better models even after being penalized for utilizing more parameters. Both the Lindblad modelM 0 and the PMME models can account for a spurious longitu- dinal field component in the qubit Hamiltonian due to the always-on ZZ interaction between neighboring qubits [62]. This spurious field effectively shifts the qubit frequency [Eq. ( 3.2)] and manifests as the oscillations in the off-diagonal elements v x (t) and v y (t) of the density matrices of the qubit state (the first and second rows of Fig. 3.2.2). However, a Hamiltonian term does not modify the purityp = Tr[ρ 2 ] since ˙ p =iTr[ρ[H,ρ]] = 0, so purity oscillations must have a different origin. The fact that M 0 is unable to capture the purity oscillations seen in the fourth row of Fig. 3.2.2, while in contrast both M 1 and M 2 do display purity oscillations, is evidence of non-Markovianity, as we explore below in greater depth. The non-monotonic envelope of the purity oscillations is consistent with the non-unitality of the Lindbladian model [63]. Overall, in comparison to the Lindbladian modelM 0 , the PMME- predicted evolution is significantly closer to the empirical data, as quantified by D(ˆ ρ exp ,ρ prd ) 35 fitting data testing data median 95th percentile median 95th percentile M 0 0.14 0.24 0.12 0.19 M 1 0.01 0.03 0.06 0.08 M 2 0.01 0.03 0.06 0.08 T able 3.2.1: T race-norm distances corresp onding to the results sho wn in Fig. 3.2.3 . (last row of Fig. 3.2.2). M 0 M 1 M 2 M 0 M 1 M 2 0.0 0.1 0.2 distance −22.5 −20.0 −17.5 −15.0 fitting AIC fitting data testing data Figure 3.2.3: The predictiv e abilit y of differen t mo dels for the fitting data set (left) and the test- ing data set (righ t) for ibmq_athens data. F or b oth datasets, w e sho w the trace-norm distance b et w een the tomographically constructed statefˆ ρ exp g and the mo del predicted statefρ prd g . T he b o x plot sho ws the 5th, 50th (median), and 95th p ercen tiles for this distance o v er t j and resp ec- tiv e initial states. The op en circles denote extremal outliers. The PMME mo delsM 1 andM 2 describ e b oth the fitting data set and the testing data set b etter than the Lindblad mo del M 0 (see T able 3.2.1 ). W e also rep ort the AIC of the mo dels on the fitting data (purple squares, righ t axis), again with b etter p erformance b y the non-Mark o vian mo dels; note that all AIC v alues are negativ e. 2 Testing set Next, we test how well the fitted models predict the evolution of states that were not used to fit these models. The last four columns of Fig. 3.2.2 represent four time series of quantum state evolutions in the testing data set with each of the different initial states fjψ 1 (0)ig 4 i=1 . 36 The goal of this ‘testing set’ is to validate whether the dynamical model is capable of de- scribing the evolution for an arbitrary single-qubit state. Rather than doing this by selecting random states, we chose a set of four states that is maximally separated on the Bloch sphere, i.e., four pure states on the vertices of a tetrahedron (see Fig. 3.1.2). We emphasize that no fitting is done on this testing data; instead, we use the fits from the fitting data set to predict the state evolution in the testing data set. Once again, we find that the Lindbladian model’s prediction provides a crude approxima- tion to the evolution. In particular, for jψ 1 (t)i, where the initial qubit state is the excited state, the amplitude damping process dominates, and the Markovian Lindblad model is suf- ficient to describe the dynamics. However, jψ 1 (t)i is an exception. For all other states, the PMME models are far more accurate, as is clear from Fig. 3.2.2 (a2-d4). This suggests that the non-Markovian effects mainly manifest in the evolution of the qubit phase coherence but not in the state populations, so the need for the more complex PMME models arises when dealing with states with coherence in the computational basis (this pointer basis – the ground and excited states – is einselected [64] due to thermal relaxation). Another observa- tion visible from Fig. 3.2.2 (row d) is that the Markovian model’s predictions become more accurate at relatively long evolution times, i.e., Markovian effects become more dominant on a timescale of 100 µs. Overall, as shown in Fig. 3.2.3 and Table 3.2.1, the median and worst-case prediction distances of the PMME models M 1 and M 2 are very close, and substantially better than those of the Lindbladian model M 0 . In particular, consider the box plot in Fig. 3.2.3 reporting the statistics for the trace-norm distance across all sampling points t j and initial states of the fitting and testing datasets respectively. For the testing data, while the Lindblad model M 0 has a median trace-norm distance of 0.12 and worst-case fidelity of 0.18, both the PMME modelsM 1 andM 2 have a median distance of 0.06 and the worst-case fidelity of 0.08. Notably, the distance does increase slightly when going from the fitting data to the testing 37 data, which can be attributed to the qubit’s environment changing during the time between measurements; for example, the qubit relaxation time T 1 fluctuates [ 65]. Still, the PMME models provide a significantly closer correspondence to the fitting data set and more accurate predictions of the testing data sets than the Lindblad model does in both the average and worst cases. In this case the best-fit model for M 2 is approximately equal to M 1 (see App. E), indicating that the simpler model is sufficient to capture the behavior of the fitting data set. However, this does not hold true for all data, as shown in App. E. Figure 3.2.2 clearly shows that these differences in distance between M 0 andM 1,2 come fromM 0 ’s inability to capture oscillations about a non-zero mean in the Bloch vector com- ponentsv x (t) andv y (t), whichM 1 andM 2 do capture. Our results validate that whileM 0 is a good first Markovian approximation, the PMME models account for non-Markovian nuances. 3 Quantifying non-Markovianity Lastly, we test whether the PMME model can capture non-Markovianity during the qubit free evolution. To quantify this, we adopt the measure in Eq. 2.37. Here, the time-integration is over time intervals whereσ is positive, and the maximum is over all pairs of initial states. Thus, this quantity measures the total increase of distinguishability over the whole evolution time. We note that it is not normalized, and hence its actual numerical values are difficult to interpret. Its main significance is in the the fact that a non-zero non-Markovian measure means non-Markovian dynamics, and, as shown in Table 3.2.2, the experimentally obtained values are in fair agreement with the values predicted by our model. The non-Markovianity of the PMME is discussed in [54] using this measure. We performed state tomography on free evolution with the qubit initialized in the follow- ing two pairs of states: fρ 1 (0) = j+ih+j,ρ 2 (0) = jihjg, and fρ 1 (0) = j+iih+ij,ρ 2 (0) = jiihijg, where the state j+i (j +ii) and the state ji (jii) are the eigenstates of σ x 38 0.00 0.50 1.00 D(ρ 1 ,ρ 2 ) a ρ A =|0 fi› 0| ⊗4 c ρ A =|1 fi› 1| ⊗4 e ρ A =| + fi› +| ⊗4 0 30 60 90 Time (μs) -0.10 0.00 dD(ρ 1 ,ρ 2 )/dt N exp =1.06±0.02 N prd M 1 =1.11±0.02 N prd M 2 =1.10±0.02 b 0 30 60 90 Time (μs) N exp =0.18±0.03 N prd M 1 =0.28±0.02 N prd M 2 =0.25±0.01 d 0 30 60 90 Time (μs) N exp =0.13±0.02 N prd M 1 =0.49±0.03 N prd M 2 =0.47±0.03 f Tomography data M 0 M 1 M 2 Figure 3.2.4: Non-Mark o vianit y of qubit free-ev olution dynamics for sp ectator qubits in the ground state (a,b), the excited state (c,d) and thej+i state (e,f ). (a,c,e) The trace-norm distance D(ρ 1 (t),ρ 2 (t)) predicted b y the b est-fit mo dels (solid lines) and exp erimen tally measured b y p er- forming free-ev olution tomograph y with a pair of initial states ρ 1 (0) =j+ih+j and ρ 2 (0) =jihj (grey circles). (b,d,f ) The deriv ativ e σ(t) , defined in Eq. ( 2.35 ), predicted b y the b est-fit mo dels (solid lines), and appro ximated exp erimen tally using forw ard differencing based on the tomogra- ph y data in (a,c,e) (grey circles). (σ y ) corresponding to the eigenvalues +1 and 1, respectively. From the time-series state tomography data, we see how the trace-norm distanceD(Φ t (ρ 1 (0)),Φ t (ρ 2 (0))), starting from D = 1 (maximally distinguishable), evolves as a function of time, as plotted in the top row of Fig. 3.2.4. Using forward differences to approximate the time derivative of our experi- mental data, we then approximate the quantity in Eq. (2.35), plotted in the bottom row of Fig. 3.2.4. By performing a linear interpolation on the discrete samples shown there, we further estimate the observed degree of non-Markovianity as defined in Eq. ( 2.37) (but ex- cluding the maximization it requires). The results we find for the degree of non-Markovianity are summarized in Table 3.2.2. Of course, N prd M 0 0 for the Markovian Lindblad model. The experimental measure is estimated from the state tomography data, while the predicted non-Markovian measure is 39 spec. qubit init. state N exp N prd M 1 N prd M 2 j0i ⊗4 1.060.02 1.110.02 1.100.02 j1i ⊗4 0.180.03 0.280.02 0.250.01 j+i ⊗4 0.130.02 0.490.03 0.470.03 T able 3.2.2: Degree of non-Mark o vianit y for three differen t initial states of the sp ectator qubits (column 1) as computed from the exp erimen tal data (column 2) and the t w o non-Mark o vian mo dels (columns 3 and 4). calculated exactly from our dynamical models, since they provide a continuous description of the qubit state evolution. We train these models on a new fitting data set with the same initial state jψ 0 (0)i as the prior fitting data (Fig. 3.1.2); the new fitting data was taken in the same batch of jobs as the trace-norm distance data in order to minimize the time delay between them, thus reducing the effect of systematic errors due to differences between batches. As seen in Table 3.2.2, the two PMME models converge to almost the same quantity in their predictions, despite the different forms of their kernels. The tomography experiments confirm that there is indeed an increase of distinguishability between quantum states during evolution. The predictions of the PMME models adequately match the observed quantity σ(t j ), with some deviations which are due to numerical errors arising from using the finite difference formula to approximate Eq. ( 2.35), and due to the run-to-run system fluctuations of the quantum processor [66]. 4 PMME model construction result for different spectator qubit states In this section we present data supplementing the results of the PMME model construction reported in Fig. 3.2.2, for the following three initial states of the spectator qubits: ground state j0i, the excited state j1i, and the j+i state. This data was also used to calculate the degree of non-Markovianity reported in Fig. 3.2.4, which used different initial states of the main qubit. Figures 3.2.5, 3.2.6 and 3.2.7 show the fitting data used to construct the 40 -1.0 0.0 1.0 v x a0 |ψ 0 (0) fi a1 |ψ +x (0) fi a2 |ψ −x (0) fi a3 |ψ +y (0) fi a4 |ψ −y (0) fi -1.0 0.0 1.0 v y b0 b1 b2 b3 b4 0.0 0.5 v z c0 c1 c2 c3 c4 0 30 60 90 0.0 0.2 D(ˆ ρ exp ,ρ prd ) d0 0 30 60 90 d1 0 30 60 90 time (μs) d2 0 30 60 90 d3 0 30 60 90 d4 fitting data testing data tomography data M 0 M 1 M 2 Figure 3.2.5: The tomograph y data set and the corresp onding mo del predictions used to calcu- late the degree of non-Mark o vianit y in Fig. 3.2.4 (a,b) w hen the sp ectator qubits are initialized in the ground state. (a0-c0) The fitting data set with the qubit initialized in jψ 0 (0)i that is used to find the b est-fit Lindblad mo del M 0 (orange lines), the PMME mo del with the t yp e 1 k ernel M 1 , and the PMME mo del with the t yp e 2 k ernelM 2 . (a1-c2) The tomograph y data set with qubit initialized injψ +x i ,jψ −x i and the prediction from the b est-fit mo dels from the fitting data set in (a0-c0). The data sets are used to ev aluate the degree of non-Mark o vianit y in Fig. 3.2.4 (a,b). (a3-c4) The tomograph y data set with the qubit initialized injψ +y i ,jψ −y i , a nd the pre- diction from the b est-fit mo dels from the fitting data set in (a0-c0). The data sets are used to ev aluate the degree of non-Mark o vianit y in Fig. 3.2.10 (a,b). (d0-d4) The distance b et w een the tomographically constructed state and the state predicted b y the b est-fit mo dels. 41 -1.0 0.0 1.0 v x a0 |ψ 0 (0) fi a1 |ψ +x (0) fi a2 |ψ −x (0) fi a3 |ψ +y (0) fi a4 |ψ −y (0) fi -1.0 0.0 1.0 v y b0 b1 b2 b3 b4 0.0 0.5 v z c0 c1 c2 c3 c4 0 30 60 90 0.0 0.1 D(ˆ ρ exp ,ρ prd ) d0 0 30 60 90 d1 0 30 60 90 time (μs) d2 0 30 60 90 d3 0 30 60 90 d4 fitting data testing data tomography data M 0 M 1 M 2 Figure 3.2.6: The tomograph y data set and the corresp onding mo del predictions used to calcu- late the degree of non-Mark o vianit y in Fig. 3.2.4 (c,d) w hen the sp ectator qubits are initialized in the excited state. (a0-c0) The fitting data set with the qubit initialized in jψ 0 (0)i that is used to find the b est-fit Lindblad mo del M 0 (orange lines), the PMME mo del with the t yp e 1 k ernel M 1 , and the PMME mo del with the t yp e 2 k ernelM 2 . (a1-c2) The tomograph y data set with qubit initialized injψ +x i ,jψ −x i and the prediction from the b est-fit mo dels from the fitting data set in (a0-c0). The data sets are used to ev aluate the degree of non-Mark o vianit y in Fig. 3.2.4 (c,d). (a3-c4) The tomograph y data set with the qubit initialized injψ +y i ,jψ −y i , and the pre- diction from the b est-fit mo dels from the fitting data set in (a0-c0). The data sets are used to ev aluate the degree of non-Mark o vianit y in Fig. 3.2.10 (c,d). (d0-d4) The distance b et w een the tomographically constructed state and the state predicted b y the b est-fit mo dels. 42 -1.0 0.0 1.0 v x a0 |ψ 0 (0) fi a1 |ψ +x (0) fi a2 |ψ −x (0) fi a3 |ψ +y (0) fi a4 |ψ −y (0) fi -1.0 0.0 1.0 v y b0 b1 b2 b3 b4 0.0 0.5 v z c0 c1 c2 c3 c4 0 30 60 90 0.0 0.1 0.2 D(ˆ ρ exp ,ρ prd ) d0 0 30 60 90 d1 0 30 60 90 time (μs) d2 0 30 60 90 d3 0 30 60 90 d4 fitting data testing data tomography data M 0 M 1 M 2 Figure 3.2.7: The tomograph y data set and the corresp onding mo del predictions used to calcu- late the degree of non-Mark o vianit y in Fig. 3.2.4 (e,f ) when the sp ectator qubits are initialized in thej+i state. (a0-c0) The fitting data set with the qubit initialized in jψ 0 (0)i that is used to find the b est-fit Lindblad mo del M 0 (orange lines), the PMME mo del with the t yp e 1 k ernelM 1 , and the PMME mo del with the t yp e 2 k ernelM 2 . (a1-c2) The tomograph y d ata set with qubit initialized injψ +x i ,jψ −x i and the prediction from the b est-fit mo dels from the fitting data s et in (a0-c0). The data sets are used to ev aluate the degree of non-Mark o vianit y in Fig. 3.2.4 (e,f ). (a3-c4) The tomograph y data set with the qubit initialized injψ +y i ,jψ −y i , and the prediction from the b est-fit mo dels from the fitting data set in (a0-c0). The data sets are used to ev aluate the degree of non-Mark o vianit y in Fig. 3.2.10 (e,f ). (d0-d4) The distance b et w een the tomograph- ically constructed state and the state predicted b y the b est-fit mo dels. 43 model and the testing data to validate it. The initial states in the testing data sets are fjψ +x i =j+i,jψ −x i =ji,jψ +y i =j+ii,jψ −y i =jiig and they are used to evaluate the degree of non-Markovianity as in Eq. (2.35), plotted in Figs. 3.2.4 and 3.2.10. The initial state pairs ρ 1 (0) =j+ih+j and ρ 2 (0) =jihj (or ρ 1 (0) =j+iih+ij and ρ 2 (0) =jiihij) are optimal pairs such that they feature a maximal flow of information from the environment back to the system [67]. The yellow, blue, and red solid lines in Fig. 3.2.5, Fig. 3.2.6 and Fig. 3.2.7 represent the constructed models M 0 , M 1 , and M 2 , respectively, with their best-fit model parameters summarized in Table E.0.1. On the fitting data set, we find that the Lindblad model M 0 does not adequately describe the data, while the PMME modelsM 1 andM 2 describe the data accurately. The kernels of the constructed PMME models are plotted in Fig. 3.2.9. Although M 2 has a more elaborate kernel with more free parameters, it does not necessarily provide a better fit to the data; M 1 and M 2 result in similar fits, which suggests that we do not need a very complicated kernel to significantly improve upon the standard Lindblad model. On the testing data set, the constructed models provide a qualitatively adequate prediction for different initial states, but deviations do arise relative to the experimentally constructed states (see Fig. 3.2.5–3.2.7 columns 1, 2, and 4). Due to system fluctuations, which the PMME models cannot capture, the model constructed from the fitting data set may lose some of its predictive power for the testing data set. Examples of such system fluctuations include fluctuations of qubit T 1 relaxation time and 1/f noise in the qubit frequency. As further evidence of such fluctuations, the data shown in Fig. 3.2.2 (a0-d0) and Fig. 3.2.5 (a0- d0) were taken under nominally identical conditions but nonetheless show different behavior. These data were taken days apart, during which time the system parameters drifted. Nonetheless, compared with the Lindblad model, the PMME models provide higher levels of agreement for the fitting data sets and a more accurate prediction for the testing data sets in all spectator qubit configurations. This can be seen in the bottom row of Figs. 3.2.5-3.2.7, 44 0.0 0.1 0.2 0.3 a ρ A (0) =|0 fi› 0| ⊗4 −16 −15 −14 0.0 0.1 0.2 0.3 distance b ρ A (0) =|1 fi› 1| ⊗4 −17 −16 −15 fitting AIC M 0 M 1 M 2 M 0 M 1 M 2 0.0 0.1 0.2 0.3 c ρ A (0) =| + fi› +| ⊗4 −16.50 −16.25 −16.00 −15.75 fitting data testing data Figure 3.2.8: Summary of the results of running the PMME tomograph y proto col on the IBMQE pro cessor ibmq_athens, sho wing ho w w ell the mo del candidates describ e the fitting data set and the testing data set in Fig. 3.2.5 for the sp ectator qubits in the ground state (a), for those in Fig. 3.2.6 for the sp ectator qubits in the excited state (b) and for those in Fig. 3.2.7 for the sp ectator qubits in thej+i state (c). The b o x plots sho w the trace norm distance (see text) and the median is rep orted as the middle v alue offD j g . The lo w er line of the b o x corresp onds to the lo w er quartile of the data (25th p ercen tile, Q1), and the upp er line of the b o x corresp onds to the upp er quartile of the data (75th p ercen tile, Q3). Let IQR denote the in terquartile range: IQR = Q3-Q1. The outliers, plotted in circles, are the data outside the range (Q1-1.5*IQR, Q3+1.5*IQR). 