Non-Gaussian Noise Spectroscopy with Superconducting Qubits by Youngkyu Sung B.A., Seoul National University (2016) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2018 ○c Massachusetts Institute of Technology 2018. All rights reserved.

Author...... Department of Electrical Engineering and Computer Science August 31, 2018 Certified by...... William D. Oliver Professor of the Practice, Physics Department Thesis Supervisor 2nd Certified by ...... Simon Gustavsson Principal Research Scientist, Research Laboratory Of Electronics Thesis Supervisor Accepted by ...... Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Non-Gaussian Noise Spectroscopy with Superconducting Qubits by Youngkyu Sung

Submitted to the Department of Electrical Engineering and Computer Science on August 31, 2018, in partial fulfillment of the requirements for the degree of Master of Science in Computer Science and Engineering

Abstract Most quantum control and quantum error-correction protocols assume that the noise causing decoherence is described by Gaussian statistics. However, the Gaussianity assumption breaks down when the quantum system is strongly coupled to a sparse environment or has a non-linear response to external degrees of freedom. Here, we experimentally validate an open-loop quantum control protocol [2] that reconstructs the higher-order spectrum of a non-Gaussian dephasing process using a superconduct- ing qubit as a noise spectrometer. This experimental demonstration of non-Gaussian noise spectroscopy protocol represents a major step towards the goal of demonstrat- ing a complete noise spectral characterization of quantum devices.

[1] E. Paladino et al. Rev. Mod. Phys. 86, 361 (2014). [2] L. M. Norris, G. A. Paz-Silva, L. Viola, Phys. Rev. Lett. 116, 150503 (2016)

Thesis Supervisor: William D. Oliver Title: Professor of the Practice, Physics Department

Thesis Supervisor: Simon Gustavsson Title: Principal Research Scientist, Research Laboratory Of Electronics

3 4 Acknowledgments

I would like to thank my advisor, Professor Oliver, who gave me the opportunity to work on this exciting project and encouraged me to pursue my own research interests. For the past two years, I am deeply inspired by his extraordinarily broad knowledge in fields of science and engineering. His kindness and generosity with time areof great help to me and make me persevere with any experimental difficulty. I am always grateful to Dr. Gustavsson for his teaching about how to perform amazing experiments. Numerous discussions with him over the course of two years in the lab and his office gave me a tremendous amount of guidance in my academic journey. I can’t forget Professor Orlando’s lectures in solid state physics, and lectures on circuit quantization in our lab. Not only an amazing lecturer and theorist, he is also a great supportive mentor; conversations with him always relieve and inspire me. This work would not have been possible without our collaborators; Dr. Beaudoin, Dr. Norris and Professor Viola in Dartmouth college. It was truly an enjoyable experience to work with amazing theorists who are always ready to discuss and provide deep insights into physics obscured by puzzling experimental data. Our Skype calls usually go longer than two hours, which sounds quiet long, but it was actually a lot of fun and has made a lot of progress; we made things clear, set up a plan to figure out problems we had, and finally resolved the problems! I hope to continue our collaboration and explore new physics and cool techniques together! Dr. Yan was a postdoc in the group pioneering this project. From late nights in the lab to phone calls while he is driving, he has always been my mentor who taught me countless things, not only details about the experiment, but also important lessons to become a better scientist. Not only a mentor, he is one of the best friends who played and watched basketball games together until late night. This work would not have been possible without his research experience and kind help. Amy and Bharath are both cool lab-mates who always make me laugh and also outstanding teachers who are always ready to help. I feel particularly fortunate to work in a research group with these vibrant spirits! Also, it is such a pleasure to

5 have a friend who has the same tastes in food, Jack. He is also smart and talented in experiments, which makes me learn many things from him. Tim and Ben are great grads as well, who are doing amazing work and making a lot of progress, which motivate me a lot. Conversations with our post-docs (Philip, Dan, Joel, Morten, Roni and Jochen) are always inspiring and intellectually pleasing. They have very solid backgrounds in various fields, so that I feel so thankful to learn a variety of things from them. I also would like to extend my thanks to Moritz, Uwe, Thorvald, Andreas, Niels Jacob and Antoine who have visited our group as visiting students and helped and taught me a lot. Additionally, I enjoyed the company of all my friends here in the MIT community. They made my passing years here an unforgettable memory. Finally, I must give my deep gratitude to my parents, and my brother, for their love and unconditional support. Their lovely messages and phone calls boost my energy more than any brand of energy drink or coffee.

6 Contents

1 Introduction 17 1.1 Quantum noise spectroscopy ...... 18 1.2 Non-Gaussian noise in superconducting qubits ...... 19 1.3 Overview of thesis ...... 20

2 Superconducting Qubits and Circuit QED 21 2.1 Superconducting Qubits ...... 21 2.1.1 Josephson junction as a non-linear inductor ...... 22 2.1.2 Cooper-pair box (CPB) ...... 23 2.1.3 Transmon ...... 24 2.1.4 Persistent-current flux qubit ...... 27 2.1.5 C-shunt flux qubit ...... 28 2.1.6 Quarton ...... 30 2.2 Cavity quantum electrodynamics ...... 34 2.2.1 Strong coupling regime ...... 34 2.2.2 Dispersive coupling regime ...... 36 2.3 Circuit QED ...... 36 2.3.1 Coupling a superconducting qubit to a coplanar waveguide res- onator ...... 36 2.3.2 Dispersive regime of circuit QED ...... 37 2.4 Coherence properties of superconducting qubits ...... 38 2.4.1 Coherence time and anharmonicity of a Quarton ...... 38

7 3 Quantum Noise Spectroscopy 41 3.1 A qubit as a noise probe ...... 41 3.1.1 Hamiltonian of open quantum system ...... 41 3.1.2 The relation between spin observables and bath-operator cu- mulants ...... 42 3.2 Quantum noise spectroscopy (QNS) protocols ...... 43 3.2.1 Bylander et al.’s protocol(2011): CPMG spectroscopy for Gaus- sian dephasing process ...... 44 3.2.2 Álvarez and Suter protocol (2011): Repetition of suitably de- signed pulse sequences ...... 46 3.2.3 Norris et al. protocol (2016): Generalization of the Álvarez and Suter protocol to Non-Gaussian dephasing processes ...... 47 3.2.4 Norris et al.’s protocol(2016) cont.: Reconstructing bispectrum 49

4 Experimental Setup and Details 57 4.1 Cryogenic setup ...... 57 4.2 Room temperature control ...... 60 4.3 Pulse control and modulation ...... 60 4.4 Materials and fabrication of superconducting qubits ...... 60 4.4.1 Growth and patterning of high-Q aluminum ...... 61 4.4.2 Patterning the qubit loop and Josephson junctions ...... 61 4.4.3 Dicing and packaging ...... 62

5 Reconstructing the Bispectrum of non-Gaussian Dephasing Pro- cesses in a Superconducting Qubit 63 5.1 Non-Gaussian dephasing in a superconducting qubit ...... 64 5.1.1 Non-linear energy dependence on external flux ...... 64 5.1.2 Squared Gaussian dephasing process ...... 65 5.2 Verifying the presence of non-Gaussianity ...... 67 5.3 Reconstruction of the bispectrum ...... 69 5.3.1 Review of the old approach (Section 3.2.4)...... 69

8 5.3.2 Reconstructing the bispectrum with new approach ...... 70

6 Conclusions and Future Work 77 6.1 Characterizing natural non-Gaussian dephasing processes ...... 77 6.2 Regularized maximum likelihood estimation for bispectrum reconstruc- tion ...... 78 6.3 Faster noise spectroscopy using active reset protocols ...... 79

A Moments and Cumulants 81 A.1 Moment and moment-generating function ...... 81 A.2 Cumulant and cumulant-generating function ...... 82

B Derivation of the Joint Hamiltonian in the Toggling Frame 83

C Timing digram of control pulse sequences 89

9 10 List of Figures

1-1 A schematic diagram showing the steps involved in quantum noise spectroscopy (QNS)...... 18

2-1 Circuit diagrams of cooper pair box(CPB) qubit and Transmon. . . . 24 2-2 Charge dispersion. The energies of the lowest 5 levels of the charge

qubit Hamiltonian in units of the charging energy 퐸C. For low 퐸J/퐸C, the qubit is in the Cooper-pair box regime, and the energies are parabolic

functions of the offset charge 푛푔, with avoided crossings. As the ratio

of 퐸J/퐸C is increased, the levels become exponentially flattened, as one enters the Transmon regime...... 25

2-3 Potential function of the persistent-current flux qubit where 훼 = 0.75, 푓 = 0.5. We can clearly see that the potential has a double-well profile,

around the points where 휑p + 휑m ≈ 2푛휋, where 푛 is an integer. . . . . 29 2-4 Near 푓 = 0.5, the potential function assumes a double well profile. For 푓 < 0.5, the left well has a lower energy (clockwise circulating cur- rent); for f > 0.5, the right well has a lower energy (counter-clockwise circulating current). At f = 0.5 the diabatic state energies are degener- ate. Tunneling energy mixes the states, forming superpositions of the circulating currents and the qubit has no magnetic polarization. Im- age courtesy of Prof. William D. Oliver in his 44th IFF spring school lecture notes [36]...... 29 2-5 Circuit diagrams of a persistent-current flux qubit and a C-shunt flux qubit...... 30

11 2-6 Circuit diagram of the generalized flux qubit ...... 31 2-7 Illustration of the dependence of key qubit features (anharmonicity

(퐴), transition frequency (휔01), charge-noise sensitivity, flux-noise sen- 훾 1 sitivity) on various circuit parameters (퐼푐, 퐶Σ, 푁, 푁 = 훼푁 ). The red- colored (blue-colored) dial in each box denotes that the corresponding circuit parameter should be increased (be decreased) to obtain the desired feature. The gray-colored means that the corresponding pa- rameter has no relevance to the feature...... 33 2-8 Illustration of cavity QED. A two-level atom passes through a Fabry-

Perot cavity over a transit time 푡푡푟푎푛푠푖푡. While the atom passes through, it interacts coherently with photons contained in the cavity with a strength 푔. Photons can leave the cavity at a rate 휅 and the atom decays via non-cavity modes at a rate 훾 ...... 35 2-9 Illustration of Circuit QED with a scanning electron microscope (SEM) image of a Quarton. The Febry-Perot cavity is now replaced by a microwave-frequency co-planar waveguide (CPW) transmission line res- onator. Near either end of the CPW, superconducting qubits are cou- pled to the transmission line resonator through an electrostatic capac- itive interaction...... 37

푒푐ℎ표 2-10 Illustration of pulse sequences used to measure 푇1 and 푇2 , and a 푒푐ℎ표 quarton’s coherence times (푇1 = 82.83 ± 0.57 휇푠, 푇2 = 129.59 ± 3.27 휇푠)...... 39 2-11 Anharmonicity (풜) of a quarton. The qubit transitions measured are the 0-1 transition in a single tone spectroscopic measurement (red), and the 1-2 transition (blue) while populating the first excited state with a second drive on the 0-1 transition. The second excited state is not populated at normal spectroscopy powers (red), indicating a low excited state thermal population. The 0-1 and 1-2 transitions are separated by 813 MHz; This is why the quarton can be treated as a two-level system even during fast control operations...... 40

12 3-1 Schematic representation of the CP(MG) sequence and the filter func- tions of different pulse sequences ...... 45

3-2 퐺2퐷−푐표푚푏(휔1, 휔2). This 2D frequeny comb samples the multiples of

harmonic frequency(2휋/푇 ) over 푓1 and 푓2. Plotted for T = 1휇푠, M =10. 51 3-3 The principal domain of a bispectrum (red dotted line). The bispec-

trum at point (휔1, 휔2) in the principal domain is equivalent to the points in other 11 regions in this figure by the symmetry arguments. 52 3-4 An octant containing the harmonic frequencies where the bispectrum will be reconstructed. The interior points in the blue region are the points which do not contain a zero frequency and the points in red region contain a zero...... 53

4-1 Schematic of cryogenic circuitry ...... 58 4-2 Room temperature control schematic ...... 59

5-1 Spectrum of the Quarton near the sweet-spot. The measurement is ob- tained by measuring the transmission at a fixed frequency close to the cavity resonance (7.348 GHz) and sweeping the qubit drive frequency(푦 axis) ...... 64 5-2 An example of the power spectrum of Gaussian noise, 휉(푡) [V] and the 2 7 bispectrum of the squared Gaussian noise 훽푉 휉 (푡) [Hz]. 훽푉 = 3.5 * 10 [Hz / 푉 2]...... 65 5-3 Illustration of the CPMG sequence with two control pulses (휋 pulses). The last 휋/2 pulse is a tomography pulse which can be either 푋 + 휋/2

or 푌 + 휋/2, which projects ⟨휎푥⟩ or ⟨휎푦⟩ onto the measurement-axis

(휎푧), respectively...... 67

5-4 Experimental result and Monte-Carlo simulation result for 휑, 휒, vs. 푃0, where the phase angle (휑) and the decay parameter (휒) are defined as 휑 = tan−1(− ⟨휎푥⟩ ), 휒 = − ln(√︀⟨휎 ⟩2 + ⟨휎 ⟩2). We ran a Monte-Carlo ⟨휎푦⟩ 푥 푦 simulation over 106 noise realizations, and the experiment over 4 × 105 noise realizations...... 68

13 5-5 Illustration of the Ramsey sequence used to estimate the mean of de- phasing process. The first 푋, +휋/2 pulse is to initialize the qubit onto the transversal plane along the y-axis and the second 푌, +휋/2 pulse

projects the x-component of spin vector (−휎^푥) onto the measurement axis (z-axis)...... 71 5-6 Robust estimation of the mean of the dephasing process. The sepa- ration between the x intercepts of two fitted lines corresponds to the