45 0.00 0.50 1.00 Kernel k(t) a c 0 30 60 90 Time (μs) 0.00 0.50 1.00 Kernel k(t) b 0 30 60 90 Time (μs) d M 1 M 2 Figure 3.2.9: The k ernels in the constructed mo delsM 1 andM 2 using the fitting datasets in Fig. 3.2.2 (a), Fig. 3.2.5 (b), Fig. 3.2.6 (c), and Fig. 3.2.7 (d). The shaded regions denote the 95% confidence region of the k ernel function due to the uncertain t y in the b est-fit k ernel parame- ters in T able E.0.1 . and is summarized in Fig. 3.2.8 which compares the models using the Akaike Information Criterion (AIC) and the trace distance metric discussed below. To compare the goodness of the fit of the models and the simplest model that best describes the fitting data set, we use the Akaike information Criterion (AIC) [ 61] which is defined as: Ξ =2ln ˆ L( ˆ θjD) +2p , (3.14) where ˆ L denotes the likelihood function and p is the number of free model parameters. The second term in the AIC (2p) is called the bias term and it penalizes the models with higher complexity. The AIC can be interpreted as the “information loss” (in the Kullback- Leibler divergence sense) of using some candidate model to approximate the “true” model. Akaike showed that the maximized log-likelihood in the first term of Eq. ( 3.14) is a biased estimator, and that under certain assumptions, the bias correction approximately equals p. The calculation of the first term in AIC depends strongly on the sample size used, and the bias-correction term becomes exact when the sample size diverges. For these reasons, the 46 numerical value of AIC has no intrinsic significance as such. With these caveats in mind, we can use the AIC metric to find a model that accurately describes the data and at the same time avoids overfitting, as lower (more negative) values indicate a better model. 0.00 0.50 1.00 D(ρ 1 ,ρ 2 ) a ρ A =|0 fi› 0| ⊗4 c ρ A =|1 fi› 1| ⊗4 e ρ A =| + fi› +| ⊗4 0 30 60 90 Time (μs) -0.10 0.00 dD(ρ 1 ,ρ 2 )/dt N exp =0.60±0.02 N prd M 1 =1.11±0.02 N prd M 2 =1.10±0.02 b 0 30 60 90 Time (μs) N exp =0.25±0.02 N prd M 1 =0.28±0.02 N prd M 2 =0.25±0.01 d 0 30 60 90 Time (μs) N exp =0.18±0.03 N prd M 1 =0.49±0.03 N prd M 2 =0.47±0.03 f Tomography data M 0 M 1 M 2 Figure 3.2.10: Non-Mark o vianit y of qubit free-ev olution dynamics for sp ectator qubits in the ground state (a,b), the excited state (c,d) and thej+i state (e,f ). (a,c,e) The trace-norm dis- tanceD(ρ 1 (t),ρ 2 (t)) predicted b y the b est-fit mo dels (solid lines) and exp erimen tally measured b y p erforming free-ev olution tomograph y with a pair of initial states ρ 1 (0) = j+iih+ij and ρ 2 (0) =jiihij (grey circles). (b,d,f ) The deriv ativ e σ(t) , de fined in Eq. ( 2.35 ), predicted b y the b est-fit mo dels (solid lines), and appro ximated exp erimen tally using forw ard differencing based on the tomograph y data in (a,c,d) (grey circles). The likelihood function ˆ L is defined over the observed dataset, and quantifies the likelihood of observing the data set as a function of the model parameters ⃗ θ. It measures how well the data supports that particular choice of parameters. Since each tomography sample ˆ ρ is independently drawn, the likelihood ˆ L is a product of the probabilities of observing each data sample given a probability distribution and distribution parameters: ˆ L( ⃗ θjD) = Y i Y k p k (t i ; ⃗ θ) , (3.15) 47 wherep k (t i ; ⃗ θ) is the probability of observing the data point in measurement basisk at time t i given model parameters ⃗ θ. We report the AIC values computed in this manner for the fitting data using the three models in Fig. 3.2.8 (purple squares), for three different initial states of the ancilla qubits. However, as mentioned above, the numerical value of the AIC is not intrinsically meaningful. In practice, it is convenient to calibrate AICs with respect to the minimum AIC value among all models: ∆ i =I i min i I i (3.16) where min i I i is the AIC value of the best model in the set. For the data in Fig. ??, the minimum is achieved for the M 1 model. The AIC difference ∆ i estimates the information loss when using model i rather than the estimated best model. Hence, the larger ∆ i , the less plausible is modeli. Some guidelines for the interpretation of AIC difference in the case of nested models are given in [68], as summarized in Table 3.2.3. Given the much larger ∆ 0 =I 0 I 1 value, we find that the data in Fig. 3.2.2 is considerably less in favor of the Lindblad modelM 0 , despite the fact that it has the smallest number of free parameters. Another metric we use here for comparison is the trace-norm distance between the ex- perimentally constructed state and the model predicted state during its evolution. Let D k j denote the distance between the experimentally constructed state ˆ ρ exp k at timet j with initial state ρ k (0) = jψ k (0)ihψ k (0)j and that predicted by the model ρ prd k (t j ), i.e., D j k = D(ˆ ρ exp k , ρ prd k (t j )), the values reported in the box plot in Fig. 3.2.8 are averaged over the test data set with different initial states: D j = 1 4 P k∈{1,2,3,4} D k j . After doing this averaging, we arrive at an array fD j g j=24 j=1 , corresponding to the trace distance, again, averaged over four different initial states in the testing data set, between the experimentally constructed states and the model’s predicted states at 24 sampled time points during the evolution. Finally, using the tomography data with the qubit initialized in ρ 0 (0) = j+iih+ij and 48 ∆ i level of empirical support for model i models ∆ i =I i I 1 0-2 substantial M 1 0 4-7 considerably less M 2 0.20 >10 essentially none M 0 9.11 T able 3.2.3: A heuristic in terpretation of AIC differences ∆ i rep orted in Fig. 3.2.3 . The larger ∆ i is, the less plausible it is that the mo delM i is the b est mo del. ρ 2 (0) =jiihij, we reevaluate the degree of non-Markovianity for different spectator qubit states in Fig. 3.2.10 according to Eqs. (2.35) and (2.37). The non-monotonic decay in the trace distance D and the estimated non-Markovian measure N in Fig. 3.2.10 agrees well with those in Fig. 3.2.4, and serves as a supplementary quantitative demonstration of non- Markovian effects present in the device. The degree of non-Markovianity N prd M 1 and N prd M 2 calculated from the constructed PMME models agrees well with that from the experimental dataN exp , showing that the PMME models have the ability to quantitatively describe and predict the degree of non-Markovianity of the dynamics during the qubit free evolution. 3.3 Discussion and Conclusion We have developed and implemented a procedure to fit a family of phenomenological quan- tum master equations to time-series state tomography data. We demonstrated this method by characterizing the free evolution of a single qubit on an IBMQE processor. From the constructed models, we conclude that the qubit Hamiltonian accounts for a residual longi- tudinal field due to the crosstalk with the neighboring qubits, and found the Lindblad rates that correspond to dephasing, spontaneous emission, and thermal excitation. However, a purely Markovian model provided a relatively poor fit to the data. We thus constructed post- Markovian master equation (PMME) models that contain a phenomenological bath memory kernel which account for the non-Markovianity of the dynamics – something a Markovian Lindblad model cannot do. These PMME models provide a closer fit to the tomography data, and also much more accurately predict the future dynamics for different and new 49 initial states that the models were not already fitted to. Our PMME construction procedure is an alternative to other methods characterizing qubit noise processes such as process tomography [69–73], machine learning (ML) [74–78], and shadow tomography [79–81]. Compared to the first of these methods, our approach is less demanding in terms of both data collection and analysis, since it relies on state tomog- raphy, which generally (but not always [72]) requires a number of measurements which is quadratically smaller in the Hilbert space dimension [82]. The ML methods usually require a large training data set for the solution to converge since the ML model is usually over- parametrized. In contrast, at least in the single-qubit setting, we have demonstrated that our method requires doing state tomography with only one initial state at multiple time points. Another advantage of our method is that the evaluation of the cost function in our problem is straightforward because the PMME is analytically solvable (see App. D). In con- trast, the noise models that have been considered in ML so far are numerically challenging because they involve solving the Nakajima–Zwanzig master equation, whose complexity is proportional to the kernel’s length [75], or solving a stochastic master equation [76]. In regards to shadow tomography methods, the comparison is less direct since the goal of these methods is not to reconstruct the dynamics at every time point, but rather to use a small set of measurements to predict a much larger set of specific observables. A similar approach to ours for constructing a dynamical model of the noise channel using the Markovian Lindblad master equation was considered in [83]. Here, in order to go beyond the Markovian approximation, a model based on a class of time-convolution-less master equations is considered, and a damping rate function describes the non-Markovian dynamics of the system at the sampling time point [84]. This method provides a discrete description of the noise dynamics but does not directly provide physical insight into quantifying the timescale of the memory effect. Likewise, a more sophisticated and systematic approach, the process tensor (PT) framework has been proposed [85] and experimentally demonstrated [86, 87] to characterize non-Markovian dynamics on actual quantum processors. The PT model, 50 once fully characterized, can be used to interpolate the dynamics between discrete times due to the containment property of the PT map. However, its construction and the interpretation of the memory effect are fairly involved. Our method provides a continuous dynamical description of the channel beyond the Markov approximation, and does not require full process tomography to construct the PMME model. However, a disadvantage of our method is that it requires one to construct a paramet- ric model; thus, it does not account for the most general noise process. Moreover, because of the specific parametrization that the PMME demands, the resulting optimization problem is no longer convex, i.e., a unique global minimum of the optimization problem is not guar- anteed. Additionally, our method requires the consideration of a hierarchy of kernels to find a model that is complex enough to describe the data accurately. But since the PMME is straightforward to solve, model estimation under different kernels is straightforward so long as there is a model selection metric under which the best model among those candidates can be selected. A promising future approach for retaining the physical interpretability and ana- lytical solvability that come with using the PMME is the use of machine learning, especially in light of the recent development of neural ODE solvers [88], in order to avoid overfitting and structural errors of the PMME model. This work is a proof-of-principle demonstration of the PMME tomography protocol on a fixed-frequency qubit. For future work, we are interested in applying our protocol to charac- terize the dynamics of a frequency-tunable qubit where dephasing noise is more pronounced, and the qubit’s sensitivity to dephasing noise can be varied as a function of the qubit fre- quency. An extension to the multi-qubit case is also a natural next step. This will require a convergence analysis to decide how the fitting data set scales as the number of qubits increases. In this work, we have focused on the free evolution channel of a single IBMQE qubit, which we have shown to be highly susceptible to non-Markovian noise. It would be interesting to extend the protocol to characterize non-Markovian effects during computa- tion, in order to understand whether such effects are significant beyond qubit idle times. By 51 explicitly including non-Markovianity in the dynamical characterization and modeling, we expect that it will become possible to use these realistic noise models to improve and tailor error suppression and correction techniques, and ultimately realize high fidelity quantum control and computation. 52 Chapter 4: Theory of bath engineering via ca vit y co oling Superconducting qubits inevitably interact with their environmental degrees of freedom, leading to the decoherence of qubit states. One route to preserve qubit coherence is to eliminate its dissipation channel to the environment. Paradoxically, by engineering a qubit’s coupling to the environment, it is possible to take advantage of a dissipation process for a quantum information task, such as preparing a target state or preserving a coherence state. In superconducting qubits, bath engineering, in general, involves controllably modifying a qubit’s microwave environment. However, the additional couplings may also hinder the coherence of some other qubit states or other logical operations, so it is often desirable to realize a dissipative coupling that can be turned on and off on demand. As discussed in Chapter 2 Sec. 3, the photon number fluctuations in the readout cavity can cause qubit frequency fluctuation via their dispersive coupling, resulting in qubit dephasing. Those photons can either be of thermal origin or can be residue photons from a previous readout if the wait time after a readout operation is not long enough for the cavity to thermalize completely. A residue thermal photon population ofn th can cause qubit dephasing at rate [89, 90]: Γ ϕ = κ 2 Re 2 4 s 1+ iχ κ 2 + 4iχn th κ 1 3 5 (4.1) whereκ is the cavity decay rate andχ is the qubit-cavity dispersive shift. Besides removing thermal photons to eliminate photon-induced dephasing, it is also desirable to speed up the photon depletion process to realize faster cavity reset. This is valuable, especially in run- ning quantum circuits with many mid-circuit measurements, for example, in quantum error correction code, where repeated syndrome measurements are required per logical operation. In this chapter, we propose such a protocol to realize active, switchable, on-demand dis- sipation on a target mode. We achieve this by coupling the cavity to a “dissipator,” an auxiliary resonant mode deliberately designed to be lossy with a tunable coupling. When the target mode is a readout cavity, the protocol can be used to speed up the cavity reset, and the dissipator acts as a refrigerator that “cools” the cavity down to lower occupancy and, thus, a lower temperature. When the target state is a qubit mode, our protocol can be used to speed up qubit reset. This is valuable because, for a long-lived qubit, if left to thermalize with its environment, it can take hundreds of microseconds for the thermalization process to take a qubit to its ground state. In the following, I will explain the mechanism of how to use the dissipator for cavity cooling. I introduce parametric coupling and its perturbation treatment and exact treatment in the interaction picture. I then explain the cooling mechanism and the transient effect under an effective model. This cooling mechanism naturally applies to the case where the target mode is a logical qubit. 4.1 Parametric driving We use parametric driving to allow for energy transfer between the subsystems. Sideband transitions under parametric driving have been shown to generate controllable interaction between superconducting qubits and cavities. The parametric drive can be imple- mented via controlled qubit frequency oscillations on a tunable qubit using a flux bias line. In the context of cavity reset, we consider a system where a readout cavity is capacitively coupled to a dissipator, as shown in Fig. 4.1.1. We consider the following Hamiltonian, where the dissipator is modeled as a frequency- 54 Figure 4.1.1: Illustration of a readout ca vit y capacitiv ely coupled to a frequency-tunable “dissi- pator” with coupling strength g . The ca vit y is mo deled as a quan tum harmonic oscillator. The dissipator is mo deled as a qubit whose frequency is parametrically mo dulated with H p . tunable qubit 1 : H(t) =ω r a † a ω d (t) 2 σ z +g a † +a (σ + +σ − ) (4.2) with ω r the cavity frequency, g the coupling strength, σ z the usual Pauli operator for the qubit, σ + (σ − ) the raising (lowering) operator for the qubit, and a (a † ) the annihilation (creation) operator for the cavity. The qubit transition frequency is assumed to be modulated periodically using an external flux control, and its average value ¯ ω d is chosen such that j∆j =j¯ ω d ω r jg. The parametric coupling can be turned on by periodically modulating the qubit transition frequency such that one of the qubit’s frequency control sidebands, which are centered on ¯ ω d and spaced by the drive frequency ω p , is in resonance with the cavity. Similar systems have been extensively studied for parametrically-activated entangle gates [91–95]. 1 Analytical solution from the perturbation theory In this dispersive regime, the cavity-qubit interaction represented by the last term of Eq. (4.2) only acts perturbatively. In this section, we derive an analytic expression for the Rabi rate between the cavity-qubit system using perturbation theory. 1 Note that ℏ = 1 throughout the deriv ation in this section. 55 Figure 4.1.2: Energy lev els of the coupled ca vit y-dissipator system: the parametric driv e turns the first-order sideband transitions and enables energy sw apping in the subspace that preserv es one-photon excitation, sho wn b y the blue arro ws. The excitation then quic kly dissipates due to the large dissipator deca y rate. Denote the unperturbed system Hamiltonian as H 0 : H 0 =ω r a † a+ 1 2 ω d σ z (4.3) The Hamiltonian is diagonal in the cavity-qubit eigenbasisj0gi,j0ei,j1gi,j1ei,…,jngi,jnei as shown in the energy diagram in Fig. 4.1.2. The interaction term between the cavity and the qubit H ′ is treated as a perturbation to the original unperturbed system and hybridizes with cavity states with the qubit states: H ′ =g a † +a (σ + +σ − ) (4.4) which can also be written in the basis spanned by the bare cavity-qubit eigenstate. We will 56 restrict ourselves in a subspace that preserves one-photon excitation, reducing the Hamilto- nian to an effective 2 2 matrix 2 : H eff = 2 6 4 nω r + 1 2 ω d p n+1g p n+1g (n+1)ω r 1 2 ω d 3 7 5 (4.5) Now we can follow the time-independent perturbation theory to find the perturbed wave- functions and energies of the system. We only consider the first-order contribution of the perturbation to the system, i.e., E n =E 0 n +E ′ n (4.6a) jψ n i = ψ 0 n +jψ ′ n i (4.6b) The perturbation correction to the system energies and wavefunctions is estimated via: E ′ n = ψ 0 n jH ′ jψ 0 n (4.7a) jψ ′ n i = X m̸=n hψ 0 m jH ′ jψ 0 n i E 0 n E 0 m ψ 0 m (4.7b) Substitute Eq. (4.4) into Eq. (4.7) and use the unperturbed eigenbasis fjn,gi,jn,ei,jn + 1,gi,jn+1,eig; we then have E ′ n+1,g = (n+1)ω r ω d 2 , (4.8a) ψ ′ n+1,g =jn+1,gi+ p n+1 g ∆ j1ei (4.8b) E ′ n,e =nω r + ω d 2 , (4.8c) jψ ′ ne i =jnei p n+1 g ∆ jn+1,gi (4.8d) where ∆ = ¯ ω d ω r is the detuning between the cavity and the average dissipator frequency 2 This is equiv alen t to the rotating-w a v e appr o ximation and neglecting the fast oscillating terms. 57 under modulation. We now introduce the frequency modulation drive on the dissipator: H ′′ =δsin(ω p t)σ z =δsin(ω p t)(jei(ejjg)hgj) (4.9) whereδ is the drive amplitude andω p is the drive frequency. The drive term can be treated as another perturbation to the cavity-qubit system. Since we are interested in the transition jn+1,gi$jn,ei, we will evaluate its Rabi frequency: Ω Rabi = 1 2 q (ω p ∆) 2 +jδω n+1,g→n,e j 2 (4.10) where the transition amplitude is evaluated via: δω n+1,g→n,e =hψ n+1,g jV ′′ jψ n,e i (4.11a) = p n+1gδ ∆ (4.11b) whereV ′′ =δ is the time-independent part of the driveH ′′ . When the drive is on resonance ω p = ∆, the Rabi frequency is: Ω R = δω n+1,g→n,e 2 = p n+1gδ 2∆ (4.12) The transition probability ofjn+1,gi!jn,ei follows: P(jn+1,gi$jn,ei) = jδj 2 Ω 2 R sin 2 (Ω R t) (4.13) From Eq. (4.13), we see that, a full population inversion betweenj0,n+1i andj1,ni at Rabi frequency 2Ω R , which is first order in g/∆. 58 2 Effective coupling strength in the interaction picture In this section, we move to the frame that diagonalizes the unperturbed Hamiltonian and shows that the role of the drive is to introduce multiple effective cavity-dissipator interactions. Choosing the appropriate drive frequency can turn these interactions on and off. We will first consider the cavity and the dissipator the same as in Fig. 4.1.1. Then we will consider a more realistic case where a filter mode is introduced to modify the dissipator’s coupling to the external environment, characterized by the dissipator decay rate κ. Cavity-dissipator We consider the evolution of the state of the cavity under the sideband transition generated by the drive on the dissipator. The analysis in this section shows that the sideband rate under the drive is of the first order in g, confirming the result in the perturbation calculation. We first move to a rotating frame defined by U(t) = exp iω r a † at exp i Z t o ω d (t ′ )dt ′ σ z 2 (4.14) whereω d (t) = ¯ ω d +δsin(ω p t) is the dissipator frequency under the drive modulation. In this rotating frame, the effective Hamiltonian can be calculated from: ˜ H =UHU † +iU † ˙ U (4.15) where ˜ H effectively generates the evolution for the state ˜ jψi =U(t)jψi under the Schrödinger equation ˜ H ˜ jψi =i∂ ˜ jψi/∂t. Substituting Eq. (4.14) into Eq. (4.15), we get ˜ H =gUa † σ − U † +h.c. (4.16) which allows us to evaluate the effective cavity-dissipator coupling in the rotating frame. 59 Denote θ r ω r t, θ d (t) = 1 2 R t 0 dt ′ ω d (t ′ ), one can show: exp iθ r a † a a † exp iθ r a † a =a † exp(iθ r ) (4.17) Similarly, exp(iθ d σ z )σ − exp(iθ d σ z ) =σ − e −2iθ d (4.18) Integrate for θ d (t), Eq. (4.16) becomes: ˜ H =ga † σ − e − iδ ωp cosωpt +h.c. (4.19) We can use the Jacobi-Anger expansion below : e izcosθ ∞ X n=−∞ i n J n (z)e inθ (4.20) where J n (z) are Bessel functions of the first kind, we arrive at the final expression: ˜ H =ga † σ − J 0 δ ω p e −i∆t +h.c. (4.21a) +ga † σ − ∞ X 1 (i) n J n δ ω p e −i(nωp−∆)t +h.c.. (4.21b) where ∆ = ¯ ω d ω r . As shown in Eq. (4.21), then-th sideband transition can be turned on if we choose a drive frequency at ω p = ∆/n; the effective qubit-cavity coupling strength is: g n =gJ n ( δ ω p ), (4.22) which corresponds to an n-photon transition process. 60 Figure 4.1.3: Setup for ca vit y co oling with a filter mo de. The filter mo de is in tro duced to mo d- ify the dissipator’s coupling to the lossy bath, hence determining the dissipator deca y rate κ . In the ca vit y-dissipator-filter system, b y c ho osing the appropriate driv e frequency , the first-order sideband transition can either turn on (a) the ca vit y-dissipator transition or (b) the ca vit y-filter transition. Cavity-filter system In the previous section, we discussed the parametric coupling between the cavity and dissi- pator modes, as shown in Fig. 4.1.3(a). That is a simplified model. In the actual design, the dissipator mode is made lossy by capacitively coupled to a lossy environment mediated by a Purcell filter. The Purcell filter introduces more transition pathways, and one can paramet- rically drive the transition between the cavity and the filter, as shown in Fig. 4.1.3(b). The role of the dissipator is to introduce parametric modulation. Intuitively, the effective cou- pling between the cavity and the filter will be smaller compared to the setup in Fig. 4.1.3(a), but the effective loss rate to the environment is larger. Similar to the derivation in Sec. 2, the total Hamiltonian can be written as: H =ω r a † a 1 2 ω d (t)σ z +ω f b † b (4.23) +g c (a † σ − +aσ + )+g f (σ − b † +σ + b) (4.24) where g c (g f ) is the coupling strength between the qubit and the cavity (filter). Move to a 61 rotating frame defined by: U = exp iω r a † a exp iω f b † b exp i 2 Z t 0 ω d (t ′ )dt ′ σ z (4.25) Using Eq. (4.15), the effective Hamiltonian in the interaction picture is: ˜ H =g c U a † σ − +aσ + U † +g f U ω f σ − b † +σ + b U † (4.26) =g c a † σ − ∞ X n=−∞ J n δ ω d i n e −i(ωr− ¯ ω d −nω d )t +h.c. (4.27) +g f σ − b † ∞ X n=−∞ J n δ ω d i n e −i(ω f − ¯ ω d −nω d)t +h.c. (4.28) Denote ˜ H(t) = P n ˜ H n (t) where ˜ H n (t) =J n δ ω d i n a † e −iωrt +b † e −iω f t σ − e i(¯ ω d +nωp)t +h.c. (4.29) Following the derivation in Ref. [95] for tunable coupler in the dispersive regime, we can obtain the effective Hamiltonian of the qubit-filter system when the dissipator-cavity de- tuning and the dissipator-filter detuning are much larger than their coupling strengths. We can eliminate the dissipator in the effective Hamiltonian using a Magnus expansion of the Hamiltonian in the interaction picture. The effective Hamiltonian in second order is: ˜ H eff (t) = ˜ H(t) i 2 Z t 0 dt ′ h ˜ H(t), ˜ H(t ′ ) i (4.30) = X n H n (t) i 2 X n,n ′ Z t 0 dt ′ [H n (t),H n ′(t ′ )]. (4.31) We keep the terms that leave the dissipator in its ground state and evaluate the direct cou- pling between cavity and filter; we can obtain the effective coupling strength corresponding to the cavity-filter coupling of the type a † b+ab † . For a comprehensive analysis under a more general setting which includes qubit anharmonicity and any-to-any static coupling between 62 three subsystems, please refer to Ref. [94]. 4.2 Cooling mechanism The cooling occurs in the following way, the drive (work) mode can swap one photon from the cavity/qubit (target) to the dissipator/filter mode. Since the dissipator/filter is coupled to a lossy bath, that photon excitation is quickly dissipated. In the limit where the dissi- pator/filter decay rate is way larger than the Rabi swap rate, the process can result in the cooling of the target as long as the dissipator/filter has a lower average excitation number than the target. We can analyze the cooling effect using the effective Hamiltonian in the basis spanned byfj0,ei,j1,gig, similar to the calculation in Ref. [96]. The cavity reset pro- tocol can be modeled by a non-Hermitian Hamiltonian accounting for the loss through the dissipator or the filter: H eff = 2 6 4 iκ/2 g n g n 0 3 7 5 (4.32) where g n = gJ n δ ω d is the effective coupling strength between cavity-dissipator, κ d is the dissipator decay rate, which provides a loss channel for the excitation from the cavity to decay. The population evolution of statesjg,1i andje,0i can be evaluated by: P ψ 0 →ψ (t) =jhψjexp(iH eff t)jψ 0 ij 2 (4.33) where ψ 0 is the initial state, and ψ is the final state. The real parts of the eigenvaluefλ k g describe the oscillatory evolution, and the imaginary part of the eigenvaluesfλ k g describes the decay process. The Hamiltonian has eigenvalues f 1 4 (iκ p 16g 2 κ 2 ), 1 4 (iκ+ p 16g 2 κ 2 )g. Depending on the relative rate between the Rabi oscillation determined by the effective coupling strength jg n j and the decay process, the system dynamics corresponds to one of the following three regimes: 63 1. ifjg n j =κ/4, a critically damped regime where the damping is just enough to prevent the system from oscillating, 2. if jg n j > κ/4, an underdamped regime where the system decays to equilibrium with oscillations 3. if jg n j < κ/4, an overdamped regime where the system does not exhibit oscillations, but the decay to equilibrium is slower than in a critically damped system We are interested in the case with ψ 0 =j1,gi for cavity reset purposes. The population of the 1-photon state is p n=1 =P |g,1⟩→|g,1⟩ (t) = 8 > > > > < > > > > : e − κ t 2 κt 4 +1 2 jg n j =κ/4 e − κt 2 cos(Mt)+ κ 4M sin(Mt) 2 jg n j>κ/4,M = p 16|gn| 2 −κ 2 4 e − κt 2 cosh(Mt)+ κ 4M sinh(Mt) 2 jg n j<κ/4,M = p κ 2 −16|gn| 2 4 (4.34) In the critically damped regime, the cavity excitation is removed from statejg,1i with a rate κ/2 without oscillations facilitated by the parametric drive. The parametric drive on the dissipator serves as a pump to remove excitation from a target mode, effectively cooling the target temperature. Next, let’s consider a toy model, depicted in Fig. 4.2.1, to derive the steady-state temperature at which the target mode that can be cooled. A similar system has been consider in Ref. [97]. The effective Hamiltonian of the target-dissipator system can be written in a general form: H eff =H target +H diss +H int , (4.35) which is written in the rotating frame with respect to the parametric drive, and the effective interaction Hamiltonian is given in Eq. (4.21). For simplicity, we consider the lowest three levels of the target-dissipator systemfjg,0i,je,0i,jg,1ig, wherejgi (jei) denotes the ground 64 Figure 4.2.1: Deca y diagrams for (a) w eak and (b) strong coupling regimes in the rotating frame of the parametric driv e. The target mo de state has its in trinsic temp erature T target due to the equilibrium of the relaxation pro cessjei!jgi with rate γ − and the excitation pro cessjgi!jei with rate γ + . (a) Whenjg n j/κ d 1 , theje,0i state can deca y bac k to the ground statejg,0i through a t w o-step pro cessje0i ! jg1i ! jg0i whic h is limited b y the slo w er pro cess of the t w o. (b) Wh enjg n j/κ d 1 , the sw ap rate is strong enough to giv e rise to a coheren t oscillation b et w eenje,0i andjg,1i states fast enough to b e view ed as an equally w eigh ted mixture of the t w o. The de ca y rate from this mixed state is effectiv ely the a v erage of γ − and κ d . (excited) state of the target mode, and jni (n = 0,1) denotes the dissipator state with n photons. The target model can either be a readout cavity mode or a qubit mode. The target mode is coupled to a bath at temperature T target . We adopt the Lindblad formalism and consider the relaxation and