푒푠푡 mean of dephasing process, 휇퐵 ...... 72 5-7 The principal domain of 2D frequency grid containing the harmonic frequencies where the bispectrum will be reconstructed and the timing diagrams for base control sequences. The length of the base control sequence (휏) is 960 ns, and Seq. 2, ..., 11 are repeated 10 times (M = 10). Note that the bispectrum on the x-axis of the principal domain (orange color) contributes to the phase accumulation of only sequence 1, ..., 5, which have non-zero values of the filter function at zero fre-

quency (퐹푝(0, 휏) ≠ 0)...... 73

5-8 휑˜푝(푀푇 ) and 휒 for 11 different control sequences...... 74 5-9 Reconstructed bispectrum based on experimental data (experiment) and theoretical calculation (theory)...... 75

C-1 Timing diagrams of the control pulse sequences. Note that Seq. 2-11 are repeated 10 times. Only 휋 pulses are shown. All 휋 pulses are around 푦-axis ...... 90

14 List of Tables

2.1 Coherence properties of the state of art superconducting qubits [55].

푒푐ℎ표 푇1 is the qubit energy relaxation time. 푇2 is the qubit coherence * time measured in a Hahn-echo (refocusing) experiment. 푇2 is the free induction decay (FID) time measured in the Ramsey interference ex- periment...... 38

* 퐸 5.1 Qubit frequency (휔01), coherence properties (푇1, 푇2 , 푇2 ) and single

qubit gate fidelity퐹 ( 푔) of the Quarton used in this work...... 65

C.1 The control pulse sequences designed for reconstructing the bispec- trum. Only 휋 pulses are shown...... 90

15 16 Chapter 1

Introduction

In the past few decades, tremendous progress has been made in the quest for quantum computing, from both the theoretical and experimental side. In particular, substan- tial progress was achieved in extending the time within which a solid-state qubit exhibits coherent dynamics [2, 57, 40, 18, 23]. This improvement is mostly attributed to clever engineering which leads to successful isolation of the qubits from external decoherence sources. Nevertheless, the efficacy of the devices for quantum informa- tion processing is still limited mostly by the finite decoherence rates of the individual qubits and quantum gate [46]. In many cases, solid state quantum devices are affected by noise that decreases with frequency 푓 approximately as 1/푓 [24, 41]. Such noise is due to material- and device- specific microscopic degrees of freedom interacting with the quantum devices. Ingen- eral, the statistical processes producing 1/푓 noise are not necessarily Gaussian [41]. In other words, statistical correlations higher than the second order should be taken into account in order to fully understand the dynamics of actual noise affecting the quantum devices. This fact can also has relevance for building a fault-tolerant quantum computer; To date, most quantum threshold theorems model noise as either Markovian or Gaus- sian, but not as a non-Gaussian process [45]. Hence, if deviations from Gaussian behavior arise in the coherent quantum dynamics of solid state qubits, the threshold estimated by existing quantum threshold theorems may not be statistically valid [45].

17 Step (1) Step (2) Step (3) Step (4)

...... Readout

Control Pulses

σˆ x

S(f)

Freq.

σˆ y

Figure 1-1: A schematic diagram showing the steps involved in quantum noise spec- troscopy (QNS).

This fact motivates us to further understand the spectral properties of noise causing the decoherence of quantum computer, in particular, beyond 2nd-order statistics.

1.1 Quantum noise spectroscopy

The sensitivity of a qubit to its surrounding environment allows one to characterize the spectral properties of a noise process affecting the quantum system by measuring its response. Many quantum noise spectroscopy (QNS) protocols, have been developed and implemented in recent years to obtain precise knowledge of the noisy environment that the qubits experience. Their objective is to characterize the actual noise affecting a quantum system of interest, regardless of its source, in terms of its correlations, or more specifically, in terms of its set of power poly-spectra. In essence, a QNS protocol entails four main steps [43]: (1) prepare the probe qubits in a known state; (2) let the qubits evolve under both bath-induced noise and external control sequences; (3) measure a set of observables of the qubits that quantifies their response to the bath and the applied control; (4) extract information about the bath

18 spectra from the measured values of the observables. To date, single-qubit QNS protocols designed to characterize a classical Gaussian noise source in the dephasing regime have been successfully demonstrated in experimental platforms including solid- state NMR [1], superconducting and spin qubits [8, 32, 60, 56, 13], as well as nitrogen vacancy centers in diamond [30]. However, no one has experimentally demonstrated a QNS protocol for reconstructing high-order spectra of non-Gaussian dephasing noise. In this work, we demonstrate ,for the first time, a QNS protocol for reconstructing the bispectrum of a non-Gaussian dephasing process.

1.2 Non-Gaussian noise in superconducting qubits

If a qubit is strongly coupled to a sparse bath of two-level fluctuators, the statistics of the noise process that the qubit experiences will likely be non-Gaussian [35]. In the case of superconducting qubits, fluctuating two-level defects (TLSs) in Joseph- son junction tunnel barriers and local dielectrics are believed to cause low-frequency noise with spectra proportional to 1/f [37, 41]. R. W. Simmonds et al. first revealed spurious microwave two level systems within Josephson tunnel junctions by reporting spectroscopic data [50]. Also Lisenfeld, et al. reported that they observed strong interaction between a superconducting qubit and single two-level defect by observing energy exchange in time-domain measurements [25]. These implies that supercon- ducting qubits could experience non-Gaussian noise processes due to its strong cou- pling with a small number of defects at the insulating barrier. In addition to a strongly coupled sparse bath, a non-linear coupling to the environ- mental degrees of freedom can lead to non-Gaussian dephasing processes in qubits. All superconducting qubits have non-linear dispersion relations with respect to both flux and charge, which converts intrinsically Gaussian flux noise [57] and charge noise [23] into non-Gaussian dephasing.

19 1.3 Overview of thesis

In this work, we successfully performed higher-order noise spectroscopy of a non- Gaussian dephasing process in a superconducting qubit and reconstructed its bis- pectrum. To be able to fundamentally understand the experimental details of this work, we will start this thesis with an introduction to the basics of superconducting qubits. There will also be discussions about some of the basics of cavity QED and a replica of cavity QED in the microwave regime, so-called circuit QED, to illus- trate the fundamental concepts about how to control and readout superconducting qubits. In Chapter 3, I will describe how quantum noise spectroscopy works, espe- cially non-Gaussian noise spectroscopy which was theoretically developed by L. M. Norris in 2016 [35]. With these theoretical concepts established, I will turn to our experimental setup in Chapter 4 to explain how the devices are cooled in a helium dilution refrigerator, how the refrigerator is cabled to control and measure the devices with high fidelity, and how the devices are fabricated. In Chapter 6, I will present the main experimental results which demonstrate the experimental validation of the non-Gaussian noise spectroscopy protocol. Finally, I will conclude with thoughts on possible future work.

20 Chapter 2

Superconducting Qubits and Circuit QED

This chapter begins with a theoretical introduction to the physics of superconducting qubits and the circuit QED architecture which enables scalable quantum computing based on superconducting circuits. I will go through different kinds of superconduct- ing qubits to give a bird’s-eye view on how superconducting qubits have been devel- oped from their embryonic stages. In the last section of this chapter, I will briefly summarize the coherence properties of state-of-art superconducting qubits including one of our lab’s best-engineered qubit, the “Quarton” qubit.

2.1 Superconducting Qubits

Contrary to other solid-state qubits that aim to confine a small number of microscopic quantum degrees of freedom such as electron spins in quantum dots, superconductors are composed of a large number of paired electrons, or Cooper pairs, all of which have condensed into a single ground state [53]. Quantum effects are then the result of macroscopic degrees of freedom of a superconducting circuit. Ideally, non-dissipative circuit elements such as inductors and capacitors can be con- structed with superconductors. However, combining these linear elements only ends up with making harmonic oscillators which have evenly spaced energy levels; there

21 would be no two-level addressability in such a system. However, there is a simul- taneously non-linear and non-dissipative circuit element, known as the Josephson junction, which introduces anharmonicity into the system. By adding Josephson junctions, one can create a qubit which possess a non-uniformly spaced set of quan- tum mechanical energy levels, which enables us to address each level. The following section will review the non-linear Josephson junction and its application in building superconducting qubits.

2.1.1 Josephson junction as a non-linear inductor

A Josephson junction consists of two superconducting electrodes separated by an insulating barrier. Cooper pairs can tunnel coherently across the insulating barrier, with a supercurrent 퐼 that is given by

퐼 = 퐼0 sin 휑(푡) (2.1)

where 퐼0 is the critical current (the maximum sustainable junction supercurrent), and 휑(푡) is the superconducting phase difference across the junction [53]. The phase difference evolves in time in the presence of a potential 푉 across the junction according to

~ 푑휑 푉 = (2.2) 2푒 푑푡

By taking the time-derivative of the supercurrent 퐼, one can find the so-called Joseph- son effect,

푑퐼 푑휑 2푒푉 퐼0 = (퐼0 cos 휑) = cos 휑 (2.3) 푑푡 푑푡 ~

푑퐼 From Faraday’s law 푉 = −퐿 푑푡 , we can define the Josephson inductance as

Φ0 퐿J = (2.4) 2휋퐼0 cos 휑

22 ℎ where Φ0 = 2푒 is the superconducting magnetic flux quantum. This non-linear induc-

tance combined with the intrinsic capacitance of the Josephson junction, given by 퐶J, thus results in an anharmonic oscillator which serves as the basis of superconducting qubits [12].

2.1.2 Cooper-pair box (CPB)

One of the simplest superconducting qubits is the Cooper-pair box (CPB) [6, 12]. This charge-based circuit is formed when a superconducting island is connected to a reservoir of Cooper pairs through a junction. The Josephson effect allows for the coherent tunneling of the Cooper pairs between the island and the reservoir, while

a gate voltage (푉B) can electro-statically modulate a Coulomb potential. Through circuit quantization [12, 17], the Hamiltonian of CPB can be found to be

2 퐻퐶푃 퐵 = 4퐸C(^푛 − 푛g) − 퐸J cos 휑^ (2.5)

where 푛^ is Cooper pair number operator defined as the number of Cooper pairs having

passed through the junction , 푛푔 is the continuously variable offset gate charge due to applied dc bias, and 휑^ is the conjugate operator to 푛^, representing a phase difference 2 operator. The first term corresponds to the charging energy where 퐸C = 푒 /2퐶Σ and

퐶Σ = 퐶g + 퐶J is the total capacitance to ground of the CPB. The second term of the Hamiltonian reflects the inductive energy across the Josepshon junction where

퐸J ≡ 퐼0Φ0/2휋 is the Josephson energy. We can explicitly write this Hamiltonian in the charge basis by using the following relations

휕 푛^ = 푖 (2.6) 휕휑 ^ ^ 푛푒^ 푖휑 = 푒푖휑(^푛 + 1) (2.7)

23 to get

퐸J ∑︁ 퐻 = 4퐸 (^푛 − 푛 )2 − (|푛⟩⟨푛 + 1| + |푛 + 1⟩⟨푛|) (2.8) C g 2 푛

The CPB can be operated as a charge qubit in the regime where 퐸J ≪ 4퐸C, such that the Josephson coupling gives a small perturbation that lifts the energy degeneracy of

integer charge states. By operating the CPB at gate charge 푛g = ±0.5, the system can be reduced into a two-level qubit system with a reduced Hamiltonian given by

퐸 퐻^ ≈ 2퐸 (1 − 푛 )^휎 − J 휎^ (2.9) C g 푧 2 푥

where 휎푥, 휎푧 correspond to the standard spin 1/2 Pauli matrices.

Figure 2-1: Circuit diagrams of cooper pair box(CPB) qubit and Transmon.

2.1.3 Transmon

By operating in a different regime of the ratio 퐸J/퐸C, it is possible to have a qubit, which is insensitive [23, 49] to 1/푓 charge noise [19]. By shunting a large capacitance

which results in 퐸J ≫ 퐸C, the CPB system switches to a system that is best described as a slightly anharmonic oscillator. As a consequence, as compared to the conventional CPB, the transmon is much less sensitive to low-frequency charge noise and, therefore, has dramatically improved coherence times.

24 Figure 2-2: Charge dispersion. The energies of the lowest 5 levels of the charge qubit Hamiltonian in units of the charging energy 퐸C. For low 퐸J/퐸C, the qubit is in the Cooper-pair box regime, and the energies are parabolic functions of the offset charge 푛푔, with avoided crossings. As the ratio of 퐸J/퐸C is increased, the levels become exponentially flattened, as one enters the Transmon regime.

To be more quantitative with respect to (퐸J/퐸C), it is useful to write down the full energy level expressions for Eq. 2.5. Switching over to the phase basis, analytical solutions to Eq. 2.5 can be expressed in terms of the special Mathieu’s functions

푚푒휈(푞, 푎) as [23],

(︂ )︂ 1 −퐸J 휑 ⟨휑|푚⟩ = √ exp[푖푛g휑]푚푒−2(푛푔+푚) , (2.10) 2휋 2퐸C 2

25 and the eigen-energies are given by

(︂ )︂ −퐸J 퐸푚(푛g) = 퐸C푎2(푛푔+푘(푚,푛푔)) , (2.11) 2퐸C

where 푎푝(푞) is Mathieu’s characteristic value and 푘(푚, 푛푔) is an integer-valued func-

tion which orders the eigenvalues. The effect of increasing 퐸J/퐸C can be seen in the level dispersion curves in Figure. 2-2. As the ratio increases, each energy level flattens

considerably and the 푛g dependence of the first few levels essentially disappears. As there is no free lunch, the level flattening is accompanied by a reduced anharmonicity. It is worth mentioning that reduced anharmonicity can be a significant problem for quantum operations because it makes harder to address quantum levels individually. Now, with regards to the small anharmonicity, one can expand the cos 휑 in Eq. 2.5 to (1 − 휑2/2 + 휑4/24). The Hamiltonian then takes the form of the Duffing oscillator

√︀ 퐸C 퐻^ = 8퐸 퐸 (^푏†^푏 + 1/2) − 퐸 − (^푏† + ^푏)4, (2.12) C J J 12 where ^푏†, ^푏 are creation and annihilation operators for the harmonic oscillator portion of the cos 휑 expansion. Using perturbation theory and keeping only quartic terms of the form (푏†푏)2, the energy of the 푚-th level can be found to be:

(︂ )︂ √︀ 1 퐸C 퐸 ≈ −퐸 + 8퐸 퐸 푚 + − (6푚2 + 6푚 + 3). (2.13) 푚 J J C 2 12

The anharmonicity of a transition between the (m+1)-th and m-th level and a tran- sition between the m-th an (m-1)-th level is given by

훼푚 = 퐸푚+1,푚 − 퐸푚,푚−1 ≈ −퐸C, (2.14)

where 퐸푚푛 = 퐸푛 − 퐸푚. By comparing this to the absolute anharmonicity of the ground to first excited state transition, we can get the relative anharmonicity:

푟 −1/2 훼푚 = 훼푚/퐸01 = −(8퐸J/퐸C) . (2.15)

26 This reflects an algebraic dependence of the anharmonicity on 퐸J/퐸C. Although as 푟 퐸J/퐸C → ∞, the anharmonicity will be reduced 훼푚 → 0, good transmon performance can be obtained without needing to reach this extreme. Since the charge dispersion reduced exponentially with increasing 퐸J/퐸C, there can be sufficient charge noise suppression before the anharmonicity becomes too small and degrades the two-level addressability.

2.1.4 Persistent-current flux qubit

The persistent-current flux qubit provides the conceptual complement to the charge qubit; The physical observable where information is encoded is the superconducting circulating current whereas it was the excess charge on a superconducting island in the case of charge qubit. The persistent-current flux qubit is a superconducting loop interrupted by three [38, 31] or four Josephson junctions. One the junction is smaller in area by a factor 훼, with a typical value 훼 = 0.75 for a three-junction qubit and 훼 = 0.5 for a four-junction qubit [36]. In this section, I will focus on a three-junction flux qubit. The small junction on the left side (called the alpha junction ortheblack- sheep junction [15]) acts as a flux-shuttle value that lets a fluxoid in and outofthe loop. The array of larger junctions on the right side (called the array junctions) serve to add loop inductance. The loop itself has very small geometric inductance and can

be neglected. Here, I will assume symmetric large junctions with 퐸J2 = 퐸J3 ≡ 퐸J

and junction capacitances 퐶2 = 퐶3 ≡ 퐶. The alpha junction has capacitance 훼퐶 and

Josephson energy 훼퐸J. A flux qubit has a potential energy 푈 equivalent to the sum of Josephson energies,

∑︁ 푈 = 퐸J푖 (1 − cos 휑푖). (2.16) 푖

27 The fluxoid quantization condition sets 휑1 − 휑2 + 휑3 = 2휋푓, where 푓 ≡ Φ푒푥푡/Φ0 is the reduced external magnetic flux, normalized by Φ0. Eliminating 휑3 yields,

푈 = 퐸J[2 + 훼 − cos 휑1 − cos 휑2 − 훼 cos(2휋푓 + 휑1 − 휑2)] (2.17)

= 퐸J[2 + 훼 − 2 cos 휑p cos 휑m − 훼 cos(2휋푓 + 2휑m)], (2.18)

where we have used the sum and difference phases 휑p,m ≡ (휑1 ± 휑2)/2 The two- dimensional potential 푈 forms an “egg carton” as a function of 휑p and 휑m, with the unit cells periodic in 2휋 (See Fig. 2-3) [38]. When the applied flux is close to half a flux quantum, 푓 ≈ 0.5, the potential (within a unit cell) assumes a two-dimensional double well profile. A slice of this double well along the 휑푝 direction is shown in Fig. 2-4. The localized states in each well correspond to diabatic states of clockwise and counter-clockwise circulating persistent current around the loop with magnitude 퐼푝. At 푓 = 0.5, the lowest-energy quantized states of the wells are energy degenerate. Quantum tunneling through the double- well barrier hybridizes the classical (diabatic) states into superpositions of circulating currents, opening an avoided crossing of strength Δ. As illustrated in Fig. 2-4, 푓 < 0.5 tilts the left well (clockwise circulating current) to lower energy, whereas 푓 > 0.5 tilts the right well (counter-clockwise circulating current) to lower energy. The two level system Hamiltonian for the flux qubit is [38],

^ 1 ~ 퐻 = − [2퐼푝Φ0(푓 − 1/2)^휎푥 + ~Δ^휎푧] ≡ − [휖휎^푥 + Δ^휎푧], (2.19) 2 2 where 휎^푥,푧 are the Pauli spin matrices. 휖 and 훿 are in units of frequency.

2.1.5 C-shunt flux qubit

As with the CPB and transmon, there is a capacitively shunted version of the flux qubit [61, 57]. Decreasing 훼 decreases the 퐸J/퐸C ratio in the flux qubit and tends to flatten the energy dispersion, thereby decreasing the qubit’s sensitivity tolow- frequency flux noise. However, decreasing the 퐸J/퐸C ratio also makes the qubit more

28 Figure 2-3: Potential function of the persistent-current flux qubit where 훼 = 0.75, 푓 = 0.5. We can clearly see that the potential has a double-well profile, around the points where 휑p + 휑m ≈ 2푛휋, where 푛 is an integer.

Figure 2-4: Near 푓 = 0.5, the potential function assumes a double well profile. For 푓 < 0.5, the left well has a lower energy (clockwise circulating current); for f > 0.5, the right well has a lower energy (counter-clockwise circulating current). At f = 0.5 the diabatic state energies are degenerate. Tunneling energy mixes the states, forming superpositions of the circulating currents and the qubit has no magnetic polarization. Image courtesy of Prof. William D. Oliver in his 44th IFF spring school lecture notes [36]. sensitive to charge noise. This can be mitigated by shunting the small junction with a capacitor. Hence, by tuning both 훼 and the shunt capacitance(퐶푠ℎ), one can balance between flux noise sensitivity and charge noise sensitivity to obtain optimal coherence. Yan, et al. experimentally demonstrated that a capacitively shunted (C-shunt) flux qubit can have long relaxation time (푇1 = 55 휇푠) with long Hahn-echo decay time

29 Figure 2-5: Circuit diagrams of a persistent-current flux qubit and a C-shunt flux qubit.

퐸 (푇2 = 80휇푠) [57].

2.1.6 Quarton

Our group (Yan et al.) investigated the implications of various parameters of a generalized flux qubit to better understand how the key features of qubits are related to circuit parameters [58]. The dependence of relavant aspects of qubit on various parameters are summarized in Figure. 2-7. Based on this framework, Yan et al. has found a special circuit parameter configuration, named the Quarton regime, which would be practically useful for gate-based quantum computing. In this thesis, I will not discuss the details or simulation results, but rather just provide insight into the utility of the quarton. The illustration of the Hamiltonian picture of the Quarton regime is as follows. Through circuit quantization [12], the Hamiltonian of a generalized flux qubit can be found to be

(︃ 푁 )︃ 1 ∑︁ 퐻^ = −4퐸 (^푛 − 푛 )2 + 훼퐸 − cos(휑^ ) − 푐표푠(휑^ + 2휋푓) , (2.20) 퐶 푔 퐽 훼 푖 푖

where 휑^푖 corresponds to a gauge-invariant phase difference operator across each i- th Josephson junction which constitutes an array, 휑^ corresponds to a phase differ-

30 Figure 2-6: Circuit diagram of the generalized flux qubit

ence operator associated with the alpha junction, and 푓 is the normalized exter- 2 nal magnetic flux. The effective charging energy is 퐸퐶 = 푒 /2퐶휎, where 퐶Σ =

퐶푠ℎ + 퐶퐽 /푁 + 훼퐶퐽 + 퐶푔 is the total capacitance across the alpha junction. Here, by assuming that the 푁-array Josephson junctions are identical, this 푁-variate problem can be reduced to quasi-one-dimensional problem due to the symmetry [15]. After simplification, we obtain the following one-dimensional Hamiltonian,

(︂ 푁 )︂ 퐻^ = −4퐸 (^푛 − 푛 )2 + 훼퐸 − cos(휑/푁^ ) − cos(휑^ + 2휋푓) (2.21) 퐶 푔 퐽 훼

At the sweet spot of the flux qubit, 푓 = 0.5, the qubit transition frequency is insen- sitive to flux fluctuations. One can expand Eq. 2.21 to fourth order:

(︂1/(훼푁) − 1 1 − 1/(훼푁 3) )︂ 퐻^ ≈ −4퐸 (^푛 − 푛 )2 + 훼퐸 휑^2 + 휑^4 . (2.22) 퐶 푔 퐽 2 24

In the quarton regime, 훼 ≈ 1/푁, vanishes the second term, which contains a second order dependence on phase operator (휑). Then, the dynamics of the system can be understood as a particle in a quartic potential, ergo, “quarton”. In this quarton regime (훾 = 푁), the eigenenergies of the Hamiltonian in Eq. 2.22 can

31 be numerically solved and yield:

(︂2 1 )︂1/3 퐸 = 휖 (1 − )퐸 퐸2 , (2.23) 푛 푛 3 푁 2 퐽 퐶

where 휖0 = 1.0604, 휖1 = 3.7997, 휖2 = 7.4557 for the lowest three levels. Note that 4 (휖2 − 휖1) ≈ 3 (휖1 − 휖0), indicating that the anharmonicity of the quarton is about 1/3 of its qubit frequency. This interesting observation turns out to be practically useful; because the qubit frequency is generally located in the 3 - 6 GHz range. This makes the anharmonicity about 1 - 2 GHz, advantageous for implementing fast (a few ns) operations without leakage to non-computational states.

32 Figure 2-7: Illustration of the dependence of key qubit features (anharmonicity (퐴), transition frequency (휔01), charge-noise sensitivity, flux-noise sensitivity) on various 훾 1 circuit parameters (퐼푐, 퐶Σ, 푁, 푁 = 훼푁 ). The red-colored (blue-colored) dial in each box denotes that the corresponding circuit parameter should be increased (be decreased) to obtain the desired feature. The gray-colored means that the corre- sponding parameter has no relevance to the feature. 33 2.2 Cavity quantum electrodynamics

In cavity quantum electrodynamics (QED), individual atoms coherently interact with the harmonic oscillator excitations, which are optical [47, 7] or microwave photons. Figure. 2-8 illustrates the atom-photon field interaction. The coupled atom-photon system is described by the Jaynes-Cummings (JC) Hamiltonian

(︂ )︂ ^ † 1 ~휔a † ^ ^ 퐻퐽퐶 = ~휔C 푎^ 푎^ + + 휎^푧 + ~푔(^푎 휎^− +푎 ^휎^+) + 퐻휅 + 퐻훾, (2.24) 2 2

where the first term corresponds to cavity photons with excitation ~휔C, the second term describes the transition energy of the individual spin-1/2 atom, and the third term represents a dipole interaction between the cavity and the atom within the ro- tating wave approximation (RWA). The interaction term (the vacuum Rabi coupling) is the result of the quantization of the electric dipole coupling, and corresponds to

† coherent absorption emission (휎−푎 )/(휎+푎) of a photon to / from the electromag- netic field at a rate 푔. Here, 퐻휅 describes the coupling of the cavity to a continuum of environmental modes which produces the cavity decay rate 휅 = 휔C/푄, while 퐻훾 describes the coupling of the atom to modes other than the cavity modes which cause the excited state to decay at rate 훾 and additional dephasing effects.

2.2.1 Strong coupling regime

When the interaction rate 푔 is much larger than the atom and cavity decay rates, 훾 and 휅, respectively, we say that the atom and the cavity are in the 푠푡푟표푛푔 푐표푢푝푙푖푛푔 regime. When the photons in the cavity and atom are in resonance with each other

(휔C = 휔a), the interaction fully hybridizes the energy levels of the combined atom and photon field system, resulting in dressed-state eigenstates in the one-excitation manifold as follows.

1 |+⟩ = √ (| ↑, 0⟩ + | ↓, 1⟩) (2.25) 2 1 |−⟩ = √ (| ↑, 0⟩ − | ↓, 1⟩), (2.26) 2

34 Figure 2-8: Illustration of cavity QED. A two-level atom passes through a Fabry- Perot cavity over a transit time 푡푡푟푎푛푠푖푡. While the atom passes through, it interacts coherently with photons contained in the cavity with a strength 푔. Photons can leave the cavity at a rate 휅 and the atom decays via non-cavity modes at a rate 훾

where ⟨↑⟩ denotes a ground state of the atom, ⟨↓⟩ corresponds an excited state of the atom and the number indicates the number of photons in the cavity. Here, the atom and cavity can freely exchange single quanta at a rate 푔, the atom-cavity coupling strength. In the strong coupling regime, there can be any number of excitations, resulting in a ladder of states with eigenstates in the n-excitation manifold given by

1 |+⟩푛 = √ (| ↑, 푛 − 1⟩ + | ↓, 푛⟩) (2.27) 2 1 |−⟩푛 = √ (| ↑, 푛 − 1⟩ − | ↓, 푛⟩). (2.28) 2 √ These states’ energies are separated by 2푔 푛 and exhibit a built-in anharmonicity that allows the strong-coupling regime of cavity QED to behave as a multi-level qubit [11].

35 2.2.2 Dispersive coupling regime

In the dispersive coupling regime, the atom does not directly exchange a quanta with the cavity; the atom is far-off resonant with the cavity, in other words, |Δ| = |휔a −

휔C| ≫ 푔. In this limit, this regime can be studied using a second-order perturbative expansion of Eq. 2.24 in 푔/Δ, to give the dispersive Jaynes-Cumming Hamiltonian,

[︂ 2 ]︂ (︂ )︂ ^ 푔 † 1 ~휔a 퐻JC,Disp = ~ 휔C + 휎^푧 푎^ 푎^ + + 휎^푧 (2.29) Δ 2 2

The first term implies that the dipole-interaction is now transferred into anatom state-dependent shift of the harmonic oscillator frequency. In other words, the reso- nant frequency of the harmonic oscillator can take either of two values dependent on

2 the qubit’s state, 휔C ± 푔 /Δ. This dispersive shift enables ’approximately’ quantum non-demolition (QND) measurement and is the basis for readout of multiple qubits later [5].