excitation processes between jgi and jei, their Lindblad rates denoted by γ − and γ + , respectively. When the target mode is at thermal equilibrium, we have the detailed balance condition: γ + γ − = exp ℏω target k B T target . (4.36) With the parametric drive on the dissipator,je,0i andjg,1i are coupled throughH int with coupling strengthg, given in Eq. (4.11). The target mode can lose its excitation and scatter a Raman photon in the dissipator mode, which is then lost through the dissipator decay channel that bringsjg,1i back tojg,0i with rate κ. The two-step processje,0i!jg,1i!jg,0i is a dissipator-assisted decay channel. The decay rate ofje,0i through this process is limited by the slower rate of the two. In the weak coupling limit, jg n j κ d , corresponding to the overdamped region in the 65 dynamical analysis above, there is no population build-up in jg,1i as photons are quickly depleted. Therefore, the population inje,0i exponential decays at rate Γ given by the Fermi golden rule [98]: Γ = g 2 κ (κ/2) 2 +∆ 2 , (4.37) where ∆ =jω target ω diss j. Now we have expressions for transition rates under the parametric drive, and we are interested in the effective temperature to which we can cool the target mode. Denote j1i jg,0i,j2ije,0i andj3ijg,1i, and the population of statejni asP n (t), we can represent the stochastic process in Fig. 4.2.1 by a system of first-order linear differential equations: ⃗ P ′ (t) = ˆ Q ⃗ P(t) (4.38) where ⃗ P(t) = (p 1 (t),p 2 (t),p 3 (t)) T and the transition matrix Q is given by Q = 0 B B B B @ γ + γ − κ γ + (γ − +Γ) 0 0 Γ κ 1 C C C C A (4.39) By solving for the steady-state population of the Eq. (4.38), we can obtain the steady-state ground-state population of the target mode: p g =p g,0 +p g,1 , and its excited-state population p e = p e,0 . The steady-state solution is given by d ⃗ P/dt = Q⃗ p = 0. By solving for the null vector of matrix Q, after vector normalization, the excited state population is found to be: p e = 1 , s 1+ Γ κ 2 + (Γ+γ − ) 2 γ 2 + (4.40) With the partition function, we can calculate the effective temperature of the target mode 66 using its excited state population: p e = exp ℏω target k B T target (4.41) In the weak coupling limit where the population ofje,0i varies at a rate much slower than κ, we can ignore higher orders of Γ/κ and Eq. (4.41) becomes p e = γ + q γ 2 + +(Γ+γ − ) 2 (4.42) For the parametric drive on the dissipator to cool the target model, the cooling condition can be derived from the following: γ + q γ 2 + +(Γ+γ − ) 2 < γ + γ + +γ − (4.43) That is to say, in order to cool the target mode below its initial temperature, the transition rates should satisfy γ − Γ γ + Γ 1 < 1 2 , (4.44) and this agrees with our intuition that, to cool the target mode to a low temperature, we want to be a regime where κγ + and a decent transition rate Γ fromje,0i tojg,1i. As for the strong coupling limit where jg n j/κ d 1, the system undergoes coherent os- cillations between the states je,0i and jg,1i hence they can be considered as a statistical mixed state of the two levels, as shown in Fig. 4.2.1(b). Its decay rate to the ground state is the mean of the dissipator decay and the target model decay rate (γ − +κ)/2. The excited steady-state population is: p e = 2γ + 2γ + +γ − +κ . (4.45) 67 Figure 4.2.2: Sc hematic represen tation of a quan tum absorption refrigerator. (a) a three-lev el quan tum refrigerator where eac h transition is coupled to a differen t bath. ω c corresp onds to the target mo del frequency to co ol ω r or ω q , ω w corresp onds to the driv e(pump) frequency ω p , and ω h corresp onds to the dissipator frequency ω d , adopted from Ref. [ 1 ]. (b) A quan tum heat pump H S that couples a w ork re serv oir with temp erature T w , a hot reserv oir with temp erature T h , and a cold reserv oir with temp erature T c . The heat and w ork curren t are indicated b yJ c ,J h andJ h . In the steady stateJ h +J c +P = 0 , adopted from Ref. [ 2 ]. The cooling condition in the strong coupling limit is κ>γ − /2. In the setup above, the cavity is assumed at zero temperature. In a more realistic setup, we can consider the case where each transition in the three-level system is coupled to a different reservoir with different temperatures, as shown in Fig. 4.2.2. A system like this has been examined previously as a quantum absorption refrigerator [1, 2, 99, 100], where heat is extracted from the target mode at T c , transferred to the dissipator bath at T h , assisted by heat input from a so-called pump reservoir at T w , completing a cooling cycle. As time goes to infinity and the system reaches its steady state, we have: J h +J c +P = 0 (4.46) where J k = D L k ( ˆ H) E is the heat flow between the system and bath k. This equation represents the conservation of energy, i.e., the first law in quantum thermodynamics. We 68 also have J h T h J c T c P T w 0 (4.47) which can be regarded as the second law. For refrigeration,T w T h T c . If the heat pump only allows for transitions with a gap of ℏω α among the eigenstates of the working material, and if the resonance condition ω h =ω c +ω w holds, we have jJ α /J β j =ω α /ω β . (4.48) This expression essentially says that, in the cooling cycle, each cold excitation is traded for one hot excitation at the expense of consuming one work excitation. Eq. (4.48) is a distinctive feature of an ideal three-level heat pump. Combing Eq. (4.48) with Eq. (4.47),the cooling condition can be found to be [101] ω w T w ω h T h + ω c T c 0 (4.49a) ) ω c ω c,max (T w T h )T c (T w T c )T h ω h . (4.49b) This condition suggests a higher dissipator frequency will allow a larger cooling window and a lower cooling temperature. 69 Chapter 5: Exp erimen t for on-demand dissipation In this Chapter, I will show the experimental realization of our proposal for on-demand dissipation. First, I will review the experimental progress where engineered dissipation is used for fast cavity reset, cavity cooling, and for fast qubit reset. I will then show the design for the dissipator device. At last, I will explain the device tune-up and characterization procedure and discuss the results of the cavity reset. As for cavity reset, previous work proposed to use advanced pulse shaping [102–104]. Instead of passively waiting for the cavity to ring down freely, microwave pulses are applied at the cavity frequency to drive the cavity to the desired 0-photon steady state. The reset protocol is qubit state independent. One needs to perform numerical optimization for pulse parameters or pulse shape to achieve desired reset performance. This method is shown to shorten the photon ring time by about five times compared to passive waiting [ 103]. However, their method requires fine parameter tuning and complicated pulse shapes and is only partially optimal to be qubit-state independent. Several approaches have been used to reduce dephasing due to thermal photons in the cavity. One can design the readout resonator with a reduced dispersive shift χ, limiting the effect of each photon on the qubit, or with a lower quality factor, resulting in an increased cavity decay rateκ, ensuring photons spend less time in the cavity. Both approaches reduce readout effectiveness. Alternately, the external electromagnetic environment can be better thermalized using a customized-made attenuator [105] or a well-thermalized dissipative cavity [106]. However, these approaches also do not help to actively reset a cavity after it has been populated with readout photons. Extra attenuation also increases heating, slows control, and can hurt the readout signal-to-noise ratio if the attenuation is right on the readout cavity. Our protocol for depleting photons from a cavity can also be used to remove excitations from a qubit. Since the qubit lifetime has been greatly improved to over 100-200 µs for fixed-frequency transmons and up to ms for fluxonium qubits [ 107], it has become imprac- tical to passively wait for a long-lived qubit to reset by waiting for a time much longer than T 1 . Several approaches have been proposed for qubit reset. One type of approach is based on measurement with post-selection or measurement with feedback control [108–111]. Some methods require tuning qubit in-band with a lossy harmonic oscillator, hence require frequency-tunability of the qubit [112]. As explained in Chapter 4, we will look into a type of approach where microwave drives are used for energy transfer. This approach has been explored using resonant driving, for example, in sideband cooling [113, 114], or using double drives to a higher excited state and then a lossy environment [115, 116]. For parametric driving protocol, preliminary works have been done in numerical study [117], and it has been demonstrated to reset a tunable qubit [96]. We want to demonstrate the on-demand dissipation technique we proposed here can be used to cool either a cavity mode or qubit mode. Compared with previously proposed methods, our method has a few advantages: (1) first, our method has no strict requirements on dissipator or filter frequencies, (2) the target qubits to be reset can be fixed-frequency, (3) the reset pulse has fast on/off, (4) and is easy to parallelize with frequency multiplexing to address multiple target modes (5) our design allows the lossy mode to be at high frequency for easier cooling and to avoid the frequency crowding issue for other circuit elements. 5.1 Device design considerations In this section, we discuss the design consideration of the device. The circuit model of the device is shown in Fig. 5.1.1. The device consists of the following main components: (1) a logical qubit (purple), (2) the readout cavity (orange), (3) the frequency-tunable dissipator (blue), and (4) the Purcell filter (green). Fig. 5.1.1(a) shows the first iteration of the design 71 where the notch-type Purcell filter acts as a band-stop filter at the frequency of the readout cavity. In the second iteration of the design, as shown in Fig. 5.1.1(b), the Purcell filter acts as a bandpass filter above the frequency of the readout cavity, and the dissipator, with its maximum frequency above the Purcell filter, can be tuned on resonance with the Purcell filter. The considerations for the design iteration will become clear later. 1 Circuit model Figure 5.1.1: Circuit mo del of the device. (a) a notc h-t yp e readout resonator (orange) capac- itiv ely coupled to the logical qubit (purple) and the flux-tunable dissipator (blue). The read- out ca vit y is on resonance with the notc h-t yp e Purcell filter (green), effectiv ely shorting out the 50-Ohm output en vironmen t at the readout frequency . (b) An up dated design based on the cir- cuit in (a) with the Purcell filter (green) acts as a bandpass filter at the dissipator op erating fre- quency . The dissipator (blue) is lossy when tuned in-band with the Purcell filter due to stronger coupling to the 50-Ohm output en vironmen t. The readout cavity in our design is a λ/2 coplanar waveguide. It is capacitively coupled to the logical qubit to read the qubit states. The coupling capacitance C q determines the coupling strengthg between the logical qubit and the readout cavity, hence determining the 72 dispersive shift according to Eq. (1.4). The other end of the readout resonator is capacitively coupled to the tunable lossy mode, which we refer to as the dissipator. The dissipator is a frequency-tunable transmon qubit, and its maximum frequency is designed to be 10-12 GHz. A high-frequency dissipator has a low excitation occupancy, hence acts as a cold source of dissipation and can provide a better cooling effect. Its frequency can be controlled via the magnetic flux Φ ext threading the SQUID loop. The coupling capacitanceC g to the readout cavity determines the coupling strength between the dissipator and the cavity [118]: g rd = C r ω r L j ω 2 d r Z c e 2 πℏ (5.1) where ω r is the cavity frequency, ω d is the dissipator frequency and Z c is the characteris- tic impedance of the cavity. This, in turn, determines the photon swap rate between the dissipator and the cavity according to Eq. 4.12. In the design, Fig. 5.1.1(a), the dissipator is made lossy by being capacitively coupled to a 50-Ohm resistor to ground. Its loss rateκ d is determined by the coupling capacitanceC κ . The Purcell filter is of the notch type and is designed to be at the frequency of the readout cavity. It acts as a band-rejection filter and impedes microwave propagation at its frequency so the readout cavity remains largely protected from the static coupling to the resistor to the ground. As we will discuss in Sec. 5.3, in this early version of the design, the static coupling via the dissipator to the resistor to the ground was too high, which caused unwanted effects on the logical qubit when using the dissipator to reset the readout cavity. In the second iteration of the design, shown in Fig. 5.1.1(b), we changed the Purcell filter to a bandpass filter. It allows microwave propagation only within its bandwidth. The Purcell bandpass filter is made of an open-end half-wave coplanar waveguide. Its bandwidth is determined by the interdigital capacitance C f to the 50 Ohm termination to ground. In this configuration, we can control the dissipator loss rate by controlling its detuning with 73 respect to the Purcell filter. The dissipator is much more lossy when it is in band with the Purcell filter, and the readout resonator and the logical qubit stay protected from the 50-Ohm loss channel throughout. In the absence of drive near the dissipator-filter detuning, and with the dissipator detuned from the filter much more than the filter linewidth, the dissipator decay rate is set by its frequency and its coupling strength to the filter similar to Eq. (5.2), we have κ d =κ f g 2 f (κ f /2) 2 +(ω d ω f ) 2 (5.2) where g f is the coupling strength between the dissipator and the filter, determined by the capacitance C κ , κ f is the Purcell filter linewidth, ω d is the dissipator frequency. When the dissipator is on resonance with the Purcell filter ω d =ω f , it has a maximum decay rate: κ d = 4g 2 f κ f (5.3) Figure 5.1.2: SEM image of the device using design in circuit mo del Fig. 5.1.1 (b). For optimal cavity reset performance, the dissipator coupling capacitance to the cavityC g 74 and the coupling capacitance to the Purcell filter C κ is designed to yield g cd κ d , which is a regime that photon decay faster from the dissipator than the parametric swap operation between the target mode and the dissipator. The device, shown in Fig. 5.1.2, is fabricated using the process detailed in App. F. The device is then packaged and mounted to the bottom of the dilution fridge for measurement. The device is connected to a microwave input line from port 3 in Fig. 5.1.2 to realize fast flux control. The dissipation channel is introduced by terminating port 4 with an external microwave termination with a 50 Ohm impedance. Alternatively, the dissipation can also be introduced by an on-chip source of loss, realized by, e.g., a normal metal thin film. 5.2 Device tune-up This section presents a set of experiments that we perform to characterize and calibrate the dissipator device for demonstrating cavity reset. The measurement setup is described in App. G. First, we use a vector network analyzer (VNA) to measure the response from the readout cavity in the frequency domain. In our measurement setup, we send the DC current through the coil to control the magnetic flux threading the SQUID loop hence controlling the dissipator frequency. For initial characterization, we perform single-tone spectroscopy to track the readout cavity response as a function of the dissipator flux bias. We observe the avoided crossing and use the avoided-crossing data to estimate relevant device parameters, as shown in Sec. 1. The parametric drive is implemented by sending microwave signals via the fast-flux line (FFL) to modulate the dissipator frequency parametrically. In Sec. 2, we show the effect of the parametric drive on the cavity response linewidth as initial confirmation of the parametric drive removing photons from the cavity. In Sec.3, we perform FFL calibration to calibrate the dissipator flux tunability via the FFL and to estimate the amplitude of the parametric drive. Then we move to pulsed measurement using an FPGA-based qubit controller with the setup in Fig. G. In Sec. 4, we perform pulsed two-tone spectroscopy to choose the DC flux bias on the coil such that the dissipator is in resonance with the Purcell 75 filter for enhanced loss rate. We then move to the time-domain measurement in Sec. 5. The ringdown measurement directly measures how fast we can use our method to reset a cavity after performing a measurement. In Sec. 5.3, we discuss the effect of cavity reset pulse on the logical qubit, which motivated our lasted iteration of the design in Fig. 5.1.2. With the updated design, we explore the parameter space of the parametric drive in Sec. 5.4.1, where we measure the cavity ringdown time as a function of the drive amplitude, drive frequency, and the dissipator flux bias and identify the optimal drive parameters. In Sec. 5.5, I discuss future work on using a parametric drive for cavity cooling and qubit reset. 