2.3 Circuit QED

2.3.1 Coupling a superconducting qubit to a coplanar waveg- uide resonator

We can understand the circuit QED (Fig. 2-9) along the same lines as what we have done for cavity QED. Here, the Fabry-Perot cavity is replaced by a microwave- frequency coplanar waveguide (CPW) transmission line resonator. One can place a superconducting qubit near either end of the resonator, thereby capacitive coupling it to a voltage anti-node of the first mode퐿 ( = 휆/2) of the resonator. The Hamiltonian for the whole system includes the qubit Hamiltonian, the res- onator Hamiltonian, and the dipole interaction Hamiltonian:

^ † ~휔푞 † 퐻 = ~휔c푎^ 푎^ + 휎^푧 + 푔휎^푥(^푎 +푎 ^ ). (2.30) 2

Here, we assume that the superconducting qubit is sufficiently anharmonic so that

36 Figure 2-9: Illustration of Circuit QED with a scanning electron microscope (SEM) image of a Quarton. The Febry-Perot cavity is now replaced by a microwave-frequency co-planar waveguide (CPW) transmission line resonator. Near either end of the CPW, superconducting qubits are coupled to the transmission line resonator through an electrostatic capacitive interaction. the qubit can be modeled as a two-level system.

2.3.2 Dispersive regime of circuit QED

By making the detuning (Δ) between the qubit’s transition frequency and the res- onator’s resonant frequency much larger than coupling strength (푔), the qubit-resonator ^ [︀ 푔 † ]︀ can be in the dispersive regime. After a unitary transformation (푈 = exp −푖 Δ (^푎휎^+ +푎 ^ 휎^−) , up to the 2nd order in 푔/Δ, the dispersive Hamiltonian can be approximated as :

^ ~ † 1 † 1 퐻퐷푖푠푝 ≈ 휔푞휎^푧 + ~휔푐(^푎 푎^ + ) + ~휒(^푎 푎^ + )^휎푧, (2.31) 2 2 2 where 휒 represents the qubit-state-dependent dispersive shift (2휒 = 푔2/Δ). The last term includes the AC Stark shift Δ푠푡푎푟푘 = 2휒푛^ due to the resonator photon number

37 † 푛^ =푎 ^ 푎^, and the Lamb shift Δ푙푎푚푏 = 휒 due to the resonator’s zero-point energy.

2.4 Coherence properties of superconducting qubits

Coherence is a fundamental property of quantum mechanics, and it is necessary for useful functioning quantum bits. However, when a quantum system is not perfectly isolated but in contact with its surroundings, the coherence decays with time via a process called decoherence. Within the standard Bloch Redfield picture of two-level

system dynamics, decoherence is characterized by two rates: Γ1 = 1/푇1 and Γ2 =

Γ1/2 + Γ휙 = 1/푇2. 푇1 is called the longitudinal coherence time, which measures the

loss of energy from the system while 푇2 is called the transverse coherence time which contains the pure dephasing attributed to the interactions with the environment. Table 2.1 shows the coherence properties of the state of the art superconducting qubits.

2D Fixed Freq. 2D Tunable 2D Cshunt 2D 3D Transmon 3D 3D Heavy Type Transmon [52] Transmon [2, 22] Flux Qubit [57] Quarton[58] [20, 39, 40] Fluxonium [28, 44] Fluxonium [14]

휔01 5.0 - 5.3 (GHz) 5.0 - 5.8 (GHz) 4.7 (GHz) 2.6 (GHz) 4.7 - 6.3 (GHz) 0.5 - 0.7 (GHz) Use meta-stable state in Λ system (fluxon 휔12 − 휔01 -330 (MHz) -220 (MHz) 500 (MHz) 813 (MHz) -300 (MHz) & 8 (GHz) transition).

푇1 70 (휇푠) 50 (휇푠) 55 (휇푠) 83 (휇푠) 100 (휇푠) 1 (ms) 8 (ms) * 퐸 푇2 , 푇2 - / 100 (휇푠) 20 / - (휇푠) 40 / 80 (휇푠) - / 130 (휇푠) 140 / 140 (휇푠) 14 / 20 (휇푠) 550 (ns)

Table 2.1: Coherence properties of the state of art superconducting qubits [55]. 푇1 푒푐ℎ표 is the qubit energy relaxation time. 푇2 is the qubit coherence time measured in * a Hahn-echo (refocusing) experiment. 푇2 is the free induction decay (FID) time measured in the Ramsey interference experiment.

2.4.1 Coherence time and anharmonicity of a Quarton

The quarton is designed to feature both long coherence and large anharmonicity at the degeneracy point. We have experimentally demonstrated its performance by mea- suring both the coherence times and the anharmonicity. Figure. 2-10 illustrates the

푒푐ℎ표 pulse sequences used to evaluate 푇1 and 푇2 , and the coherence times of a quarton 푒푐ℎ표 (푇1 = 82.83 ± 0.57 휇푠, 푇2 = 129.59 ± 3.27 휇푠). We also evalulated the anharmonic- ity of the Quarton at the degeneracy point by measuring its spectrum as depicted

38 in Figure. 2-11. It is immediately apparent that the transitions 0→1 (red) and 1→2 (blue) are well resolved in frequency space. The observed large anharmonicity (813 MHz) allows one to drive the single-qubit gates much faster using shorter pulses (a few ns) compared to transmon qubits with low anharmonicity (200 - 300 MHz).

T1 Measurement

X +pi Readout

tau

echo T2 Measurement

X +pi/2 X +pi X +pi/2 Readout

tau/2 tau/2

푒푐ℎ표 Figure 2-10: Illustration of pulse sequences used to measure 푇1 and 푇2 , and a 푒푐ℎ표 quarton’s coherence times (푇1 = 82.83 ± 0.57 휇푠, 푇2 = 129.59 ± 3.27 휇푠).

39 Figure 2-11: Anharmonicity (풜) of a quarton. The qubit transitions measured are the 0-1 transition in a single tone spectroscopic measurement (red), and the 1-2 transition (blue) while populating the first excited state with a second drive onthe 0-1 transition. The second excited state is not populated at normal spectroscopy powers (red), indicating a low excited state thermal population. The 0-1 and 1-2 transitions are separated by 813 MHz; This is why the quarton can be treated as a two-level system even during fast control operations.

40 Chapter 3

Quantum Noise Spectroscopy

This chapter will discuss the fundamental ideas of quantum noise spectroscopy. It will serve as an important background for the experiments presented in chapter 5. Section 3.1 illustrates how a qubit serves as a noise probe with Hamiltonian formulation. That is followed by section 3.2, which describes quantum noise spectroscopy (QNS) protocols, especially one for characterizing non-Gaussian dephasing processes.

3.1 A qubit as a noise probe

3.1.1 Hamiltonian of open quantum system

We consider a single qubit S as a noise probe coupled to a bath with a purely dephasing interaction. In other words, no energy exchange takes place between the system and an bath (environment) modeled by a continuum of classical and/or quantum modes. The general form of the relevant open system Hamiltonian is [42]

^ ^ ^ ^ ^ ^ ^ ^ 퐻 = 퐻S ⊗ IB + IS ⊗ 퐻B + 퐻SB + 퐻ctrl(푡) ⊗ IB (3.1)

where 퐻^S and 퐻^B denote the internal Hamiltonian for the system (here, a qubit) and the bath respectively, and 퐻^SB describes the interaction between them. An open-loop control on the qubit is implemented via the Hamiltonian 퐻^ctrl(푡). For a purely dephasing model, [퐻^SB, 퐻^S] = 0. Without loss of generality, we will take the

41 ^ 1 quantization axis to be the z-axis and write the system Hamiltonian as 퐻S = 2 ~휔0휎^푧. The open loop control consists of sequences of 휋-pulses about some axis orthogonal to the z-axis. which means the control Hamiltonian 퐻ctrl(푡) applies a pi-rotation around

푦-axis to the qubit at 푡 = 푡푖

3.1.2 The relation between spin observables and bath-operator cumulants

The joint Hamiltonian after transforming into the interaction picture associated with ^ ^ 1 ^ bath Hamiltonian, 퐻B, the qubit Hamiltonian, 퐻S = 2 ~휔0휎^푧, and 퐻ctrl(푡) is (See Appendix B.) [35]

^ ~ 퐻̃︀(푡) = 푦(푡)퐵(푡)^휎푧 (3.2) 2

The effect of the external open-loop control is described by the “switching function” 푦(푡), which changes sign between ±1 with every control pulse (휋-pulse) applied to the qubit. In this frame, the dephasing and open-loop control leave 휎^푧 invariant but cause the transverse spin components to evolve. For a state prepared in the transverse plane along 푛^ = cos 휃0푥^ + sin 휃0푦^ [35]

(︃ ∞ )︃ (︃ ∞ )︃ ∑︁ (−1)푙 ∑︁ (−1)푘 ⟨휎^ (푡)⟩ = exp ϒ(2푙)(푡) cos ϒ(2푘+1)(푡) + 휃 (3.3) 푥 (2푙)! (2푘 + 1)! 0 푙=1 푘=1

and

(︃ ∞ )︃ (︃ ∞ )︃ ∑︁ (−1)푙 ∑︁ (−1)푘 ⟨휎^ (푡)⟩ = exp ϒ(2푙)(푡) sin ϒ(2푘+1)(푡) + 휃 (3.4) 푦 (2푙)! (2푘 + 1)! 0 푙=1 푘=1

In both of these expressions,

∫︁ 푡 (푘) (푘) ϒ (푡) = 푑푡1 . . . 푑푡푘푦(푡1) . . . 푦(푡푘)퐶 (푡1, . . . , 푡푘), (3.5) 0

(푘) with 퐶 (푡1, . . . , 푡푘) denoting the k-th order cumulant of the dephasing process (See

42 (푘) Appendix A). The cumulant 퐶 (푡1, . . . , 푡푘) depends on multi-time correlations of the bath ⟨퐵(푡1) . . . 퐵(푡푗)⟩ for 푗 ≤ 푘, where ⟨·⟩ denotes an ensemble average over the dephasing process.

From Eqs. 3.3 and 3.4, we can see that the odd and even cumulants have different effects on the dynamics of the spin. The even cumulants are responsible for thedecay of the transverse spin component

(︀ 2 2)︀1/2 −휒(푡) ⟨휎^푥(푡)⟩푞 + ⟨휎^푦(푡)⟩푞 = 푒 , (3.6)

where ⟨·⟩푞 denotes a quantum expectation value. The decay parameter in this expres- sion is

∞ ∑︁ (−1)푙 휒(푡) = ϒ(2푙)(푡) (3.7) (2푙)! 푙=1

On the other hand, the odd cumulants are responsible for rotating the spin in the transverse plane with the angle of rotation (phase angle) given by

(︂ )︂ ∞ 푙 −1 ⟨휎^푦(푡)⟩푞 ∑︁ (−1) (2푙+1) 휑(푡) = tan − 휃0 = ϒ (푡), (3.8) ⟨휎^푥(푡)⟩푞 (2푙 + 1)! 푙=0

where 휃0 is the initial phase angle of the qubit in the transverse plane.

3.2 Quantum noise spectroscopy (QNS) protocols

(푛+1) For a statistically stationary noise process, where 퐶 (푡1, . . . 푡푛+1) is a function of

the time separations 휏푗 ≡ 푡푗+1 − 푡1, 푗 ∈ 1, . . . , 푛, the noise spectra are fully character- ized by the Fourier transform of the cumulants with respect to the time separations

{휏푗}. Then the nth-order polyspectrum is defined:

∫︁ −푖⃗휔푛·⃗휏푛 (푛+1) 푆푛(⃗휔푛) ≡ 푑⃗휏푛푒 퐶 (⃗휏푛), 푛 ≥ 1, (3.9) R푛

43 where ⃗휔푛 ≡ (휔1, . . . , 휔푛), 푆1(⃗휔1) ≡ 푆(휔) is the power spectral density(PSD), and

푆2(⃗휔2) and 푆3(⃗휔3) are known as the “bi-spectrum” and “tri-spectrum”. For all orders,

푆푛(⃗휔푛) is a smooth n-dimensional surface when the noise is classical and ergodic.

3.2.1 Bylander et al.’s protocol(2011): CPMG spectroscopy for Gaussian dephasing process

For a Gaussian dephasing process, the decay parameter (휒) is solely determined by (2) second-order cumulant of the dephasing process 퐶 (푡1, 푡2) and switching function 푦(푡)

∫︁ 휏 ∫︁ 휏 1 (2) 1 (2) 휒(휏) = − ϒ (휏) = − 푑푡1 푑푡2푦(푡1)푦(푡2)퐶 (푡1, 푡2) (3.10) 2 2 0 0

If the dephasing process is stationery, we can rewrite Eq. 3.10 in terms of the power spectral density, 푆(휔):

2 ∫︁ ∞ 푆(휔) 2 ∫︁ ∞ 휒(휏) = 2 퐹 (휔, 휏)푑휔 = 푆(휔)푔(휔)푑휔, (3.11) 휋 0 휔 휋 0

∫︀ ∞ −푖휔푡 2 where 푆(휔) is the power spectral density and 퐹 (휔, 휏) ≡ | −∞ 푦(푡)푒 푑푡| denotes the filter function. Bylander et al. used the filtering property of the CPMG sequence[10, 29], which is one of the dynamic decoupling (DD) sequences, to characterize the Gaussian dephasing process (see Figure 3-1). If 푔(휔, 휏) ≡ 퐹 (휔, 휏)/휔2 is narrow enough compared to the spectral features of the noise, that one can treat 푔(휔, 휏) as a rectangle function peaked at 휔, and regard the noise in transition frequency of the qubit as constant within its bandwidth [9]. This provides a means of characterizing the noise at a frequency 휔 by varying 휏.

44 ∘ (a) Schematic representation of the CP(MG) sequence; the 휋-pulses are shifted 0 (푋휋) ∘ and 90 (푌휋) from the 2-pulses (푋휋/2) for CP and CPMG, respectively.