1 Avoided crossing from single-tone spectroscopy In this section, we perform the initial characterization of the cavity-dissipator system. Since the dissipator qubit is deliberately made lossy, it can be challenging to measure it in direct spectroscopy directly due to the short lifetime of its states. The procedure explained below will allow us to extract relevant device parameters from the cavity-dissipator avoided-crossing pattern, which we observe in the single-tone spectroscopy measurement as a function of dissipator flux bias. Let’s consider the cavity-dissipator system described by the Hamiltonian below: H =ω r a † a ω d (ϕ) 2 σ z +g cd a † +a σ x (5.4) Note that the dissipator frequency ω d (ϕ) is flux-tunable; we assume it starts above the cavity and will be tuned across the cavity frequency as a flux function. However, due to the coupling between the cavity and the dissipator, the dissipator-cavity states hybridize; hence its energy spectrum exhibits an avoided crossing when they are on resonance. To see this explicitly, suppose we are only interested in the low-energy states of the Hamiltonian and truncate and express it in the basis that spansfj0gi,j0ei,j1gi,j1eig, we have 76 H = 0 B B B B B B B @ ω d 2 0 0 g cd 0 ω d 2 g cd 0 0 g cd ω r ω d 2 0 g cd 0 0 ω r + ω d 2 1 C C C C C C C A (5.5) To find the energy levels of the hybridized states, we diagonalize the Hamiltonian; the eigenvalues of the system are: E 0 = 1 2 ω r q 4g 2 cd +(ω r +ω d ) 2 (5.6a) E 1 = 1 2 ω r q 4g 2 cd +(ω r ω d ) 2 (5.6b) E 2 = 1 2 ω r + q 4g 2 cd +(ω r ω g ) 2 (5.6c) E 3 = 1 2 ω r + q 4g 2 cd +(ω r +ω d ) 2 (5.6d) f + (ϕ)/2π = 1 2 q 4g 2 cd +(ω r ω d (ϕ)) 2 + q 4g 2 cd +(ω r +ω d (ϕ)) 2 (5.7a) f − (ϕ)/2π = 1 2 q 4g 2 cd +(ω r ω d (ϕ)) 2 + q 4g 2 cd +(ω r +ω d (ϕ)) 2 (5.7b) where ω r is the bare resonator frequency. When the dissipator is tuned to be on resonant with the cavity ω r =ω d , the splitting between the lower and upper branches is 2g cd . Hence we use this information to extract cavity-dissipator coupling strength g cd . We can also use the avoided crossing pattern to obtain information about the dissipator frequency. Recall that the dissipator is a frequency-tunable transmon with the external flux threading the SQUID loop modifies the effective Josephson energy according to Eq. ( 1.2). The frequency of the transmon can be approximated byω d ' p 8E C E J , hence the frequency 77 tunability takes the formω d (ϕ)ω max d p jcos[π(ϕϕ 0 )]j, so the dissipator frequency have a period ofΦ 0 and spans from 0 to its maximum frequencyω max d . Accounting for the junction asymmetry in the SQUID loop, the dissipator frequency is modeled by ω d (ϕ) =ω max d cos 2 [π(ϕϕ 0 )]+d 2 sin 2 [π(ϕϕ 0 )] 1 4 (5.8) where ϕ = Φ/Φ 0 is the reduced flux, and d is the junction asymmetry d = E J1 −E J2 E J1 +E J2 . It is important to account for junction asymmetry here since it changes the frequency range that the dissipator can tune across. For an ideal SQUID with identical junctions, the frequency of the flux tunable dissipator can be tuned all the way down to zero. However, if the junction asymmetry is non-negligible, the dissipator minimum frequency is p dω max d . As shown below, this will change how the avoided crossing pattern looks in the single-tone spectroscopy mea- surement. The avoided crossing data is obtained by measuring the resonator transmission S 21 using a VNA. For a notch-type resonator used in our design, the cavity responseS 21 has the following form [119]: S 21 (ω) =ae iα e −iωτ " 1 Qe iϕ jQ c j 1+2iQ ω−ωr ωr # (5.9) wherea denotes overall attenuation in the transmission lines,α is the frequency-independent phase shift in the transmission lines,τ is the propagation delay away from the resonance,Q is the total quality factor of the resonator given by 1 Q = 1 Q c + 1 Q i (5.10) Q i denotes the internal quality factor, and Q c is the resonator’s external quality factor, accounting for all the loss channels through the coupling to the feedline. When on resonance f =f 0 , the transmissionjS 21 j is at its minimum; when far off-resonance f f 0 or f f 0 the transmissionjS 21 j is close to 1. 78 Using a Keithley 2400 source meter, we source a DC current I coil to a coil mounted on the device package to control the external flux bias Φ ext threading the SQUID loop. We measure the cavity response S 21 as a function of flux bias to tune the dissipator frequency across its frequency range. Fig. 5.2.1 shows the measured S 21 traces as a function of flux bias current at the coil. Each vertical slice in the figure corresponds to jS 21 j at a given flux bias and shows a period in the flux bias that corresponds to Φ 0 , the period of the dissipator frequency ω d (ϕ). We have observed three types of qualitatively different avoided crossing data depending on the maximum dissipator frequency ω max d . Figure 5.2.1: The Resonator resp onse as a function of flux bias in single-tone sp ectroscop y mea- suremen t(a) ω max d < ω r , no a v oided c rossing pattern (b) ω max d ≳ ω r , the dissipator minim um frequency is close to resonator at the a v oided crossing, and the dissipator is punc hed out as a re- sult (c) ω max d > ω r , the dissipator tuned through the resonator and the a v oided c rossing pattern is observ ed. When the dissipator frequency is entirely below the resonator throughout the flux tuning, as in the case in Fig.5.2.1(a), we do not see the dissipator tuned through the resonator. In another case in Fig. 5.2.1(b), the dissipator’s minimum frequency is close to the resonator frequency, hence ”punched out” near the avoided crossing due to high readout power. Only when the dissipator can be tuned below or above the resonator can we see both branches of the avoided crossing pattern, as shown in Fig. 5.2.1(c). The fitting procedure is the following: we first identify the period of the avoided crossing dataI 0 to convert bias current to flux unit by the conversion ϕ i =I i /I 0 . Then, from theS 21 79 Figure 5.2.2: A v oided crossing pattern in single-tone sp ectroscop y measuremen t for device Diss08_09C. The data to fit the mo del in Eq. ( 5.7a ) and e xtract the relev an t parameters as sum- marized in T able 5.2.1 . The white dashed line sho ws the fit from the mo del traces at each flux bias current ϕ i , we can identify the resonance frequencyf i by fitting jS 21 j from the single-tone spectroscopy to the Lorentzian function in Eq. (5.9). Then, we fit the dataff i (ϕ i )g for both the upper and lower branches to Eq. (5.7a) to extract the parameters fω r ,ω max d ,g,d,ϕ 0 g. The fitted parameters are summarized in Table 5.2.1. 80 Parameters Device08_07A Diss08_09C bare cavity frequency ω 0 r /2π (GHz) 5.681 6.0576 maximum dissipator frequency ω max d (GHz) 11.1 12.680 cavity-dissipator coupling strength g/2π (MHz) 111 104 junction asymmetry d 6.96% 8.83% T able 5.2.1: Fitted parameters for device Device08_07A and Diss08_09C 2 Cavity response with parametric drive on After initial characterization for device parameters, we now set up the parametric drive by sending a microwave signal through the FFL to implement the drive Hamiltonian in Eq. (4.9) that modulates on the dissipator frequency with a modulation frequency ofω drive /2π. As an initial demonstration of the effect of the parametric drive, we measure the cavity response as a function of readout power, and we do this with and without the parametric drive on. In standard qubit characterization, a ”punchout” measurement is used to confirm the presence of a qubit and determine the maximum readout power. In a punchout measurement, we measure the cavity responseS 21 as a function of readout power. At low readout power (i.e., low average photon number ¯ n), the cavity frequency is dressed by the qubit in its ground state, hence acquires a dispersive shift according to Eq. (1.4): χ 01 = g 2 ∆ (5.11) As the readout power increases, the cavity inherits nonlinearity from the qubit. At very high readout power, the qubit is ”punched out,” i.e., the qubit junction is ionized, so the qubit mode disappears, and the cavity responds as if the qubit is not there. Hence its frequency is the bare cavity frequency [120, 121]. Fig. 5.2.3(a) shows the result of such a ”punchout” scan. The punchout shift corresponds to the cavity Lamb shift due to the presence of the logical qubit. When we drive the dissipator with a parametric (flux) drive at the detuning between the 81 Figure 5.2.3: Ca vit y magnitude resp onse as a function of readout p o w er. (a) FFL driv e off: ca v- it y resp onse exhibit ”punc h out” shift. (b) FFL driv e on: ca vit y line width broadens, and the dip b ecomes shallo w er, suggesting a reduced ca vit y in ternal Q as photons are pump ed out of the ca vit y b y the parametric driv e. cavity frequency and the dissipator at ω drive =6.6 GHz, instead of a punchout shift, we see that the cavity frequency remains at its dressed frequency throughout. The cavity linewidth becomes broader and shallower, which is an indication of increased cavity internal loss due to the dissipation channel induced by the drive. At high power, the drive can not keep up to remove the photons fast enough to the external environment. As a result, the dip in the cavity response becomes deeper, indicating a higher internal quality factor Q i . This is a smoking gun signature of the dissipation induced by our parametric drive. 3 FFL calibration from resonator response The parametric drive is sent via the FFL with microwave signals modulated by the two output channels of the Quantum Machine’s OPX control unit, as shown in the measurement setup in Fig. G. As a result, the dissipator SQUID loop is threaded by the external flux ϕ ext , which is modulated in time with a modulation amplitude δϕ: φ ext (t) = ¯ φ ext +δφsin(ω p t). (5.12) 82 The parametric drive resulting from the flux modulation is modeled by a modulation of the dissipator frequency at a frequency ω p : ω d (t) =ω d +δsin(ω p t) (5.13) Note that Eq. (5.13) is an approximation, and we have simplified the problem and neglected all higher harmonics of the drive, and we do not consider the ac-Stark shifts of the transmon normal modes due to the drive. A more comprehensive analysis is done using the full circuit Hamiltonian in Ref. [94]. This section explains the procedure we use to estimate the flux modulation amplitude δϕ and the frequency modulation amplitude δω on the dissipator due to the parametric drive. In Sec. 1, we show that, through Eq. (5.8), we can estimate dissipator frequency as a function of flux bias at the coil. To estimate frequency modulation δ at a given drive amplitude δ V from the flux tuning curve in Eq. ( 5.8), we need to characterize the relation between flux tuning via the FFL and flux tuning via the coil. In the FFL calibration, we start by changing the flux bias at the coil and recording how much the resonance frequency in the cavity response moves accordingly. As shown in Fig. 5.2.4(a), we start with an initial coil bias, move it by δI coil and record the cavity response at each stepfω i r (I i coil )g. Then we change the current bias at the FFLI FFL such that the flux bias from the coil is offset and the cavity response moves step by step back to the starting point, as shown in Fig. 5.2.5(b). We can extract the resonance frequency at each flux bias at the coil ( I i coil ) or at the FFL (I i FFL ) by fitting the magnitude of the cavity response S 21 to a Lorentzian function, which is shown in Fig. 5.2.4(c), shown in red and blue circles, respectively. We start with an initial flux bias at the coil I 0 coil =80µA, change the flux bias by ∆I coil , and see the cavity’s resonance frequency moves accordingly. We then change the flux bias at the FFL and ask: how much bias current∆I FFL is needed to move the readout cavity response back to the same resonance frequency. We plot ∆I (i) FFL and ∆I (i) coil in Fig. 5.2.4(d) and extract the conversion relation of 83 the flux control between the coil and the FFL by fitting the data to a straight line. The fitting result suggests that the flux bias at the coil is equivalent to 50 times that at the FFL. Figure 5.2.4: FFL calibration: (a) readout ca vit y resp onse as a function of the coil bias I coil , (b) readout ca vit y resp onse as a function of the FFL bias I FFL , (c) the extracted ca vit y resonance frequencies at eac h coil biasfω i r (I i coil )g in orange and at eac h FFL bias ∆I (i) FFL in blue, (d) the con v ersion relation of the flux con trol b et w een the coil and the FFL. After establishing the relation between the flux tuning between the coil and the FFL, we can estimate the frequency modulation with a given microwave drive signal using Eq. (5.8). 84 4 Two-tone fast-flux line spectroscopy After extracting relevant parameters from single-tone spectroscopy, we can estimate the dis- sipator frequency as a function of the flux bias current using Eq. 5.8. Next, we perform a two-tone spectroscopy measurement to choose an appropriate drive frequency for the para- metric driving. Using the measurement configuration in Fig. G, we send a probe signal at the cavity frequencyω r /2π to port 1 of the device and measure the transmitted signal from port 2. Additionally, we send a drive tone via the fast-flux line control and vary the drive frequency. As we vary the drive frequency, we look for the drive frequency corresponding to a two-photon process where the cavity probe photon and the drive photon combine, resulting in a filter/dissipator excitation. We expect to find the drive frequencies corresponding to cavity-dissipator and cavity-filter transitions. Fig. 5.2.5 shows the measurement result as a function of flux bias current. We saw two main features: the Purcell filter at 2.55 GHz and the dissipator feature at around 3.8 GHz at flux bias 175 µA. As the flux bias decreases to 90 µA, the dissipator spectroscopic feature moves down in frequency, and the filter feature broadens due to stronger dissipator-filter coupling. The spectroscopy measurement allows us to determine the FFL drive frequency for the cavity reset experiment. 