1.0 Ramsey (Free Evolution) Spin Echo CPMG (N=2) 0.8 CPMG (N=5) ) CPMG (N=10) ( F

, 0.6 n o i t c n u F

0.4 r e t l i F

0.2

0.0 0 2 4 6 8 10 Frequency, /2 (MHz)

(b) Filter functions of Ramsey, spin-echo, and CPMG sequences for a total pulse-sequence length 휏 = 1 휇푠. N corresponds to the number of 휋 pulses in each sequence. Figure 3-1: Schematic representation of the CP(MG) sequence and the filter functions of different pulse sequences

45 3.2.2 Álvarez and Suter protocol (2011): Repetition of suit- ably designed pulse sequences

Álvarez and Suter extended the DD noise spectroscopy method by employing the idea of sequence repetition [1]. If the pulse sequence consists of many repetitions of a basic cycle, the resulting exponential decay functions are determined by the spectral density at discrete frequencies. This allows one to build a linear system of equations that can be solved to obtain the unknown spectral density function.

If the pulses are applied during the interval 휏푐, the resulting decay after M cycles becomes:

√︂ ∫︁ ∞ 1 2 휋 2 ⟨휑 (푡)⟩ = 휒 = 푅(푡)푡 = 푑휔푆(휔)|퐹푁 (휔, 푀휏푐)| , (3.12) 2 2 −∞ where 퐹푁 (휔, 푀휏푐) is the filter function. For 푡 = 푀휏푐 ≫ 휏퐵, where 휏퐵 is the noise correlation time, 퐹푁 (휔; 푡) can be represented by a series of 훿 functions centered at

푘휔0 neglecting the contributions from the secondary maxima. Thus, in the limit of many cycles, the decay is exponential and R(t) becomes a constant.

∞ ∑︁ 2 푅(푡) = 푅 = 퐴푘푆(푘휔0), (3.13) 푘=1

2 2휋 2 where 퐴 = 2 |퐹푁 (푘휔0, 휏푐)| . The main idea entails the following two steps. 푘 휏푐 (1) Apply 푀 different pulse sequences which have different pulse delays so that they probe the spectral density function at a discrete set of harmonic frequencies with different sensitivity amplitudes퐴 ( 푘). This yields a square matrix that we can invert to obtain the values of 푆(푗휔푚푖푛), (푗 = 1, 2, ..., 푚) (2) From the resulting spectral density function, estimate a functional form for the tail of the distribution and correct the data for the contributions from the tail. In- verting the matrix again, with the corrected values, gives the final spectral density distribution.

46 3.2.3 Norris et al. protocol (2016): Generalization of the Álvarez and Suter protocol to Non-Gaussian dephasing processes

In the noise spectroscopy protocol implemented by Álvarez and Suter for Gaussian noise, a fixed CPMG sequence is used with cycle times varying from 푇 to 푇/푛 = 2휏, where 푛 ∈ Z+ and 휏 is the minimum time spacing between pulses [1]. While this protocol produces a well-conditioned linear inversion, it has a following disadvantage: because the CPMG sequence filters out zero-frequency noise (퐹퐶푃 푀퐺(0) = 0), it precludes reconstruction at any point in frequency space containing a zero frequency, which brings a substantial information loss for higher-dimensional polyspectra [35].

Non-Gaussian noise spectroscopy requires a number of sequences with distinct filter functions, including some with zero filtering퐹 order( 푝(0) ≠ 0) [35]. Again, the control sequences with zero filtering order enables reconstructing the polyspectra at points containing a zero frequency. Norris et al. implemented a generalized version of the Álvarez and Suter protocol using different orders of concatenated dynamic decoupling (CDD) sequence, not only CPMG (filtering order, 푚 = 2), but also free evolution (푚 = 0), and even up to higher filtering order푚 ( = 5) base sequences.

Following Ref. [42], the effect of a base control sequence 푝 in the frequency do- main is characterized by a single “fundamental” filter function, namely, 퐹푝(휔) ≡ ∫︀ 푇 +푖휔푡 0 푑푡푒 푦푝(푡). 푀 repetitions of base sequence 푝 yield

∫︁ 푘−1 (푘) ∏︁ sin(푀휔푗푇/2) ϒ[푝]푀 = 퐹푝(휔푗) 푘−1 sin(휔 푇/2) R 푑⃗휔푘−1 푗=1 푗

sin(푀|⃗휔푘−1|푇/2) 푆푘−1(⃗휔푘−1) × 퐹푝(−|⃗휔푘−1|) 푘−1 , (3.14) sin(|⃗휔푘−1|푇/2) (2휋)

where |⃗휔푘−1| ≡ 휔1 +···+휔푘−1. The key to extending the protocol in Ref. [42] beyond Gaussian noise (k=2) is to realize that the repetition produces a multidimensional

47 frequency comb for all orders, namely,

⎡ ⎤ 푘−1 [︂ ]︂ 푘−1 ∞ (︂ )︂ ∏︁ sin(푀휔푗푇/2) sin(푀|⃗휔푘−1|푇/2 ∏︁ 2휋 ∑︁ 2휋푛푗 ≈ 푀 ⎣ 훿 휔푗 − ⎦ , sin(휔푗푇/2) sin(|⃗휔푘−1|푇/2) 푇 푇 푗=1 푗=1 푛푗 =−∞

푀 ≫ 1, ∀푘 (3.15)

provided that 푆푘−1(⃗휔푘−1) in Eq.(3.14) is a smooth function.

Thanks to the “hypercomb” in Eq.(3.15), the reconstruction problem becomes an inverse problem: substituting Eq.(3.15) into Eq.(3.14) produces a linear equation that couples the polyspectra and the filter functions evaluated at the harmonic frequencies

푗 ℋ푗 ≡ {2휋⃗푛푗/푇 |⃗푛푗 ∈ Z },

푘−1 (푘) ∑︁ 푀 ∏︁ ϒ = 퐹 (ℎ )퐹 (−|⃗ℎ |)푆 (⃗ℎ ) (3.16) [푝]푀 푇 푘−1 푝 푗 푝 푘−1 푘−1 푘−1 ⃗ 푗=1 ℎ푘−1∈ℋ푘−1

A finite linear equation can be acquired by truncating the above sum to afiniteset

Ω푘−1. With no prior knowledge of the noise, it suffices to consider

Ω푘−1 ⊂ 풟푘−1 ∩ ℋ푘−1, (3.17)

where 풟푘−1 denotes the principal domain of the polyspectrum and ℋ푘−1 denotes the set of harmonic frequencies. Using the truncated expression in Eq.(3.16) enables us to relate the sampled polyspectra to experimentally observable dynamical quantities,

∞ ∑︁ (−1)푙푀 ∑︁ 휒 푀 ≈ 푚 (⃗ℎ ) [푝] (2푙)!푇 2푙−1 2푙−1 2푙−1 푙=1 ⃗ ℎ2푙−1∈Ω2푙−1 2푙−1 ∏︁ × 퐹푝(ℎ푗)퐹푝(−|⃗ℎ2푙−1|)푆2푙−1(⃗ℎ2푙−1) (3.18) 푗=1

48 ∞ ∑︁ (−1)푙푀 ∑︁ 휑 푀 ≈ 푚 (⃗ℎ ) [푝] (2푙 + 1)!푇 2푙 2푙 2푙 푙=1 ⃗ ℎ2푙∈Ω2푙 2푙 ∏︁ × 퐹푝(−|⃗ℎ2푙|)푆2푙(⃗ℎ2푙) (3.19) 푗=1

⃗ ⃗ 푛 ⃗ where the multiplicity 푚푛(ℎ푛) ≡ card{ℎ푛 ∈ R |푆푛(ℎ푛) = 푆푛(⃗휔푛), ∀휔푛 ∈ 풟푛} accounts for the symmetry of the poly spectrum (card ≡ cardinal number). If high-order cor- relation terms have negligible contributions to both 휒 and 휑, the cumulant expansion in Eqs. 3.18, 3.19 may be truncated at 푙 = 퐿. If 푁 terms are retained, measuring

휒[푝]푀 (휑[푝]푀 ) for N different control sequences creates a system of linear equations that can be inverted to obtain the even (odd) polyspectra up to the order 2퐿 − 1(2퐿).

3.2.4 Norris et al.’s protocol(2016) cont.: Reconstructing bis- pectrum

Effect of sequence repetition

Consider the phase angle 휑푝(푀푇 ) after 푀 cycles of the control sequence 푝 with associated filter function 퐹푝(휔, 푀휏) up-to third orders,

sin(푀휔푇/2) 휑푝(푀푇 ) ≈ 퐹푝(0) lim ⟨퐵휇(푡)⟩ 휔→0 sin(휔푇/2) ∫︁ ∞ ∫︁ ∞ 1 sin(푀휔1푇/2) sin(푀휔2푇/2) sin[푀(휔1 + 휔2)푇/2] − 2 푑휔1 푑휔2 3!(2휋) −∞ −∞ sin(휔1푇/2) sin(휔2푇/2) sin[(휔1 + 휔2)푇/2]

× 퐹푝(−휔1)퐹푝(−휔2)퐹푝(휔1 + 휔2)푆2(휔1, 휔2) (3.20)

For 푀 ≫ 1, the function of sines can be viewed as a 2D frequency comb function (See Fig 3-2 for example), i.e.

sin(푀휔1푇/2) sin(푀휔2푇/2) sin(푀(휔1 + 휔2)푇/2) 퐺2퐷−푐표푚푏(휔1, 휔2) = sin(휔1푇/2) sin(휔2푇/2) sin((휔1 + 휔2)푇/2) 2 ∞ ∞ (︂2휋 )︂ ∑︁ ∑︁ ≈ 푀 훿(휔 − 2푞 휋/푇 )훿(휔 − 2푞 휋/푇 ). (3.21) 푇 1 1 2 2 푞1=−∞ 푞2=−∞

49 Substituting the expression of 퐺2푑−푐표푚푏 in Eq. 3.21 into Eq. 3.20 produces a linear equation relating the phase angle to the filter function and bispectrum at the harmonic frequencies,

∞ ∞ (︂ )︂ (︂ )︂ 푀 ∑︁ ∑︁ 2휋푞1 2휋푞2 휑 (푀푇 ) ≈ 푀퐹 (0)⟨퐵 (푡)⟩ − 퐹 − 퐹 − 푝 푝 휇 3!푇 2 푝 푇 푝 푇 푞1=−∞ 푞2=−∞ (︂2휋(푞 + 푞 ))︂ (︂2휋푞 2휋푞 )︂ × 퐹 1 2 푆 1 , 2 . (3.22) 푝 푇 2 푇 푇

Therefore, measuring the phase angles for different control sequences produces a system of linear equations. To make the system of linear equations invertible, we must truncate the infinite sums in Eq. 3.22. Here, we truncate the sums to asubset of harmonics in the “principal domain” of the bispectrum.

Principal domain

The principal domain is a subspace of polyspectra that contains all information about the polyspectrum. If the polyspectrum is known over the principal domain, it can be fully reconstructed due to the following symmetry relations:

1. Permutation symmetry. 푆2(휔1, 휔2) = 푆2(휔2, 휔1)

* 2. Conjugation symmetry. 푆2(휔1, 휔2) = 푆2 (−휔1, −휔2) = 푆2(−휔1, −휔2)

3. Stationarity symmetry. 푆2(휔1, 휔2) = 푆2(−(휔1 + 휔2), 휔2)

From these symmetries, we can show that 푆2(휔1, 휔2) = 푆2(휔2, 휔1) = 푆2(−휔2, 휔1 +

휔2) = 푆2(−휔1, 휔1 + 휔2) = 푆(−휔1 − 휔2, 휔1) = 푆(−휔1 − 휔2, 휔2) = 푆(−휔1, −휔2) =

푆(−휔2, −휔1) = 푆(휔2, −휔1 − 휔2) = 푆(휔1, −휔1 − 휔2) = 푆(휔1 + 휔2, −휔1) = 푆(휔1 +

휔2, −휔2). Let’s truncate the sums of Eq.3.22 to a subset of harmonics contained in the principal domain. After truncation, Eq. 3.22 becomes

휑푝(푀푇 ) ≈ 푀퐹푝(0)⟨퐵휇(푡)⟩ 푀 ∑︁ − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ) (3.23) 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2

50 Figure 3-2: 퐺2퐷−푐표푚푏(휔1, 휔2). This 2D frequeny comb samples the multiples of har- monic frequency(2휋/푇 ) over 푓1 and 푓2. Plotted for T = 1휇푠, M =10.

51 Figure 3-3: The principal domain of a bispectrum (red dotted line). The bispectrum at point (휔1, 휔2) in the principal domain is equivalent to the points in other 11 regions in this figure by the symmetry arguments.

52 Interior

Zeros

Figure 3-4: An octant containing the harmonic frequencies where the bispectrum will be reconstructed. The interior points in the blue region are the points which do not contain a zero frequency and the points in red region contain a zero.

where ℎ푖 = 2휋푛/푇 are harmonic frequencies in the principal domain, 푚(ℎ1, ℎ2) is the multiplicity of the point (ℎ1, ℎ2), and ℋ2 is a set of paired harmonic frequencies(ℎ1, ℎ2) in the principal domain.

Matrix inversion

Matrix inversion entails the following two steps: 1) reconstructing the bispectrum at points containing only non-zero frequencies (i.e. 푆2(ℎ1, ℎ2), where ℎ1 ≠ 0, ℎ2 ≠ 0). 2) reconstructing the bispectrum at points containing a zero (i.e. 푆2(0, 0) or 푆2(ℎ1, 0)).