85 Figure 5.2.5: FFL sp ectroscop y as a function of dissipator flux bias. The dissipator frequency is tuned do wn as the flux bias c hanges from 90 µ A to 175 µ A. W e see feature 1 at around 2.8 GHz at the ca vit y-filter detuning frequency . This feature gets broader as the dissipator is tuned closer. W e see feature 2 starting at 3.8 GHz when flux bias is 90 µ A, corresp onding to ca vit y- dissipator detuning frequency , and this feature mo v es do wn in frequency as flux bias increases. W e d o not kno w what transition corresp onds to feature 3, but it gets broader as the dissipator is tuned do wn. 5 Cavity ringdown The readout resonator has its intrinsic decay constant κ, which characterizes how fast pho- tons leak from a resonator. It is determined by the strength of the resonator-environment coupling. The intrinsic resonator κ can be measured from the cavity frequency response in the frequency domain. However, to measure the dynamics of how a cavity relaxes, a time-domain measurement is required. Hence, we perform a resonator ringdown measurement. We do this with and 86 without the FFL drive at the frequencies expected to vacuum photons out of the resonator. When the FFL drive is on, we expect faster resonator ringdown time. Fig. 5.2.6 shows the pulse sequence we used for the ringdown measurement. We first drive photons into the resonator with a pulse at the readout frequency. And then, the readout pulse is turned off, and the resonator can ring down freely. During the free ringdown time, we turn on the FFL drive at the detuning frequency of the cavity and dissipator to swap photons from the resonator to the dissipator. At the same time, photons quickly decay from the lossy dissipator to the external environment. After the cavity reset pulse, we ”listen” to the voltage signal from the readout resonator to determine the remaining photon population. Figure 5.2.6: Ringdo wn time measuremen t: (a) pulse sequence to measure ca vit y ringdo wn time, (b) ringdo wn time with and without the ca vit y-reset driv e. Device: Diss08_09C. 87 5.3 Logical qubit T 1 We are also interested in the effect of the parametric drive on the logical qubit. Ideally, the logical qubit T 1 should stay unaffected when using the parametric drive to remove photons from the readout cavity. To check this, we use the pulse sequence in Fig. 5.3.1 to measure logical qubitT 1 while the parametric drive for cavity reset is on during the qubit free evolution time. Fig. 5.3.2(a) shows the measured logical qubit T 1 as a function of parametric drive Figure 5.3.1: Pulse sequence to measure logical qubit T 1 with the parametric driv e on. Driv e frequency at ca vit y-dissipator detuning. Device: diss08_07A. amplitude and shows that the qubit T 1 decreases as the parametric drive becomes stronger. Also, as shown in Fig. 5.3.2(b), the excited state population in the qubit steady state also increases at high drive powers. This shows that our parametric drive can not selectively introduce dissipation on the cavity mode without affecting the qubit mode. To realize better selectivity, we change the Purcell filter design from the notch type in Fig. 5.1.1(a) to the bandpass filter type in Fig. 5.1.1(b). In the latter design, the dissipator is only lossy when on resonance with the Purcell filter, and we can achieve better selectivity by engineering the Purcell filter linewidth. 88 Figure 5.3.2: Logical qubit T 1 with the ca vit y-reset driv e on. (a) Logical qubit T 1 with w eak ca vit y-reset driv e (orange) and strong ca vit y-reset driv e (blue); driv e frequency at ca vit y- dissipator detuning, (b) logical qubit T 1 degrades as the ca vit y-reset driv e b ecomes stronger, (c). the ca vit y-reset driv e also effectiv ely mak es the logical qubit steady state hotter, p ossibly b ecause it is driving higher-lev el transitions in the logical qubit. Device: diss08_07A. 5.4 Optimal drive parameters With the upgraded design, we further explore the parameter space of the drive and obtain the optimal drive parameters. This section measures the ringdown time, varying the drive amplitude (δ) and drive frequency (ω p ) at different flux biases. In principle, the dissipator loss rate is controlled by the flux bias: when the dissipator is tuned in-band with the Purcell filter, the dissipator has the maximum loss rate, given by Eq. ( 5.3); when the dissipator tuned out of band of the Purcell filter, its loss rate is set by Eq. ( 5.2). Fig. 5.4.1 shows the measured cavity ringdown time as a function of drive frequency and the drive power (normalized) at different flux biases. The feature we see at all flux biases is around ω p /2π = 2.8 GHz, corresponding to the cavity-filter transition ω p = ω f ω r . As the dissipator frequency is tuned down in frequency, the same drive power on the fast-flux line corresponds to a larger frequency modulation as the dissipator is more sensitive to flux bias current at a lower frequency, giving a larger effective drive amplitude δ. At the same time, the effective static coupling between the dissipator and the cavity also increases since the dissipator-cavity and the dissipator-filter detunings are smaller. This is qualitatively in agreement with the fact that the cavity-filter feature gets broader and can be turned on with smaller drive power as 89 the dissipator moves down in frequency (flux current increases from 140 µA to 165µA). We do not know which transition corresponds to the feature around 3.3 GHz in Fig. 5.4.1, but this feature also gets broader as the dissipator moves down in frequency. Figure 5.4.1: Ringdo wn time as a function of driv e frequency and amplitude at differen t dissipa- tor flux biases: (a) flux bias = 140 µ A, (b) flux bias = 155 µ A, ( c) flux bias = 165 µ A. Device: diss08_09C. Fig. 5.4.2 shows line cuts of the 3D sweep in Fig. 5.4.1. For the cavity-filter transition at ω p /2π = 2.85 GHz, there is an optimal drive amplitude: the ringdown time first decreases as the drive amplitude increases, then increases again as the drive gets stronger. This is likely 90 Figure 5.4.2: Ringdo wn time as a function of driv e amplitude for differen t driv e frequencies: (a) flux bias = 140 µ A, (b) flux bias = 1 55 µ A, (c) flux bias = 165 µ A. Device: diss08_07A. due to the underdamped regime discussed in Sec. 4.2, where the photon excitation gets swapped back to the cavity before decaying to the environment through the filter. As the dissipator frequency tunes down, the optimal normalized drive amplitude gets smaller for two reasons: first, less drive amplitude is needed to achieve optimal frequency modulation since the flux tuning curve ∂ω q /∂Φ ext becomes steeper; second, for a given frequency modulation amplitude, the cavity-filter swaps happen faster at smaller detunings ∆ 1 and ∆ 2 , so less drive amplitude is needed to produce the desired optimal cavity-filter swap rate. The best ringdown time reduction is achieved withω drive /2π = 2.85 GHz, at flux bias = 140µA. With the drive on, we demonstrate we can reset the cavity at a rate of around 25 MHz. 5.5 Future work So far, we have shown that we can use the parametric drive to speed up cavity reset. As the next steps, we want to demonstrate that the parametric drive can reduce the logical qubit dephasing rate by removing residue photons in the cavity. The most straightforward measurement one can perform is to measure qubit T ∗ 2 time with or without cavity reset pulse, as shown in Fig. 5.5.1(a). First, a readout pulse is applied to populate the readout cavity with photons. A cavity reset pulse is then applied 91 with the optimal parametric drive parameters found in Sec. 5.4. Then, a standard Ramsey measurement is performed, and we compare the measuredT ∗ 2 with and without cavity reset time. This allows us to quantify the effectiveness of the reset pulse in removing residue readout photons after a readout of performed. In a slightly different setting, we want to test the effectiveness of the parametric drive for cavity cooling. We can perform a similar Ramsey-like experiment where the parametric drive for cavity cooling is turned on during the qubit free evolution time, and we compare theT ∗ 2 for the cases where the cavity cooling tone is on and off, as shown in Fig. 5.5.1(b). To make the effect of the cavity cooling tone more evident, we can inject photon shot noise by turning on a weak readout tone during the qubit’s free evolution. The strength of the background photon noise can be controlled by the pulse amplitudeA. Alternatively, one can use a Ramsey-like sequence to construct the noise spectrum and compare the reconstructed noise amplitude with and without the parametric drive for cavity cooling. As a more direct measure, one can also use the logical qubit as a photon counter and directly probe photon number reduction due to the cavity cooling drive using photon-number solved spectroscopy [122]. Figure 5.5.1: (a) The pulse sequence to test the efficacy of the ca vit y-reset pulse on logical qubit coherence preserv ation b y remo ving residue readout photons after a readout. (b) The pulse se- quence to test the efficacy of the ca vit y-co oling driv e on logical qubit coherence preserv ation b y remo ving thermal photons from the readout ca vit y . When the frequency of the parametric drive is at the detuning frequency of the dissipator and the qubit, we can use the same mechanism to introduce on-demand dissipation on the logical qubit mode, as shown in Fig. 5.3.2. 92 Figure 5.5.2: The qubit-dissipator t ransition for qubit reset To test the efficacy of the parametric drive on qubit reset, we can use measure logical qubit T 1 as shown in Fig. 5.5.3(a), and compare the measuredT 1 with and without the parametric drive for qubit reset. By performing a parameter sweep similar to Sec. 5.4, one can find the optimal drive parameters for the fastest qubit reset rate 1/T ∗ 1 . To test the operational improvement from the qubit reset pulse, we will perform a Rabi-like measurement, as shown in Fig. 5.5.3(b). First, a scrambling pulse is applied to prepare the qubit into a mixed state. A scrambling pulse can be a long and strong pulse at the qubit frequency such that the qubit will be fully decohered and scrambled. Alternatively, the scrambling pulse can also be a rotation gate with random axis and angles such that the qubit is put in a mixed state under an ensemble average. After the state preparation, a qubit reset pulse is applied, which ideally initializes the qubit in its ground state. Followed by the qubit reset pulse, a Rabi drive at the qubit frequency is applied, and the Rabi contrast, i.e., the amplitude of the Rabi oscillation, is a good metric to quantify the ground state population at the start of the Rabi drive, after the qubit reset pulse. Furthermore, we want to test if we can use the parametric drive to cool the qubit mode below its thermal temperature. We can perform a Rabi population measurement to measure the qubit temperature, previously demonstrated in Ref. [115]. The qubit temperature is estimated by measuring the ratio of the initial ground state population P g and the initial 93 Figure 5.5.3: (a) The pulse sequence to measure logical qubit T 1 and test the efficacy of qubit- reset driv e at the dissipator-qubit detuning frequency . (b) The pulse sequence to test the efficacy of the qubit-reset pulse on preparing the qubit ground state. excited state population P e by driving the Rabi oscillations betweenjei andjfi states with and without π-pulse in state preparation, as shown in Fig. 5.5.4. In the first sequence, Fig. 5.5.4(a), the initial state has most of its population in the ground state. The drive at the transition jei $ jfi with varying drive amplitude produces a Rabi oscillation with an amplitude A e proportional to the initial excited state population. The π-pulse at the end transfers the excited state population back to the ground state for readout, and the readout signal will have an oscillating amplitude ofA e . The second sequence in Fig. 5.5.4(b) differs in the state preparation step. It has a π pulse to invert the initial population of the jgi and the jei states, yielding a Rabi oscillation with an amplitude A g corresponding to the initial ground state population. We can estimate the initial ground state population P g = A g /(A g +A e ) and the initial excited state population P e = 1P g from the two Rabi oscillation amplitudes. Assuming the initial state is under equilibrium, we can use the partition function, as in Eq. (4.41), to estimate the qubit temperature and compare the effective qubit temperature with or without the qubit cooling pulses. 94 Figure 5.5.4: The pulse sequences to measure the initial ground state p opulation (a) and the ex- cited state p opulation (b) to estimate the qubit temp erature with or without the qubit-co oling driv e. 95 Chapter 6: Conclusion In summarizing the work discussed in previous chapters, I have demonstrated that non- Markovian effects are non-negligible when characterizing the decoherence of superconduct- ing qubits. I have introduced a method termed PMME tomography, which provides a phenomenological master equation to describe and predict the free evolution dynamics of a qubit more accurately than the standard Lindblad model. Moreover, I have revealed that, by engineering the environment, dissipation can be utilized as a resource to remove entropy from a system with unwanted excitation, thereby facilitating on-demand qubit or cavity re- set. This can be achieved through parametric coupling of the system to a frequency-tunable dissipator, demonstrating that we can reset a readout resonator at a rate of 20 MHz. Moving forward, I would like to review recent advancements in noise modeling and en- gineering in superconducting qubits. I will reflect on the impact of our work and discuss future directions, specifically focusing on how we might devise a scalable solution for noise modeling and engineering. 6.1 Recent progress in noise characterization and modeling The study of open quantum systems offers potent tools for quantifying and scrutinizing the noise present in a quantum computer. However, the characterization of this noise and the verification of devices are not without their challenges. Quantum tomography, a prevalent method for noise characterization, demands a significant amount of resources. It necessitates an exponential number of measurements for characterizing a quantum state and a double exponential number for a quantum process. Another challenge lies in accounting for temporal correlations within the noise model. This thesis brings to light the non-Markovian effects that are observable at short time scales, often overlooked but far from negligible. It discusses methods to incorporate these effects and explores potential underlying physical mechanisms. An intriguing extension of this work would be to derive different forms of kernels from various noise mechanisms. The third challenge comes from the need to describe noise in conjunction with control. Accurately tracking the impact of noise becomes a complex task when the control filters and selects the noise spectral density using the filter function formalism [ 123, 124]. Over recent years, substantial efforts have been made to tackle these challenges. In the subsequent sections, I aim to provide a concise overview of the progress made in several theoretical frameworks. These frameworks, developed recently, are designed to account for non-Markovian noise. The discussion will be kept at a high level, without going into technical details, to provide a clear understanding of the advancements in this field. The process tensor is a framework designed to describe quantum systems. It is an extension of the superchannel [125] concept, which is a completely positive (CP) map that transitions from the set of potential control operations to output states. This framework takes into account two-time correlations between preparations at the initial time and measurements at the final time. The construction of a process tensor involves an experiment where the experimenter performs control operations at multiple times and measures the corresponding output state. This allows for the accounting of multistep temporal correlations, a significant feature of the process tensor. For a comprehensive review of the formalism of the process tensor, please refer to Ref. [85]. The importance of the process tensor formalism lies in its provision of an operationally well-defined quantum Markov condition and a measure for non-Markovianity. The method used to characterize the process tensor is known as process tensor tomography. This technique involves a series of experimental steps designed to provide a detailed understanding of the process tensor. The number of experiments required scales as O N k oc for a single qubit, where N oc represents the number of unitary bases and k denotes the number of time steps [126]. This framework provides a robust 97 approach to understanding and managing quantum systems, particularly in the context of non-Markovian memory strength [127] and effective environmental degrees of freedom [ 128]. The tensor network is a powerful computational tool, particularly for efficiently simulating quantum systems and their dynamics. This framework represents quantum states and their evolution as matrix product states (MPSs) and matrix product operators (MPOs). In a two-dimensional tensor network, the horizontal dimension signifies time, while the vertical dimension represents space. Singular value decomposition is employed to streamline the description of the state as time progresses. The tensor network formalism utilizes high- dimensional tensors, offering an efficient means to encode the restricted correlations. The bond dimension of the tensor can be interpreted as a measure of the complexity of open- system dynamics [129]. It has been demonstrated that the tensor-network formalism can proficiently describe a spin coupled to a bosonic bath [ 130] as well as a spin bath [131]. Furthermore, the tensor network formalism bears a close relationship to the process tensor and can be employed to represent a process tensor through contraction or matrix-product approximation. This paves the way for the experimental reconstruction of the process tensor [131]. Neural networks (NNs) have emerged as a powerful tool for characterizing complex inter- actions between systems and their environments. Among them, recurrent neural networks (RNNs) stand out as a type of artificial neural network specifically designed to handle se- quential data. RNNs have found wide applications in voice recognition, machine translation, and other domains. One key advantage of RNNs, like other machine learning methods, is their ability to handle noisy experimental records and extract meaningful quantum informa- tion. In the context of time series modeling, a one-to-many RNN [132] consists of RNN cells that process the input state at timet1, the hidden layer weights, and the output state at time t. The weights are updated iteratively during training to optimize the network’s per- formance. The update rule for the hidden layers at time t not only depends on the current state but also on the states at previous time steps, enabling the retention of memory across 98 different time steps. However, due to the vanishing gradient problem, standard RNNs can only capture short-term dependencies. To address this limitation and incorporate long-range dynamical correlations, long short-term memory (LSTM) networks [74, 76, 133] have been adopted. Despite their effectiveness, one limitation of the machine learning approach, in- cluding RNNs, is the lack of physical interpretation. One promising approach to overcome this limitation is integrating physical mechanisms into the description of neural networks [134]. By incorporating physical insights into the design and training process, the resulting models can offer a deeper understanding of the underlying physical processes and enhance their interpretability. To scale up noise modeling for device-specific applications, we require both an efficient representation of the noise model and a scalable noise characterization method. For noise characterization, scaling up to large system sizes can be achieved through the use of shadow tomography [135]. This technique is based on a recently proposed methodology for the simul- taneous evaluation of several quantities for NISQ devices and is more efficient than traditional tomography. When combined with process tensor tomography with classical shadow tomog- raphy, one can estimate the marginals of a process tensor with only a logarithmic overhead [136]. This makes it a highly efficient and scalable tool for characterizing non-Markovian dy- namics. Classical shadow tomography is also advantageous for witnessing non-Markovianity through moments of the Choi state [137] and does not necessitate the evaluation of the full spectrum of the evolution. Detecting and characterizing non-Markovian effects are crucial for harnessing them as a resource in quantum information processing tasks. Shadow tomog- raphy, therefore, provides a powerful tool for the efficient characterization of non-Markovian dynamics, thereby enabling the scaling up of noise modeling for large quantum processors. 6.2 Scalability of hardware The experimental work of this thesis is performed on a single qubit. In this section, I want to briefly remark on the main challenges in scaling up a practical quantum computer. Finally, I 99 want to remark on how our proposal for noise engineering can fit in as the system size scales up. There are many challenges in scaling up a quantum computer and integrating tens and thousands of coherent qubits with their control and readout. As of July 2023, several groups and institutions have reported quantum computers based on superconducting qubits of size around tens of hundreds. Among them, there is the MIT quantum processor with 16 qubits [138], the Google Sycamore Processor with 72 qubits [139], and the IBM Osprey processor with 433 qubits [140]. Scaling up quantum hardware is the first step to scalable quantum error correction and fault tolerance. The effort to scale up a quantum computer, broadly speaking, involves scaling up two primary parts: the qubits, which hold the quantum in- formation at a low temperature, and the classical control electronics that manipulate the quantum information and read out qubit states to room temperature. The control electronics are comprised of FPGA, signal generators, digitizers, etc. To scale up quantum hardware, one needs to have not only the quantum circuit chip design in mind but also the whole control pipeline. Frequency-selective readout has been proposed to scale up qubit readout, allowing multiple qubits to share a single measurement line [141]. This approach incorporates λ/4 resonators and requiresN individual microwave frequencies to read outN qubits, each using its unique microwave frequency. Subsequently, frequency-division multiplexing was proposed [142] to facilitate simultaneous readout of multiple qubits, all sharing the same reference microwave source. This considerably reduces the overhead when scaling up the qubit readout scheme. Next, I’ll touch on the considerations for scaling up the qubit and cavity reset we proposed in Chapter 4 via parametric coupling to the dissipator. From a design perspective, each qubit and cavity will be capacitively coupled to an individual tunable dissipator, either directly or indirectly (as illustrated in Fig. 4.1.3 and Fig. 5.5.2) to enable parametric coupling. A challenge we might encounter when scaling up involves modifying the readout circuit from λ/4 resonators toλ/2 resonators, which occupy more chip space. For a more compact design, 100 having multiple dissipators using a joint Purcell filter is possible, similar to the design in Ref. [143, 144]. It is also possible to couple one dissipator mode with multiple qubits and utilize frequency multiplexing to reset several qubits simultaneously. As for control electronics, each dissipator will likely necessitate its FFL for flux modula- tion. Flux control integration, as discussed in the single qubit case [145], and flux control multiplexing have been explored numerically in the context of single and two-qubit gates in a qubit architecture with tunable coupling [146]. As tunable parametric couplers become more prevalent, we anticipate increasing discourse on a scalable flux control solution. Finally, scaling up is limited by the total available bandwidth when it comes to qubit control multiplexing. Given that the dissipator operates at high frequencies and their control uses separate flux control lines, we do not expect the additional dissipator to conflict with the qubit and readout frequency ranges. 6.3 Future directions In addition to scaling up the characterization, simulation, and hardware, as previously dis- cussed, several other intriguing avenues exist. For noise modeling that accounts for non-Markovian effects, it could be compelling to extend non-Markovian models to driven cases. Recent research has scrutinized the non- Markovian gate error in fixed-frequency transmons by employing a continuous phase am- plification sequence [ 147]. These studies suggest that the coherent nature of the error is due to an off-resonant drive on a control qubit in a two-qubit gate, on a spectator qubit, or due to a spurious TLS. Developing noise characterization methodologies that account for non-Markovian noise could assist in pinpointing error mechanisms. In addition, learning noise models that accurately represent non-Markovian dynamics could facilitate the creation of suitable error mitigation and optimal control strategies. For example, using the process tensor formalism, one could find optimal controls for quantum systems coupled to non-Markovian environments. As shown in Ref. [148], slower processes 101 might achieve higher fidelity by exploiting non-Markovian effects in a driven qubit cou- pled to a bosonic environment. This approach underscores the utility of the process tensor as an efficient and accurate representation of the noise model, enabling optimization over a large number of control parameters. Moreover, it demonstrates that performance im- provements are possible when qubits experience non-Markovian decoherence, showing that non-Markovianity can be harnessed as a resource for quantum information tasks. Beyond the endeavor to eliminate unwanted decoherence, another intriguing topic is to utilize bath engineering such that dissipation assists with specific quantum information tasks. One instance of this is performing quantum state preparation through engineered dissipation. By leveraging a cooling process via coupling to dissipative auxiliary qubits, it has been shown that one can prepare low-energy states of the transverse Ising model [149]. Bath engineering introduces an alternative method for preparing entangled many-body states and exploring nonequilibrium quantum phenomena. With additional control, a qubit can also be used to manage and interrogate the environment. A recent study has demonstrated that an active feedback protocol implementing a quantum Szilard engine allows a qubit to polarize all the active TLSs in its environment [150]. The experiment reveals that TLSs and qubits serve as each other’s environment and are the dominant loss mechanisms for each other. These directions in bath engineering highlight the close interconnection among qubit decoherence, open quantum system science, and quantum thermodynamics. 102 List of Publications The work presented in this thesis contains material from the following publications and preprints: 1. Haimeng Zhang, Bibek Pokharel, EM Levenson-Falk and Daniel Lidar, “Predicting Non-Markovian Superconducting Qubit Dynamics from Tomographic Reconstruction,” Phys. Rev. Appl. 17, 054018 (2022). Other publications and preprints completed during the duration of the PhD but not included in this thesis are: 1. Oles Shtanko, Derek S. Wang, Haimeng Zhang, Nikhil Harle, Alireza Seif, Ramis Movassagh and Zlatko Minev, “Uncovering Local Integrability in Quantum Many- Body Dynamics,” arXiv:2307.07552 [quant-ph] (2023). 2. Evangelos Vlachos, Haimeng Zhang, Vivek Maurya, Jeffrey Marshall, Tameem Albash and Eli M Levenson-Falk, “Master Equation Emulation and Coherence Preservation with Classical Control of a Superconducting Qubit,” Phys. Rev. A 106, 062620 (2022). 3. James T Farmer, Azarin Zarassi, Darian M Hartsell, Evangelos Vlachos, Haimeng Zhang and Eli M Levenson-Falk, “Continuous Real-Time Detection of Quasiparticle Trapping in Aluminum Nanobridge Josephson Junctions,” Appl. Phys. Lett. 119, 122601 (2021). 4. Haimeng Zhang and Han Wang, “Two-dimensional materials for electronic applica- tions,” Advanced Nanoelectronics, M.M. Hussain (Ed.). doi:10.1002/9783527811861.ch3 (2018). 5. 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(3.3a): L(A) =i[H,A]+ X k γ k V k AV † k 1 2 n V † k V k ,A o , (A.1) where A,V k 2B(H), H =H † , and γ k > 08k. LetfF i g d 2 −1 i=0 denote an orthonormal, Hermitian operator basis forB(H): Tr(F i F j ) =δ ij , F i =F † i , F 0 =I/ p d . (A.2) For example, when H = C 2 (a qubit), we can choose the normalized Pauli matrices as the operator basis, i.e.,fF i g =fI,σ x ,σ y ,σ z g/ p 2. For any A2B(H) we can then expand A = P i a i F i , and in particularL(F i ) = P j ℓ ji F j (note the transposed index order), where ℓ is the matrix representation of L in the given basis, with the matrix elements given by ℓ ij = Tr[F i L(F j )]. The basisfF i g “coordinatizes” bothL and the operators inB(H). Assume that the superoperatorL satisfies the Hermiticity-preservation condition [L(A)] † =L A † . (A.3) This is true, in particular, for the Lindbladian (A.1), as is easily checked. Let us show that then the matrix ℓ representingL in the chosen basis is real: ℓ ∗ ij = Tr [L(F i )] † F † j = Tr(L(F i )F j ) =ℓ ij . (A.4) Thus, after coordinatization the superoperatorL can be seen as ad 2 d 2 dimensional matrix ℓ2R d 4 . Now assume that A is a right eigenoperator of L with eigenvalue λ, i.e., L(A) = λA. Then: L(A) = X ij a i ℓ ji F j =λA =λ X i a i F i . (A.5) Taking the trace of both sides after multiplying from the right by F k yields P i a i ℓ ki =λa k , i.e., ℓ⃗ a =λ⃗ a , (A.6) where⃗ a = (a 0 ,...,a d 2 −1 ) t is a column vector (the superscriptt denotes the transpose). Thus, if R i is a right eigenoperator ofL then its coordinates-vector ⃗ r i in the expansion R i = X j (⃗ r i ) j F j . (A.7) is a right eigenvector ofℓ. Conversely, by solving the linear algebra problem of finding the set of right eigenvectorsf⃗ r i g ofℓ, we can construct the right eigenoperators ofL using Eq. (A.7). Now consider the set of left eigenvectors ⃗ l i t of ℓ: ⃗ l i t ℓ = λ i ⃗ l i t . These are also the right 122 eigenvectors ofℓ t : ℓ t ⃗ l i =λ i ⃗ l i . Note that the left and right eigenvalues ofℓ are identical since the determinant of a matrix equals the determinant of its transpose. We define L † as usual via the inner product relation hL † (A),Bi =hA,L(B)i , (A.8) where we use the Hilbert-Schmidt inner product hA,Bi Tr(A † B) . (A.9) Specifically, for the Lindbladian in Eq. ( A.1), this implies that L † (A) =i[H,A]+ X k γ k V † k AV k 1 2 n V † k V k ,A o , (A.10) as can easily be verified by direct substitution of this form of L † (A) into Eq. (A.8). Let us show that ℓ t is the matrix representation ofL † . To do so, consider the expansion L † (F i ) = P j ˜ ℓ ji F j ; we will show that in fact ˜ ℓ =ℓ t . Indeed, on the one hand we have from Eq. (A.1): (ℓ t ) ji =ℓ ij = Tr[F i L(F j )] =iTr(F i [H,F j ]) (A.11) + X k γ k Tr[F i V k F j V † k ] 1 2 Tr[F i fV † k V k ,F j g] , and on the other hand we have from Eq. (A.10): ˜ ℓ ji = Tr[F j L † (F i )] =iTr(F j [H,F i ]) (A.12) + X k γ k Tr[F j V † k F i V k ] 1 2 Tr[F j fV † k V k ,F i g] , which is easily checked to be equal to the expression for (ℓ t ) ji in Eq. (A.11) by cycling 123 operators under the trace. Thus, L † (F i ) = X j ℓ ij F j , (A.13) and the same reasoning that we used above for the right eigenvectors and eigenoperators now leads to the conclusion that ifL i is a right eigenoperator ofL † , i.e., a left eigenoperator ofL, then its coordinates-vector ⃗ l i in the expansion L i = X j ( ⃗ l i ) j F j . (A.14) is a left eigenvector ofℓ. Conversely, by solving the linear algebra problem of finding the set of left eigenvectorsf ⃗ l i g of ℓ, we can construct the left eigenoperators ofL using Eq. (A.14). Finally, it is well known that each left eigenvector is orthogonal to all right eigenvectors except its corresponding one (the one is shares an eigenvalue with), and vice versa [152]. By choice of normalization, the inner products of corresponding left and right eigenvectors can always be made unity for any matrix with nondegenerate eigenvalues. Assume nondegeneracy and that we have normalizedℓ’s inner products of corresponding left and right eigenvectors, i.e., ⃗ l i ⃗ r j =δ ij . Let us show in which sense this property is inherited by the left and right eigenoperators ofL: Tr(L i R j ) = X kl ( ⃗ l i ) k (⃗ r j ) l Tr(F k F l ) = X k ( ⃗ l i ) k (⃗ r j ) k = ⃗ l i ⃗ r j =δ ij . (A.15) Note that Tr(L i R j )6=hL i ,R j i since we do not take the Hermitian conjugate ofL i under the trace [this only becomes possible ifℓ is symmetric, since we would need its eigenvalues to be real in order for Tr(L i R j ) = Tr(L † i R j ) to hold]. 