The first step requires a set of control sequences with non-zero filtering orderto reconstruct the bispectrum at points that do not contain a zero. To reconstruct the bispectrum at these points, let’s apply 푁 different control sequences with non-zero

53 filtering order. From Eq. 3.23,

푀 ∑︁ 휑 (푀푇 ) = − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ) 1 3!푇 2 1 2 1 1 1 2 1 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2−풵2 푀 ∑︁ 휑 (푀푇 ) = − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ) 2 3!푇 2 1 2 2 1 2 2 2 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2−풵2 . . 푀 ∑︁ 휑 (푀푇 ) = − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ) 푁 3!푇 2 1 2 푁 1 푁 2 푁 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2−풵2 (3.24)

where 풵2 = {(휔, 0) ∈ ℋ2|휔 ≥ 0} denotes the set of points in ℋ2 containing a zero frequency. Note that ⟨퐵휇(푡)⟩, 푆2(0, 0), 푆2(ℎ1, 0) terms are omitted because 퐹푝(0) = 0 for 푝 = {1, 2, . . . , 푁}. Here we can represent the system of linear equations ⃗ ⃗ ⃗ ⃗ 3.24 as a matrix equation (휑 = A푆2 where 휑 = {휑1(푀푇 ), . . . , 휑푁 (푀푇 )} and 푆2 =

{푆2(ℎ1, ℎ2)|(ℎ1, ℎ2) ∈ ℋ2 − 풵2}) In the second step, we reconstruct ⟨퐵휇(푡)⟩ and the bispectrum at the points containing a zero. We can rewrite Eq. 3.23 as

푀 ∑︁ 휑 (푀푇 ) = 푀퐹 (0)⟨퐵 (푡)⟩ − 푚(푧 , 푧 )Re[퐹 (−푧 )퐹 (−푧 )퐹 (푧 + 푧 )]푆 (푧 , 푧 ) 푝 푝 휇 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (푧1,푧2)∈풵2 푀 ∑︁ − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ). 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2−풵2 (3.25)

Note that the last line of Eq. 3.25 contains the bispectrum at the non-zero frequencies, which were reconstructed in the first step. Let’s rearrange the equation as follows:

푀 ∑︁ 휑 (푀푇 ) + 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ) = 푝 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2−풵2 푀 ∑︁ 푀퐹 (0)⟨퐵 (푡)⟩ − 푚(푧 , 푧 )Re[퐹 (−푧 )퐹 (−푧 )퐹 (푧 + 푧 )]푆 (푧 , 푧 ). 푝 휇 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (푧1,푧2)∈풵2 (3.26)

The LHS in Eq.3.26 consists of known quantities and the RHS is the phase angle due

54 only to the mean and the bispectrum at points containing a zero. Let’s define the

푧 LHS as 휑푝(푀푇 ). To reconstruct both the bispectrum at the points in 풵2 and the mean of 퐵(푡), we need |풵2| + 1 different control sequences with zero filtering order, in total. In other words, the bispectrum at the points containing a zero and ⟨퐵휇(푡)⟩ can be evaluated by inverting the following set of linear equations:

푀 ∑︁ 휑푧(푀푇 ) = 푀퐹 (0)⟨퐵 (푡)⟩ − 푚(푧 , 푧 )Re[퐹 (−푧 )퐹 (−푧 )퐹 (푧 + 푧 )]푆 (푧 , 푧 ) 1 1 휇 3!푇 2 1 2 1 1 1 2 1 1 2 2 1 2 (푧1,푧2)∈풵2 푀 ∑︁ 휑푧(푀푇 ) = 푀퐹 (0)⟨퐵 (푡)⟩ − 푚(푧 , 푧 )Re[퐹 (−푧 )퐹 (−푧 )퐹 (푧 + 푧 )]푆 (푧 , 푧 ) 2 2 휇 3!푇 2 1 2 2 1 2 2 2 1 2 2 1 2 (푧1,푧2)∈풵2 . . 푀 ∑︁ 휑푧 (푀푇 ) = 푀퐹 (0)⟨퐵 (푡)⟩ − 푚(푧 , 푧 )Re[퐹 (−푧 )퐹 (−푧 )퐹 (푧 + 푧 )] |풵2|+1 1 휇 3!푇 2 1 2 |풵2|+1 1 |풵2|+1 2 |풵2|+1 1 2 (푧1,푧2)∈풵2

× 푆2(푧1, 푧2). (3.27)

Eq 3.27 can be represented as a matrix equation as we did in the first step.

⃗푧 푧 ⃗푧 휑 = A 푆2 , (3.28)

⃗푧 (0) (0) (|풵2|+1) (|풵2|+1) 푇 where 푆2 = [휇퐵, 푆2(푧1 , 푧2 ), . . . , 푆2(푧1 , 푧2 )] .

55 56 Chapter 4

Experimental Setup and Details

This chapter will give a brief discussion of hardware setup used for the experiments and fabrication of superconducting qubits. I will review our cryogenic circuitry and room temperature control scheme so that one can easily understand how we con- trol and readout superconducting qubits. Then, a brief summary on materials and fabrication of superconducting qubits will be followed.

4.1 Cryogenic setup

The experiments were performed using a Leiden CF-450 dilution refrigerator, capable of cooling to a base temperature of 15mK. The samples were magnetically shielded with a superconducting can surrounded by a Cryoperm-10 cylinder. The schematic of the cryogenic circuitry is shown in Figure 4-1. There are two RF lines for the input and the output of the samples; applying microwave readout tone and measuring the transmission of sample respectively. Thermal noise from room temperature on the RF drive lines is attenuated with 20dB at the 3K stage, followed by 6dB at still, and 26dB at the 20mK stage. All attenuators in the cryogenic samples are made by XMA. Note that there is one additional RF line for pumping the Josephson traveling wave parametric amplifier (JTWPA) [26] used as a first-stage pre-amplifier to amplifythe readout signal at base temperature. The effective noise temperature is determined primarily by the JTWPA, which has a about 600mK. To avoid any back-action of

57 Figure 4-1: Schematic of cryogenic circuitry

58 Figure 4-2: Room temperature control schematic

the pump-signal from TWPA, we added a microwave isolator between the samples and the TWPA. On the RF output line, there is a high-electron mobility transistor (HEMT) amplifier (Cryo-1-12 SN508D) at the 3K stage. Two microwave isolators allow for the signal to pass through to the amplifier without being attenuated, while taking all the reflected noise off of the amplifier and dumping itina50Ω termination instead of reaching the sample. There are two additional lines for qubit flux bias: one is for DC flux bias, which is applied globally through a coil installed in the device package, the other is to apply magnetic flux to the qubit thorough a local antenna. The primary requirement of the DC flux bias line is the ability to tune through at least a single flux quantum on the SQUID of each qubit with low noise. Thelocal flux bias line is attenuated by 20dB at the 3K stage, 6dB at the still, 20dBatthe 20mK stage to remove excess thermal photons from higher-temperature stages.

59 4.2 Room temperature control

Outside of the cryostat, we have all of the control components which allow us to apply microwave signals that address the cavity and the qubits, as well as the components necessary to resolve the readout signal. All of the signals are added using microwave power splitters (Marki PD0R413) used in reverse. Pulse-shaping of the qubit signals is done by high-speed AWG (Keysight M8195A), which has 65GS/sec sampling rate. The output line is further amplified outside of the cryostat with an amplifier (MITEQ AMF-5D-00101200-23-10P) with a quoted noise figures of 2.3dB, and a preamplifier (Stanford Research SR445A). A detailed schematic is given in figure 5.16. Since the state of the qubits are encoded in the phase and amplitude of the transmitted cavity signal, we can use an IQ demodulation technique in heterodyne detection scheme; IQ mixer (Marki Microwave IR 4509LXP) is used to mix down the signal entering the RF port with a reference signal detuned by 40 MHz applied to the LO port. This results in down-converted signals at 40 MHz appearing on the I/Q ports of the mixer. The I/Q signal’s phase and magnitude contain the information about the state of the qubits and one can extract the relative phase information induced by the qubits by comparing with the reference signal (yellow colored in Figure. 4-2). All components are frequency-locked via a common SRS rubidium frequency standard (10MHz).

4.3 Pulse control and modulation

Qubit control pulse generation is performed via a Keysight M8195A AWG. The pulses are programmed in Labber and then imported into Keysight M8195A.

4.4 Materials and fabrication of superconducting qubits

The fabrication steps of the quarton qubits are as follows [57]: 1. Growth and pat- terning of high-quality factor (high-Q) aluminium films using molecular beam epitaxy (MBE). 2. Patterning and evaporation of the superconducting qubit loop and Josephson junc-

60 tions. 3. Dicing and packaging.

4.4.1 Growth and patterning of high-Q aluminum

High-Q aluminium films were deposited on 50-mm C-plane sapphire wafers in aVeeco GEN200 MBE system with a growth chamber base pressure of 10−11 torr. The wafers were cleaned in piranha solution (sulfuric acid and hydrogen peroxide) prior to load- ing into the MBE system. The wafers were annealed in the MBE system at 900∘C to facilitate outgassing and sapphire surface reconstruction, after which 250 nm of aluminium was deposited at a growth rate of 0.025 nm/s and a substrate temperature of 150∘C The high-Q aluminium was patterned using contact lithography and wet-etched using Aluminium Etchant - Type A (Transene Company, Inc.) into the following device features: shunt capacitors, coplanar waveguide (CPW) resonators, ground planes, and optical alignment marks.

4.4.2 Patterning the qubit loop and Josephson junctions

The qubit loop and junctions were formed using double-angle evaporation of alu- minium through Dolan-style bridges. The free-standing bridges were realized using a bilayer mask comprising a germanium hard mask on top of a sacrificial MMA/MAA layer [MicroChem methyl methacrylate (MMA (8.5)/MAA EL9)]. The qubit loop and junctions are patterned using electron-beam lithography (Vistec EBPG5200) and ZEP520A resist (ZEONREX Electronic Chemicals). This pattern was trans- ferred into the Ge layer using a CF4 plasma, and the underlying MMA/MAA resist was under-etched using an oxygen plasma to create free-standing bridges. Prior to the aluminium evaporation, an in situ argon ion milling was used to clean exposed contact points on the MBE alumnium to ensure superconducting contact with the the evaporated aluminum. The qubit loops and junctions were realized with two separate angle-evaporated aluminum layers; between the two aluminum evaporation

61 steps, static oxidation conditions were used to prepare junctions with a certain critical current density.

4.4.3 Dicing and packaging

Devices were diced into 2.5 × 5 mm2 chips that were mounted into gold plated copper packages. Aluminum wirebonds were used for both signal and ground connections between the device and package, as well as to connect the ground planes of the CPW resonator to prevent slotline modes.

62 Chapter 5

Reconstructing the Bispectrum of non-Gaussian Dephasing Processes in a Superconducting Qubit

Thus far, we have presented a solid theoretical background in superconducting qubits and quantum noise spectroscopy protocols, as well as the details about the exper- imental setup to control/readout superconducting qubits. Now, I will present the main experimental result of this work: reconstruction of bispectrum of a specific non- Gaussian dephasing process. Section 5.1 describes how we created the non-Gaussian dephasing process using a flux-tunable superconducting qubit. Section 5.2 will dis- cuss the experimental result which proves the non-Gaussianity in dephasing noise by measuring spin observables after a CPMG sequence. Then, section, 5.3 will show how we experimentally reconstructed the bispectrum using the protocol theoretically developed by L. M. Norris [35].

63 5.1 Non-Gaussian dephasing in a superconducting qubit

5.1.1 Non-linear energy dependence on external flux

In general, a flux-tunable superconducting qubit such as flux qubits is operated at the “degeneracy point” (sweet spot), where the qubit has an approximately quadratic energy dependence on the external magnetic flux. Figure. 5-1 shows the measured spectrum of the qubit demonstrating the quadratic dependence of the transition fre- quency on the flux-bias. Using the two-level system approximation, one can relate

Figure 5-1: Spectrum of the Quarton near the sweet-spot. The measurement is ob- tained by measuring the transmission at a fixed frequency close to the cavity resonance (7.348 GHz) and sweeping the qubit drive frequency(푦 axis)

the transition frequency 휔01 of the qubit to the external flux bias:

√ 1 휔 = Δ2 + 휖2 ≈ Δ + 휖2 = Δ + 훽 (Φ − Φ /2)2 = Δ + 훽 (Φ )2 (5.1) 01 2Δ Φ 0 푉 푉

64 * 퐸 Table 5.1: Qubit frequency (휔01), coherence properties (푇1, 푇2 , 푇2 ) and single qubit gate fidelity (퐹푔) of the Quarton used in this work.

Transition frequency, 휔01/2휋 2.920 GHz Relaxation time, 푇1 27.03 휇푠 * Free induction decay time, 푇2 12.16 휇푠 퐸 Echo time, 푇2 35.89 휇푠 Gate fidelity, 퐹푔 > 99.9 %

5.1.2 Squared Gaussian dephasing process

2 Spectrum of Gaussian noise, (t) Bispectrum of Squared Gaussian noise, V (t) 10 7

107 10 8

6 ) 10 z )

H z (

) H

2 / 5 f

9 10 , W 10 1 (

f )

( f ( 4 S S 10

10 10 . . . 4 2 4 0 z) 2 H . . . 0 2 (M 4 2 0 2 4 f f 2 1(MHz 2 4 f1(MHz) ) 4

Figure 5-2: An example of the power spectrum of Gaussian noise, 휉(푡) [V] and the 2 7 2 bispectrum of the squared Gaussian noise 훽푉 휉 (푡) [Hz]. 훽푉 = 3.5 * 10 [Hz / 푉 ].