124 App endix B: Analytical solution of the PMME Here we present the analytical solution of the PMME, Eq. (2.22), for our model. We take the Laplace transform, and the PMME becomes: s˜ ρ(s)ρ(0) =L 0 ˜ ρ(s)+L 1 Lap[k(t)exp(L 0 +L 1 )t]˜ ρ(s) . To deal with e Lt :=e (L 0 +L 1 )t , it is convenient to work in the damping basis ofL, as defined in Appendix A. Recall that the sets of right and left eigenoperators of L, fR i g and fL i g, are complete and mutually orthonormal in the sense of Eq. (A.15). We therefore expand ρ in the basis of right eigenoperators ofL: ρ(t) = X i µ i (t)R i , (B.1) where the expansion coefficients are µ j (t) = X i µ i (t)Tr(L j R i ) = Tr[L j ρ(t)] . (B.2) Substituting Eq. (B.1) into the PMME Eq. (2.22), we obtain: X i ∂µ i (t) ∂t R i = X i µ i (t)L 0 R i X i Z t 0 dt ′ k(t ′ )exp(λ i t)µ i (tt ′ )L 1 R i . Notice that if we assume that [L 0 ,L 1 ] = 0 (as is the case for us), then L 0 and L 1 both commute with L = L 0 +L 1 and hence share the same set of left and right eigenoperators with it, i.e.,L 0 (R i ) =λ 0 i R i ,L † 0 (L i ) =λ 0 i L i ,L 1 (R i ) =λ 1 i R i ,L † 1 (L i ) =λ 1 i L i . Multiplying both sides of Eq. (B.3) by L j from left and taking the trace, we obtain, under this assumption: ∂µ i (t) ∂t =λ 0 i µ i (t)+λ 1 i Z t 0 dt ′ k(t ′ )exp[λ i t ′ ]µ i (tt ′ ) . (B.3) Take the Laplace transform of both sides and use the shifting property of the Laplace trans- form, we have: s˜ µ(s)µ i (0) =λ 0 i ˜ µ i (s)+λ 1 i Lap k(t)e λ i t ˜ µ i (s) (B.4a) =λ 0 i ˜ µ i (s)+λ 1 i ˜ k(sλ i )˜ µ i (s). (B.4b) Therefore, ˜ µ i (s) = 1 sλ 0 i λ 1 i k(sλ i ) . (B.5) Taking the inverse Laplace transform: µ i (t) =ξ i (t)µ i (0), (B.6) where: ξ i (t) = Lap −1 1 sλ 0 i λ 1 i k(sλ i ) (B.7a) µ i (0) = Tr[L i ρ(0)]. (B.7b) 126 App endix C: Complete p ositivit y of the PMME The complete positivity of the PMME is not guaranteed because of the freedom in choosing the kernel k(t). A complete positivity test for the PMME was provided in Ref. [28]. Below we apply this test to the kernels specified in the model M i . The solution of the PMME can be viewed as a map Φ acting on the operators represented by dd matrices, where d is the dimension of the Hilbert space H = spanfjiig d i=1 . Using Eq. (B.1), ρ(t) = X i µ i (t)R i = X i ξ i (t)µ i (0)R i (C.1a) = X i ξ i (t)Tr[L i ρ(0)]R i = Φ[ρ(0)], (C.1b) where Φ[X] X i ξ i (t)Tr[L i X]R i . (C.2) Let jϕi = P i jii jii be a maximally entangled state in H H. According to the Choi’s theorem [30], Φ is CP if and only if the Choi matrix C 0, where C = (I Φ)jϕihϕj = X ij jiihjj Φ[jiihjj] . (C.3) We construct the Choi matrix for the PMME. Let us pick the basis statejii to be a column vector of zeros, except for a 1 in position i; we have: C = X ij jiihjj X k ξ k (t)Tr[L k jiihjj]R k (C.4a) = X k ξ k (t) X ij jiihjj hjjL k jiiR k (C.4b) = X k ξ k (t) X ij jiihjj L T k ij R k (C.4c) Hence, C = X k ξ k (t)L T k R k > 0 (C.5) is the complete positivity condition for the kernel, under a given LindbladianL and its set of left and right eigenvectors. For the LindbladianL in Eq. (3.3a) and its set of left and right eigenvectors in Eq. (D.3), the Choi matrix is: C = 0 B B B B B B B @ 1+Γrξ 4 1+Γr 0 0 ξ 2 0 Γr(1−ξ 4 ) 1+Γr 0 0 0 0 1−ξ 4 1+Γr 0 ξ 3 0 0 Γr+ξ 4 1+Γr 1 C C C C C C C A . (C.6) Its eigenvalues are found to be: λ c 1 = 1ξ 4 1+Γ r , (C.7a) λ c 2 = Γ r (1ξ 4 ) 1+Γ r (C.7b) λ c 3,4 = 1+ξ 4 2 s ξ 4 +1 2 2 Γ r +ξ 4 +Γ 2 r ξ 4 +Γ r ξ 2 4 jξ 2 j 2 (Γ r +1) 2 . (C.7c) 128 Therefore, the PMME in this case corresponds to a CP map iff: jξ 4 j< 1, (C.8a) jξ 2 j =jξ 3 j< p (Γ r +ξ 4 )(1+Γ r ξ 4 ) 1+Γ r < 1 , (C.8b) which is a condition on the problem parametersγ z ,γ + ,γ − and the kernel parameters⃗ a and ⃗ b. 129 App endix D: PMME solution with the sp ecific Lindbladian and k er- nels Choosing the operator basis as fF i g = fI,σ x ,σ y ,σ z g/ p 2, we follow the methodology of App. A and find the matrix representation of L =L 0 +L 1 of the specific PMME model we seek to construct in Sec. 2 to be ℓ = 0 B B B B B B B @ 0 0 0 0 0 Γs 2 2γ z ω z 0 0 ω z Γs 2 2γ z 0 γ − γ + 0 0 Γ s 1 C C C C C C C A . (D.1) The eigenvalues of ℓ are: fλ i g =f0, 1 2 Γ s 2γ z +iω z , 1 2 Γ s 2γ z iω z ,Γ s g . (D.2) It is straightforward to compute and normalize the corresponding right and left eigenvectors f⃗ r i , ⃗ l i g 4 i=1 such that they are mutually orthonormal, i.e., ⃗ l i ⃗ r j =δ ij . This allows us to find the right and left eigenoperators ofL using Eqs. (A.7) and (A.14), which yields: R 1 = 0 B @ 1 Γr+1 0 0 Γr Γr+1 1 C A , R 2 = 0 B @ 0 1 0 0 1 C A , R 3 = 0 B @ 0 0 1 0 1 C A , (D.3a) R 4 = 0 B @ 1 Γr+1 0 0 1 Γr+1 1 C A (D.3b) L 1 = 0 B @ 1 0 0 1 1 C A , L 2 = 0 B @ 0 0 1 0 1 C A , L 3 = 0 B @ 0 1 0 0 1 C A , (D.3c) L 4 = 0 B @ Γ r 0 0 1 1 C A . (D.3d) It is simple to verify that this set satisfies Eq. ( A.15) as required. ApplyingL 0 andL 1 tofR i g, we find the corresponding sets of eigenvalues: fλ 0 i g =f0, Γ s 2 +iω z , Γ s 2 iω z ,Γ s g (D.4a) fλ 1 i g =f0,2γ z ,2γ z ,0g (D.4b) Next, we need to evaluate Eq. (B.7a) with the specific forms of the kernels we have chosen. Regardless of the kernel, ˜ ξ 1 (s) = 1 s ()ξ 1 (t) = 1 (D.5a) ˜ ξ 4 (s) = 1 s+Γ s ()ξ 4 (t) =e −Γst , (D.5b) while for i = 2,3: f(t)ξ 2 (t) =ξ ∗ 3 (t) (D.6a) = Lap −1 1 sλ 0 2 λ 1 2 ˜ k(sλ 2 ) . (D.6b) 131 For the exponentially decaying kernel in Eq. (3.7), we make the parameter substitutionx 2+ b 0 γz andy γs 2γz i ωz γz and transform the variables in the Laplace transform correspondingly as s =γ z z and τ =γ z t, to get: f(τ) = Lap −1 h ˜ f(z) i (D.7a) ˜ f(z) = z +x+y (z +y) 2 +x(z +y)+2 . (D.7b) The analytical solution can be found by using the residue theorem [153]: f(τ) = sum of residues of e zt ˜ f(z) at poles of ˜ f(z) . (D.8) The rational function ˜ f(z) has two poles z 1 and z 2 z 1,2 = 1 2 x2y p D , Dx 2 8 , (D.9) and using Eq. (D.8): f(τ) =e z 1 τ z 1 +y +x z 1 z 2 +e z 2 τ z 2 +y +x z 2 z 1 . (D.10) For the other kernel in Eq. (3.8), we make the parameter substitution a 0 γ 2 z x, y γs 2γz i ωz γz + 2, a 1 = γ z w,b 0 γ 2 z u, b 1 γ z v, and transform the variables in the Laplace transform correspondingly as s =γ z z and τ =γ z t, to get: f(τ) = Lap −1 h ˜ f(z) i (D.11a) ˜ f(z) = p 1 (z) (z +y2)p 1 (z)+2w(z +y)+2x , (D.11b) where p 1 (z) = (z +y) 2 +v(z +y) +u. The analytical solution can therefore be found in 132 terms of the roots of the cubic polynomial (z +y2)p 1 (z)+2w(z +y)+2x (D.12a) =z 3 +c 2 z 2 +c 1 z +c 0 = 0 , (D.12b) where the coefficients are c 2 = 3y +v2 (D.13a) c 1 = 3y 2 +2vy4y +3w2v (D.13b) c 0 =y 3 +vy 2 2y 2 +2wy2vy +2x2w . (D.13c) The corresponding depressed cubic is found by the substitution z =z ′ c 2 /3, r(z ′ ) =z ′3 +pz ′ q = 0 (D.14) where the coefficients are p = 3c 1 c 2 2 3 (D.15) q = 9c 1 c 2 27c 0 2c 3 2 27 (D.16) which yields the cubic discriminant D = p 3 3 + q 2 2 (D.17) Further denote S = 3 r q 2 + p D, T = 3 r q 2 p D, (D.18) 133 The zeros of the cubic are z 1 = 2x 3 +(S +T) (D.19) z 2 = 2x 3 1 2 (S +T)+ i 2 p 3(ST) (D.20) z 3 = 2x 3 1 2 (S +T) i 2 p 3(ST) (D.21) This completes the exact solution of the PMME. 134 App endix E: Best-fit parameters of the nested PMME mo dels ω z (MHz) γ z (MHz) γ − (MHz) γ + (MHz) b 0 a 0 b 1 Fig. 3.2.2 M 0 9.610 −2 (310 −4 ) 1.210 −2 (810 −4 ) 1.910 −2 (810 −4 ) 1.910 −3 (310 −4 ) M 1 5.610 −2 (210 −4 ) 3.310 −3 (210 −5 ) 1.710 −2 (310 −4 ) 1.110 −3 (110 −4 ) 9.610 −5 (310 −6 ) M 2 5.610 −2 (210 −4 ) 3.310 −3 (210 −5 ) 1.710 −2 (310 −4 ) 1.110 −3 (110 −4 ) 0 5.510 −5 (910 −5 ) 0 Fig. 3.2.5 M 0 5.310 −2 (310 −4 ) 1.210 −2 (710 −4 ) 1.310 −2 (810 −4 ) 1.510 −3 (410 −4 ) M 1 1.710 −2 (210 −4 ) 2.810 −3 (310 −5 ) 1.410 −2 (810 −4 ) 1.810 −3 (410 −4 ) 2.310 −3 (110 −3 ) M 2 1.710 −2 (210 −4 ) 3.010 −3 (910 −5 ) 1.310 −2 (810 −4 ) 1.310 −3 (410 −4 ) 8.610 −2 (210 −2 ) 9.710 −2 (210 −2 ) 0 Fig. 3.2.6 M 0 1.210 −1 (510 −3 ) 4.310 −2 (210 −3 ) 1.810 −2 (910 −4 ) 2.410 −3 (410 −4 ) M 1 1.310 −1 (710 −4 ) 3.710 −3 (210 −4 ) 1.910 −2 (110 −3 ) 2.810 −3 (4times10 −4 ) 4.710 −2 (410 −3 ) M 2 1.310 −1 (710 −4 ) 1.010 −2 (110 −3 ) 1.810 −2 (910 −4 ) 2.410 −3 (410 −4 ) 1.010 −1 (210 −2 ) 5.210 −1 (110 −1 ) 0 Fig. 3.2.7 M 0 0 6.510 −2 (210 −3 ) 3.210 −2 (110 −3 ) 4.010 −3 (310 −4 ) M 1 2.510 −2 (910 −4 ) 8.610 −3 (310 −4 ) 3.210 −2 (110 −3 ) 4.110 −3 (310 −4 ) 3.110 −2 (310 −3 ) M 2 2.510 −2 (910 −4 ) 1.110 −2 (910 −4 ) 3.210 −2 (110 −3 ) 4.010 −3 (310 −4 ) 1.110 −1 (410 −2 ) 2.010 −1 (710 −2 ) 0 T able E.0.1: Best-fit parameters of the mo dels M 0 ,M 1 andM 2 constructed using the fitting datasets in Fig. 3.2.2 , Fig. 3.2.5 , Fig. 3.2.6 and Fig. 3.2.7 . The v alues i n the paren theses corre- sp ond to the 2σ error bars (95% confidence in terv als) estimated using the b o otstrap metho d. P arameters with b est-fit v alues smaller than 1e-5 are expressed as zero as their effects on the dy- namical predictions are negligible and w ell b elo w the measuremen t precision of the tomograph y exp erimen t. F or the same reason, w e rep ort the non-zero b est-fit parameters with t w o significan t figures and their confidence in terv als with one significan t figures. 136 App endix F: F abrication Recip e 1 Chip cleaning Use a Si(111) wafer. Gather beakers for Acetone, Methanol, IPA, and DI water at the solvent fume hood. Check that the sonicator is set to 100% power at 72 kHz. Perform the following cleaning procedure: Sonicate in Acetone for 6 minutes, Sonicate in Methanol for 3 minutes, Sonicate in IPA for 3 minutes, Swirl in DI for 30 seconds to rinse, Dry with N2, Place on a hotplate at 110-180 ◦ C for 20 seconds. Spin resist with the recipe below: It is important to ensure that the chip is not allowed to dry between steps 1-4. In practice, this means pulling the chip from one beaker and place in the next as fast as possible. It is best to do the cleaning procedure immediately before spinning to minimize the chance that debris finds its way to the silicon surface. 2 Spin coat Check the hotplates are set to 180 ◦ C. Set the spinner recipe for MMA EL13 as follows: Ramp 100/s, speed 500, duration 5 seconds, Ramp 2000/s, speed 3000, duration 60 seconds, Ramp 2000/s, speed 0, duration 0. After spin coating MMA, bake at 180 ◦ C for 5 minutes. Set spinner recipe PMMA A6 as follows: Ramp 100/s, speed 500, duration 5 seconds, Ramp 2000/s, speed 4000, duration 60 seconds, Ramp 2000/s, speed 0, duration 0. After spin coating PMMA, bake at 180 ◦ C for 5 minutes. 3 E-beam lithography Be careful with choosing the dose for each layer. Check under the microscope that the chip is clean (no visible dust) within the writing field. Align and load the sample. Generate .gdf files from .gds using Beamer, separate layers using large beam current (100 nA) and medium beam current (50 nA) for features close to small features, small beam current (200 pA) for small features, specify dose ratio and proximity effect correction. Restore beams and measure beam current, from large beam current to small beam current. Submit jobs to write. 4 Development Set up the ice bath, and let ice and water sit for 25 minutes until the temperature drops to 2 degrees ◦ C, Gently agitate the chip in the MIPK:IPA for 80 seconds , Immediately transfer to IPA beaker for another 60 seconds, Immediately transfer to DI water for another 30 seconds, Nitrogen blow dry. 138 5 Pre-ash Oxygen flow rate = 110 sccm, Power = 60 W, pressure = 300 mTorr, time = 30 sec 6 Evaporation 7 Liftoff Soak the chip in acetone with a water bath at 45 degrees C for 3 hours. Procedures for taking the chip out of the acetone: Towards the end of the soak, gather up the beakers for methanol, IPA, and DI water at the solvent fume hood. Fill each beaker with the respective solvent. Check the sonicator settings: 40% power, 72 kHz. When the soak is complete: Sonicate the chip in the acetone liftoff beaker for 30 seconds (more time if the liftoff has not been completed.) With the acetone wash bottle in the other hand, take the chip out of the acetone slowly, maintaining a stable jet of Acetone to the chip to make sure there are no liftoff metal chips left on the device; immediately dip the chip in the next beaker before the solvent dries out. Sonicate in methanol for 10s, similarly to step 2, transfer the chip from the methanol beaker to the IPA beaker. Sonicate in IPA for 10s, and transfer the chip from the IPA beaker to the DI water beaker with the same technique. Sonicate in DI water for 20 sec. Rinse with DI water. Dry with N2. 139 8 SEM Make sure to image the SQUIDs for the dissipators and check if the SQUID loops are connected before dicing and packaging. 9 Probe station measurement 10 Dicing To fit in the package, the chip should be diced to (and no more than) 5 by 5 mm. The ground plane is 4.8 by 4.8 mm, which leaves a 100-um margin for each cut. Before dicing: spin coat PMMA and bake for 5 min to protect the surface. 11 Post ash Oxygen flow rate = 110 sccm, Power = 60 W, pressure = 300 mTorr, time = 60 sec. 12 Packaging Thoroughly clean the circuit board with IPA (use the cotton tip); check under the microscope to make sure the bonding surface is as smooth and as clean as possible. 13 Device mounting Check the orientation of the device. Connect 2 SMA ports, 4 SSMB ports (2 ports for FFL and 2 ports terminated), and the DC pin for the coil; connect straps for better device thermalization. Check the coil is connected using Keithley. 140 App endix G: Measuremen t setup Figure G.0.1: Measuremen t setup with VNA Figure G.0.2: Measuremen t setup with Quan tum Mac hine OPX con troller 142
Abstract (if available)
Abstract
A central challenge in controlling and programming quantum processors is to overcome noise that arises from the system’s unwanted interaction with the environment. From the system’s perspective, the open system effects alter the desired system evolution, leading to decoherence and dissipation. From the environment’s perspective, an engineered controllable Hamiltonian modifies the effective environment that the qubit undergoes, and the open system effects can be used as a resource.
In the first part of the thesis, I discuss how to model noise in a superconducting qubit. I focus on non-Markovian noise, that is, temporally correlated noise, and it is challenging to mitigate. I show how to construct a simple phenomenological dynamical model known as the post-Markovian master equation to accurately capture and predict non-Markovian noise in superconducting qubits. The model also allows the extraction of information about cross-talk from near-neighboring qubits and measures of non-Markovianity.
The system’s coupling to the environmental degree of freedom can also be used to our advantage if we can control it. In the second part of the thesis, I discuss how to engineer a system’s coupling to a cold source of dissipation to remove entropy from a system with unwanted excitations. This is useful to speed up readout cavity reset, qubit reset, or preserve qubit coherence by eliminating unwanted thermal residue photons in the readout cavity. I show the design and operation of an active, on-demand source of dissipation via parametric coupling. Our experiment demonstrates it can be used to reset the readout resonator at a rate greater than 20 MHz through a quantum refrigeration process.
This thesis addresses the interplay between system dynamics and environment fluctuation for a small system size with the outlook that noise modeling and engineering techniques discussed can be extended to larger system sizes.
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Zhang, Haimeng
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Core Title
Modeling and engineering noise in superconducting qubits
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Viterbi School of Engineering
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Doctor of Philosophy
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Electrical Engineering
Degree Conferral Date
2023-08
Publication Date
08/15/2023
Defense Date
07/17/2023
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OAI-PMH Harvest,quantum computing,quantum noise,superconducting qubits
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Levenson-Falk, Eli (
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quantum computing
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