Therefore we can consider a classical “squared Gaussian dephasing process” arising from a quadratic response to a Gaussian flux noise source. We model the dephasing process 퐵(푡) as:

2 퐵(푡) = (1 − 푎)훼푉 휉(푡) + 푎훽푉 휉(푡) , (5.2) where 휉(푡) is a zero-mean Gaussian flux noise process, and 푎 ∈ [0, 1] interpolates between purely Gaussian (푎 = 0) and fully non-Gaussian (푎 = 1) regimes. 훼푉 cor- responds to the linear coefficient, 훽푉 corresponds to the quadratic coefficient. If the flux bias of qubit is exactly located at Φ = Φ0/2, the dephasing process will be in the

65 fully non-Gaussian regime (푎 = 1). Following Ref. [33], the power spectrum and bispectrum of a dephasing process 퐵(푡) are given by

∫︁ ∞ −푖휔1휏1 푆1(휔1) = 푑휏1푒 퐶2(0, 휏1) −∞ 2 ∫︁ ∞ 2 2 2푎 2 =(1 − 푎) (훼푉 )푆휉(휔1) + (훽푉 ) 푆휉(푢)푆휉(휔1 − 푢). (5.3) 2휋 −∞

∫︁ ∞ ∫︁ ∞ −푖휔1휏1 −푖휔2휏2 푆2(휔1, 휔2) = 푑휏1 푑휏2푒 푒 퐶3(0, 휏1, 휏2) −∞ −∞ 2 2 =2(1 − 푎) 푎(훼푉 훽푉 )(푆휉(휔1)푆휉(휔2) + 푆휉(휔1 + 휔2)푆휉(휔2) + 푆휉(휔1 + 휔2)푆휉(휔1)) 3 ∫︁ ∞ 8푎 3 + (훽푉 ) 푑푢푆휉(푢)푆휉(휔1 + 푢)푆휉(휔2 − 푢), (5.4) 2휋 −∞

(2) (3) where 퐶 (0, 휏1) and 퐶 (0, 휏1, 휏2) correspond to the second-order cumulant and the third-order cumulant of the dephasing process 퐵(푡), respectively. Here, 푆(휔) is the (2) (2) (2) power spectrum associated with 휉(푡). Let 퐶휉 (푡1, 푡2) = 퐶휉 (0, 푡2 − 푡1) ≡ 퐶휉 (0, 휏) be the second order cumulant of 휉(푡). Then

∫︁ ∞ −푖휔휏 (2) 푆휉(휔) = 푑휏푒 퐶휉 (0, 휏). (5.5) −∞

Figure. 5-2 shows an example of bispectrum of a squared Gaussian dephasing process. In the experiments, we will apply Gaussian flux noise with a Lorentzian power spectral density,

푃0 1 1 푆휉(휔) = × ( 2 + 2 , (5.6) 2휋휔푐 1 + [(휔 − 휔0)/휔푐] 1 + [(휔 + 휔0)/휔푐] where 푃0 is noise power, 휔푐 is the HWHM (half-width-half-maximum), and 휔0 is the center frequency of the Lorentzian function. Due to the quadratic response to external flux, the qubit will experience a squared Gaussian dephasing process, for whichthe bispectrum has the same shape as shown in Figure. 5-2.

66 5.2 Verifying the presence of non-Gaussianity

X +pi/2 or X +pi/2 Y +pi Y +pi Y +pi/2

Readout 1000 (ns)

Figure 5-3: Illustration of the CPMG sequence with two control pulses (휋 pulses). The last 휋/2 pulse is a tomography pulse which can be either 푋 + 휋/2 or 푌 + 휋/2, which projects ⟨휎푥⟩ or ⟨휎푦⟩ onto the measurement-axis (휎푧), respectively.

Following Ref. [33], for sufficiently weak noise or at small enough time scales, the cumulant expansion for the phase angle (휑) in Eq. 3.17 can be truncated at 푙 = 1. After a control sequence at time 푡 = 푇 , the phase angle rotation(휑) is:

∫︁ 푇 ∫︁ 푇 ∫︁ 푇 ∫︁ 푇 (1) 1 (3) 휑(푇 ) ≈ 푑푡푦(푡)퐶 (푡) − 푑푡1 푑푡2 푑푡3푦(푡1)푦(푡2)푦(푡3)퐶 (푡1, 푡2, 푡3). 0 3! 0 0 0 (5.7)

Assuming that dephasing process is stationary, the cumulants depend solely on the

time separations 휏푖−1 = 푡푖 − 푡푖−1, rather than the absolute times. Then, by taking the Fourier transform of the terms in Eq. 5.7 with respect to the time separations, one can relate 휑(푇 ) to the spectra of the dephasing noise:

1 ∫︁ ∞ ∫︁ ∞ 휑(푇 ) ≈ 퐹 (0)휇퐵 − 2 푑휔1 푑휔2퐹 (−휔1)퐹 (−휔2)퐹 (휔1 + 휔2)푆2(휔1, 휔2), 3!(2휋) −∞ −∞ (5.8)

∫︀ 휏 −푖휔푡 where 퐹 (휔, 휏) = 0 푦(푡)푒 푑푡 is the filter function of the control sequence and

푆2(휔1, 휔2) is the bispectrum of the dephasing process, 퐵(푡). Note that the first cumulant of the noise process is equivalent to temporal mean of the noise, 퐶(1)(푡) =

⟨퐵(푡)⟩ ≡ 휇퐵, because the noise process is assumed to be stationary.

67 1 1 10 10 3 3rd cumulant, (P0) Experiment Monte Carlo Simulation

0 0 10 10 ) d

, a r r ( e

t e

, m e l 1 a 1 r g

10 a 10 n p a

y e a s c a e h D P

2 2 10 10

2 2nd cumulant, (P0) Experiment Monte Carlo Simulation 3 3 10 10 6 5 6 5 10 10 10 10 2 2 Noise Power, P0 ( 0 ) Noise Power, P0 ( 0 )

(a) 휑 vs. 푃0 (b) 휒 vs. 푃0

Figure 5-4: Experimental result and Monte-Carlo simulation result for 휑, 휒, vs. 푃0, where the phase angle (휑) and the decay parameter (휒) are defined as 휑 = tan−1(− ⟨휎푥⟩ ), 휒 = − ln(√︀⟨휎 ⟩2 + ⟨휎 ⟩2). We ran a Monte-Carlo simulation over 106 ⟨휎푦⟩ 푥 푦 noise realizations, and the experiment over 4 × 105 noise realizations.

Likewise, for sufficiently weak noise, we can approximate the decay parameter 휒 by truncating at 푙 = 1,

∫︁ 푇 ∫︁ 1 푇 (2) 휒(푇 ) ≈ 푑푡1 0 푑푡2푦(푡1)푦(푡2)퐶 (푡1, 푡2) 2! 0 ∫︁ ∞ 1 2 = 푑휔1|퐹 (휔1)| 푆1(휔1), (5.9) 2!(2휋) −∞ where 푆1(휔1) is the power spectrum of the dephasing process, B(t).

Here, we apply a 1 휇푠-long CPMG sequence (See Figure. 5-3) to filter out the first term of Eq. 5.8. Then, from Eqs. 5.8 and 5.9, one can see that if thedephasing process is fully non-Gaussian (푎 = 1), then 휑(푇 ) and 휒(푇 ) approximately have a cubic dependence and a quadratic dependence on noise power, respectively. Therefore one

68 can verify whether the dephasing is non-Gaussian by measuring 휑 and 휒 with respect

to noise power 푃0. Figure. 5-9 shows the change of 휑 and 휒 with respect to the applied noise power. In this figure, both the Monte-Carlo simulation and the experimental

−5 2 results follow the expected dependence for small noise power (푃0 < 5 * 10 Φ0). Note that, for larger noise power, higher-order cumulants become non-negligible and the full cumulant expansion can no longer be truncated at 푙 = 1.

5.3 Reconstruction of the bispectrum

5.3.1 Review of the old approach (Section 3.2.4).

The old approach entails the two steps for reconstructing bispectrum:

1. Estimate the bispectrum at interior points using M repetitions of sequences with non-zero filtering order.

2. Using the reconstructed bispectrum in step (1), estimate the bispectrum at zero frequency and the mean by measuring phase angle rotations after M repetitions of sequences with zero filtering order.

However, there are two issues with this old approach [34]:

Issue 1. Ill-conditioned reconstruction matrix, A푧

The reconstruction matrix, A푧 in Eq. 3.26 is often ill-conditioned, because the first 푧 column of A is proportional to 퐹1(0) ∼ 풪(푇 ), and it is much bigger than the other 3 columns, which are proportional to 푅푒(퐹푝(−푧1)퐹푃 (−푧2)퐹푝(푧1 + 푧2)) ∼ 풪(푇 ). This implies a small error in the first element of 휑⃗푧 will give rise to much larger errors in ⃗푧 푆2

Issue 2. Error compounding

To estimate the mean and the bispectrum at zero frequencies, one should estimate the bispectrum at the interior points first. This means that errors in the previous

69 estimate of interior points will be compounded when estimating the bispectrum at zero frequencies. This compounded error combined with the ill-conditioned reconstruction matrix leads to a poor reconstruction.

5.3.2 Reconstructing the bispectrum with new approach

To resolve the issues that the old approach has, we introduce a new approach de- veloped by L. M. Norris, F. Beaudoin, and L. Viola [34, 4, 51]; First, to avoid the ill-conditioned reconstruction matrix, we will estimate the mean of dephasing process independently and in advance. Then, from Eq. 3.20, one can define a “modified phase” 휑˜:

휑˜푝(푀푇 ) ≡휑푝(푀푇 ) − 푀퐹푝(0)휇퐵 푀 ∑︁ = − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ). 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2 (5.10)

Note that for a set of control sequences, the above system of the equations can be represented as a following matrix equation:

[휑˜]푇 = 풜[푆]푇 , (5.11) where all of the columns in the reconstruction matrix 풜 are expected to be of the same order of magnitude (∼ 풪(푇 3)). This implies that we can estimate the interior points and the zeros with a reconstruction matrix which is well-conditioned. Further- more, estimating the bispectrum in one go of matrix inversion will avoid the error compounding that we observed when using the old approach [34].

Step 1. Robust estimation of the noise mean

According to Refs. [4, 51], we know that the only effect of frequency detuning 푓푑푟푖푣푒 −

푓푞푢푏푖푡 will be a shift of the mean. This fact suggests the following robust method of estimating the noise mean; First, the Ramsey sequence described in Figure. 5-5 is

70 X + /2 Y + /2 Readout

π π

50 (ns)

Figure 5-5: Illustration of the Ramsey sequence used to estimate the mean of dephas- ing process. The first 푋, +휋/2 pulse is to initialize the qubit onto the transversal plane along the y-axis and the second 푌, +휋/2 pulse projects the x-component of spin vector (−휎^푥) onto the measurement axis (z-axis). applied with a fixed pulse interval 푇 = 50 ns, leading to a constant filter function ′ 퐹퐹 퐼퐷(0). Note that the pulse interval should be chosen to be small enough that higher-order cumulants are negligible, but large enough to avoid overlap between the

pulses. Then, second, by varying the drive frequency, the expectation value of 휎^푧 is measured. If we approximate a cumulant expansion in 퐵(푡) with the truncated

expression to first order, the expectation value of 휎^푧 is:

′ ⟨휎^푧(푡푓 )⟩ ≈ 퐹퐹 퐼퐷(0, 푡푓 )(휇퐵 − 퐷) (5.12)

Therefore, as a final step, if we plot the expectation value of 휎푧 (y-axis) versus de-

tuning (퐷, x-axis), we will find the x-intercept to be 휇퐵. This estimation of the noise

mean (휇퐵) is very robust in the sense that any effect of finite pulse width does not impact the x-intercept but only slope.

71 z , without noise 0.05 z , with noise

0.00 z

0.05

est B /2 = 125.8 (kHz)

200 100 0 100 200 Detuning, fdrive fqubit (kHz)

Figure 5-6: Robust estimation of the mean of the dephasing process. The separation between the x intercepts of two fitted lines corresponds to the mean of dephasing 푒푠푡 process, 휇퐵

Step 2. Apply the control sequences measure the qubit’s response

We used 11 different control sequences to reconstruct 11 different points on theprin- cipal domain (Fig. 5-7) [34, 51]. Note that due the symmetries of the bispectrum (see Section 3.2.4 and Figure 3-4), one only needs to reconstruct the bispectrum over this octant. Seq. 2, ..., 11 are repeated M = 10 times to approximate the product of filter as “frequency comb” (see Section 3.2.3 for details).

72 Principal Domain

Free Evolution

Figure 5-7: The principal domain of 2D frequency grid containing the harmonic frequencies where the bispectrum will be reconstructed and the timing diagrams for base control sequences. The length of the base control sequence (휏) is 960 ns, and Seq. 2, ..., 11 are repeated 10 times (M = 10). Note that the bispectrum on the x-axis of the principal domain (orange color) contributes to the phase accumulation of only sequence 1, ..., 5, which have non-zero values of the filter function at zero frequency (퐹푝(0, 휏) ≠ 0).

Step 3. Reconstruct the bispectrum through linear inversion

We consider the following quantity that depends on the measured phase in Step 2., and estimated mean in Step 1.

˜ 푒푠푡 휑푝(푀푇 ) = 휑푝(푀푇 ) − 푀퐹푝(0)휇퐵 . (5.13)

These “modified phases”, 휑˜푝(푀푇 ) for each sequence (푝 = 1, 2, ..., 11) are plotted in Figure. 5-8. Note that sequence 2, 3, ... 11 are repeated M = 10 times, but sequence 1 is implemented only once M = 1. From Eq. 3.21, one can relate the modified phase angles with the bispectrum in the octant as following:

푀 ∑︁ 휑˜ (푀푇 ) = − 푚(ℎ , ℎ )Re[퐹 (−ℎ )퐹 (−ℎ )퐹 (ℎ + ℎ )]푆 (ℎ , ℎ ). 푝 3!푇 2 1 2 푝 1 푝 2 푝 1 2 2 1 2 (ℎ1,ℎ2)∈ℋ2 (5.14)

73 One can represent Eq. 5.14 as a matrix equation:

[휑˜]푇 = 풜[푆]푇 , (5.15)

˜ 푇 ˜ ˜ ˜ 푇 푇 (1) (1) (2) (2) (11) (11) 푇 where [휑] = [휑1, 휑2, ..., 휑11] , [푆] = [푆2(ℎ1 , ℎ2 ), 푆2(ℎ1 , ℎ2 ), ..., 푆2(ℎ1 , ℎ2 )] , and the reconstruction matrix 풜 has elements

푀 퐴 = − 푚(ℎ(푞), ℎ(푞))Re[퐹 (−ℎ(푞))퐹 (−ℎ(푞))퐹 (ℎ(푞) + ℎ(푞))]. (5.16) 푝,푞 3!푇 2 1 2 푝 1 푝 2 푝 1 2

Then, one can straightforwardly estimate the bispectrum in the principal domain [푆]푇 by inverting the reconstruction matrix 풜:

[푆]푇 = 풜−1[휑˜]푇 . (5.17)

1 10 1 , Experiment. 10

2 10 0 10

3 10 ) ) . d A .

a 1

r 0 N ( 10

(

3 10

2 2 10 10 , Theory 1 10 , Experiment. 3 10 1 2 3 4 5 6 7 8 9 10 11 Sequence Index

Figure 5-8: 휑˜푝(푀푇 ) and 휒 for 11 different control sequences.

74 Experiment Theory

5 5 7 7

10 ) 10 )

0 z 0 z

6 H 6 H ( (

10 ) 10 )

5 2 5 2 f f

5 , 5 ,

10 1 10 1

5 0 5 f 5 0 5 f ( (

4 2 4 2

10 S 10 S ...... 5 5

5 0 z) 5 0 z) 0 H 0 H 5 (M 5 (M f1 ( 5 f 2 f1 ( 5 f 2 MHz) MHz)

< 103 104 105 106 > 107 < 103 104 105 106 > 107 S2(f1, f2) S2(f1, f2)

(a) Reconstructed bispectrum (experiment) (b) Reconstructed bispectrum (theory)

Figure 5-9: Reconstructed bispectrum based on experimental data (experiment) and theoretical calculation (theory).

75 76 Chapter 6

Conclusions and Future Work

The precise knowledge of the spectral properties of environmental noise enables a better understanding of the qubit environment and the abilities to better isolate the qubit from this environment. In this work, by using the filtering property of dynamic decoupling sequences, we have shown that it is possible to characterize higher-order cumulants of a non-Gaussian dephasing process. This experimental demonstration of non-Gaussian noise spectroscopy protocol represents a major step towards the complete noise spectral characterization of quantum devices. In the following sections, I present a few potential directions to extending this work including non-Gaussian noise spectroscopy for natural noise process (Section 6.1), a theoretically proposed idea to further reduce the effect of experimental errors for the reconstruction (Section 6.2), and high-speed noise spectroscopy by using active reset protocols (Section 6.3).

6.1 Characterizing natural non-Gaussian dephasing processes

There are natural noise sources which may cause non-Gaussian dephasing in supercon- ducting qubits. Of course, a squared Gaussian dephasing process originating from 1/f flux noise applied to the qubit biased at its degeneracy point is one mechanism. Based

77 on the previous studies on natural 1/f type flux noise in flux tunable qubits [59, 8,3],

2 the noise power of natural flux noise is known to be “universally” around (1 휇Φ0) at 1 Hz. As future work, we will theoretically and experimentally study whether our protocol is able to detect the higher-order cumulants of the non-Gaussian dephasing from the natural 1/f flux noise. Another potential source of non-Gaussian noiseis a few two level defects (TLS) that are strongly coupled to a superconducting qubit as stated in Chapter 1. This phenomena has been observed in superconducting cir- cuits [25] for a long time, but the spectral properties of noise process are not well studied. Hence, it would be interesting to characterize higher order spectral proper- ties of this noise process, where a few two level defects at Josephson junctions are strongly coupled to a superconducting qubit [25].

6.2 Regularized maximum likelihood estimation for bispectrum reconstruction

In this work, we reconstructed the bispectrum with a least-square estimate,

푇 푇 푇 [푆퐿푆] = min ||풜[푆] − [휑˜] || (6.1) [푆]푇 where [푆]푇 denotes a column vector of bispectrum values at a set of harmonics in the principal domain, 풜 is the reconstruction matrix, and [휑˜]푇 is a column vector of experimentally measured “modified phase angles”. For an invertible matrix 풜, the least square estimate is equivalent to the estimate obtained through matrix inversion,

푇 − 푇 [푆퐿푆] = 풜 1[휑˜] (6.2)

A drawback of this approach is that the reconstruction matrix 풜 is often ill-conditioned.

푇 푇 Even small errors in [휑˜] can lead to significant errors in the estimation of [푆퐿푆] . Furthermore, this does not take into account measurement error and phase estimates with large error bars contribute to the estimate of bispectrum as much as those with

78 small error bars. To improve this estimation, L. M. Norris suggested an alternative approach using a regularized maximum likelihood (RML) estimate as following:

푇 (︁ 푇 푇 −1 푇 푇 )︁ 푇 [푆푅푀퐿] = min 풜[푆] − [휑˜] )Σ (풜[푆] − [휑˜] + 푄([푆] )) (6.3) [푆]푇 where 푄([푆]푇 ) is a regularization term that helps avoid overfitting. One can use a variation of Tikhonov as a regularization term, where 푄([푆]푇 ) = ||풟[푆]푇 ||2. Here, by designing the smoothing matrix 퐷 to favor smaller values of the bispectrum at the border of its principal domain, one can take advantage of prior knowledge of the spectral cutoff frequency. Another advantage of RML estimate is that it accounts for

푇 experimental error due to the presence of the covariance matrix Σ, so that [푆푅푀퐿] depends preferentially on measured phase angles with small error bars. Although we have not studied this approach in detail yet, we expect this will make the QNS protocol more robust to experimental errors.

6.3 Faster noise spectroscopy using active reset pro- tocols

In general, noise spectroscopy, especially for reconstructing the bispectrum, requires a number of projective measurements to obtain statistical validity. For coherent qubits, without active reset protocols, “qubit initilization” takes substantial time because one needs to wait more than multiples of 푇1 for the qubit to fully initialize to its ground state. However, by using active reset protocol [54, 48, 16, 27], one can reduce the reset time, thereby enabling faster noise spectroscopy.

79 80 Appendix A

Moments and Cumulants

In statistics, moments and cumulants are specific quantitative measures used to char- acterize statistical distributions. Here, we review the mathematical definition of mo- ments and cumulants.

A.1 Moment and moment-generating function

In statistics, the 푘-th moment of a random variable 푋 with probability density func- tion 푓(푥) is defined as:

∫︁ ∞ 푀 (푘) ≡ 퐸(푋푘) = 푥푘푓(푥)푑푥. (A.1) −∞ for integer 푘 = 0, 1, .... The moment generating function 푔푀 is defined as:

휉2 휉푘 푔 (휉) ≡ 퐸(푒휉푋 ) = 퐸(1 + 휉푋 + 푋2 + ... + 푋푘) 푀 2! 푘! ∞ ∑︁ 휉푘 = 푀 (푘) . (A.2) 푘! 푘=0

Note that the moment generating function encodes the moments in its Taylor expan- sion around the origin.

81 A.2 Cumulant and cumulant-generating function

The cumulant generating function is defined as the logarithm of the moment-generating function as follows:

푔퐶 (휉) ≡ log(푔푀 (휉)). (A.3)

The cumulants 퐶(푘) are obtained from a power series expansion of the cumulant generating function 푔퐶 (휉),

∞ ∑︁ 휉푘 푔 (휉) ≡ 퐶(푘) . (A.4) 퐶 푘! 푘=0

From Eqs. A.2, A.3 and A.4, the relationship between the first few moments and cumulants can be found as follows:

퐶(1) = 푀 (1)

퐶(2) = 푀 (2) − (푀 (1))2

퐶(3) = 2(푀 (1))3 − 3푀 (1)푀 (2) + 푀 (3)

퐶(4) = −6(푀 (1))4 + 12(푀 (1))2푀 (2) − 3(푀 (2))2 − 4푀 (1)푀 (3)푀 (4) . . (A.5)

82 Appendix B

Derivation of the Joint Hamiltonian in the Toggling Frame

Here, I derive the joint Hamiltonian 퐻˜ stated in Eq. 3.3 in Section 3.1. Consider a ^ 1 ^ single qubit (퐻푆 = 2 ~휔0휎^푧) coupled to a bath (퐻B) via pure dephasing interaction

(퐻^푆퐵 = 푏SB휎^푧 ⊗ 퐸^). 퐸^ is some operator of the bath and 푏푆퐵 is the coupling strength between the qubit and the bath. An open-loop control on the qubit is implemented via

the Hamiltonian 퐻^ctrl(푡) acting on the system alone. Here, we assume that the open- loop control is implemented in terms of sequences of perfect instantaneous (bang- bang) 휋 pulses.

^ ^ ^ ^ ^ ^ ^ ^ 퐻 = 퐻S ⊗ IB + IS ⊗ 퐻B + 퐻SB + 퐻ctrl(푡) ⊗ IB. (B.1)

Because the interaction is pure dephasing (no energy exchange), the bare qubit Hamil- ^ ^ ^ tonian 퐻S ⊗ IB commutes with interaction Hamiltonian 퐻SB as follows:

^ ^ ^ [퐻S ⊗ IB, 퐻푆퐵] = 0. (B.2)

83 In the interaction picture associated with the qubit Hamiltonian 퐻^S, the full Hamil- tonian becomes

^ (S) ^ † ^ ^ ^˙ † ^ 퐻 = 푈S(푡)퐻푈S(푡) − 푖푈S(푡)푈S(푡), (B.3) where

^ 푖 ^ ^ 푈S(푡) = exp[− 퐻S ⊗ IB푡]. (B.4) ~

^ ^ Note that the second term in Eq. B.3 cancels the 퐻S ⊗ IB part of the first term, leaving

^ (S) ^ ^ ^ ^ 푅퐹 ^ 퐻 = IS ⊗ 퐻B + 퐻SB + 퐻ctrl (푡) ⊗ IB, (B.5)

푖 ^ 푖 ^ ^ 푅퐹 − 퐻S푡 ^ 퐻S푡 where 퐻ctrl (푡) ≡ 푒 ~ 퐻ctrl(푡)푒 ~ is the control Hamiltonian in the rotating frame,

which rotates at the qubit frequency (휔0). We now eliminate the bath Hamiltonian ^ ^ IS ⊗ 퐻B by using an interaction representation with respect to the evolution of the ^ ^ isolated bath IS ⊗ 퐻B,

^ (S,B) ^ † ^ (S) ^ ^˙ † ^ 퐻 = 푈B(푡)퐻 푈B(푡) − 푖푈B(푡)푈B(푡), (B.6)

where

^ 푖 ^ ^ 푈B(푡) = exp[− IS ⊗ 퐻B푡]. (B.7) ~

The full Hamiltonian in the interaction picture then becomes

푖 ^ 푖 ^ ^ (S,B) − 퐻B푡 ^ 퐻B푡 ^ 푅퐹 ^ 퐻 = 푏푆퐵휎^푧 ⊗ 푒 ~ 퐸푒 ~ + 퐻ctrl (푡) ⊗ IB. (B.8)

Since 퐻^B doesn’t commute with 퐸^ in general, the effective system-bath interaction 푖 푖 − 퐻^B푡 퐻^B푡 푏푆퐵휎^푧 ⊗ 푒 ~ 퐸푒^ ~ is time-dependent and the system experiences a fluctuating 푖 푖 − 퐻^B푡 퐻^B푡 coupling with the bath. Tracing over the bath variables replaces 푒 ~ 퐸푒^ ~ by

84 the stochastic function 푏푆퐵퐸(푡) as follows:

(S,B) ~ 푅퐹 ~ 푅퐹 퐻^ (푡) = 푏 퐸(푡)^휎 + 퐻^ (푡) ≡ 퐵(푡)^휎 + 퐻^ (푡), (B.9) 2 SB 푧 ctrl 2 푧 ctrl where 퐵(푡) is stochastic process. As a final step, let’s move onto the interaction ^ 푅퐹 picture associated with the open-loop control Hamiltonian 퐻ctrl (푡) (called the toggling frame). We introduce the control propagator 푈^ctrl(푡),

푖 ∫︁ 푡 푈^ctrl(푡) = 풯+ exp[− 푑푠퐻^ctrl(푠)], (B.10) ~ 0 where 풯+ is the time ordering operator [21]. Then, Eq. B.9 can be written in the toggling frame as:

† (S,B) ~ 퐻˜^ (푡) = 푈^ (푡)퐻^ (푡)푈^ (푡) = 푦(푡)퐵(푡)^휎 , (B.11) ctrl ctrl 2 푧 where 푦(푡) is a “switching function” induced by the control, the exact form of which ^ 푅퐹 depends on 퐻ctrl (푡). Assuming that the open loop control is implemented as sequences of perfect instantaneous 휋-pulses, 푦(푡) can be thought of as a function which changes its sign between ±1 with every 휋-pulse applied to the qubit.

85 86 87 88 Appendix C

Timing digram of control pulse sequences

The set of control pulse sequences designed for reconstructing the bispectrum are summarized in Table. C.1. These control pulses are designed by Leigh Note that all control pulse sequences start and end with a 휋/2 pulse for the purposes of preparation and tomography.

89 Figure C-1: Timing diagrams of the control pulse sequences. Note that Seq. 2-11 are repeated 10 times. Only 휋 pulses are shown. All 휋 pulses are around 푦-axis

Table C.1: The control pulse sequences designed for reconstructing the bispectrum. Only 휋 pulses are shown.

Seq. Index, p Position of control pulses (ns) Repetitions, M 퐹푝(0, 휏) 1 No pulses (free evolution) 1 ≠ 0 2 105, 240, 345, 480, 585, 720, 825, 960 10 ≠ 0 3 90, 235, 410, 555, 730, 875 10 ≠ 0 4 80, 150, 205, 355, 560, 630, 685, 835 10 ≠ 0 5 125, 175, 225, 275, 325, 610, 820, 875 10 ≠ 0 6 85, 135, 185, 240, 455, 775, 825, 880 10 0 7 130, 180, 285, 335, 475, 765, 870, 960 10 0 8 90, 150, 200, 305, 500, 715, 860, 960 10 0 9 80, 320, 370, 425, 600, 650, 720, 855 10 0 10 205, 310, 360, 545, 645, 725, 850, 960 10 0 11 145, 365, 425, 495, 600, 680, 850, 960 10 0

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