Characterizing single-qubit gate fidelity on superconducting qubits

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研究生：	林維恩 Wei-En Lin
論文名稱：	Characterizing single-qubit gate fidelity on superconducting qubits 超導電路單量子位元邏輯閘效度之表徵
指導教授：	陳永富 Yung-Fu Chen
口試委員:
學位類別：	碩士 Master
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2021
畢業學年度：	109
語文別：	英文
論文頁數：	146
中文關鍵詞：	transmon 量子位元、量子非破獲性讀取、量子邏輯閘效度、量子過程斷層掃描、隨機基準分析
外文關鍵詞：	quantum non-demolition readout, gate fidelity, quantum process tomography, randomized benchmarking
相關次數：	點閱：6 下載：0
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我們在超導電路系統研究原子與光的相互作用。在我們實驗室，我們製造量子超導電路，例如在一維傳輸線共振器嵌入一個小芯片的transmon 量子位元。為了獲取量子位元之系統的特性，列如共振器頻率、量子位元躍遷頻率和量子位元與共振器之耦合強度，我們藉由量子非波壞性(QND) 讀取，在強色散的區域下進行頻譜實驗。基於頻譜實驗的結果，我們利用微波脈衝序列進行單一量子位元狀態之控制實驗。最近，我們專注於單量子位元邏輯閘效度的研究。我們採取兩種方法估計邏輯閘的效度：量子過程斷層掃描和隨機基準分析。當邏輯閘的長度增加，由於量子位元的去同調性，我們發現從量子過程斷層掃描得到的效度會降低。對於20 微秒之X 跟Y 閘，藉由量子過程斷層掃描的方法，我們得到了平均邏輯閘效度約為88.58% 跟87.16%；利用隨機基準分析的方法，我們估計的平均邏輯閘效度約為97.49% 跟97.38%。由於量子過程斷層掃描的方法多估計了狀態準備以及測量上的錯誤，因此它的平均邏輯閘效度比隨機基準分析估計的還要小。

In our lab, we fabricate the quantum superconducting circuit such as the single transmon qubit embedded in a one-dimensional transmission line resonator on the small chip.
To obtain the characteristics of superconducting qubit system, such as the resonant frequency of a resonator, qubit transition frequency, and qubit-resonator coupling strength,
we perform spectroscopy experiments in strong dispersive regime. Based on the characteristics of the system, we carry out various qubit state control experiments with microwave
pulse sequences. Recently, we focus on investigations of single qubit gate fidelity. We estimate gate fidelity based on two metrics: quantum process tomography (QPT), and randomized benchmarking (RB). When the gate length increases, we find that the fidelity from the QPT decrease due to the qubit decoherence effect. For 20 ns X and Y gates, we obtain the average gate fidelity 88.58% and 87.16% by the QPT ; we also estimate the average gate fidelity 97.49% and 97.38% by the RB. The fidelity via the QPT is smaller than that of the RB due to the over-estimated errors from the state preparation and measurement errors.

摘要ix
Abstract xi
Contents xiii
Introduction 1
1 Cavity Quantum Electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Circuit Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Cavity quantum electrodynamics 3
1 The Jaynes-Cummings model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Strong coupling limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Dispersive limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 The coherent drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Superconducting qubit and circuit QED 15
1 The quantization of the transmission line resonator . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Cooper pair box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Transmon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1 Charge dispersion of transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 The Josephson oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Transmon coupling with coplanar waveguide resonator . . . . . . . . . . . . . . . . . . . . . . 28
5.1 The generalized Jaynes-Cummings Hamiltonian. . . . . . . . . . . . . . . . . . . . . . 29
5.2 Dispersive regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Simulation of qubit spectroscopy 35
1 Qubit decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.1 Density matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.2 Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3 Bloch Redfield model for decoherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.4 Master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 One tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1 Flux-dependent spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 Vacuum Rabi supersplitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Cavity power dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Two tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Dispersive readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Overlapped CW spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Separated pulse spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Simulation of quantum gate operation 65
1 Optical Bloch equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2 Rabi oscillation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.1 Time and power Rabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Measurements for decoherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 T1 relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.2 Ramsey experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 Spin echo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1 Gate fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Quantum process tomography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Randomized Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Experimental setup and results 99
1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
1.1 Superconducting circuit design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
1.2 Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
1.3 Homodyne demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2 Measurement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.1 One tone spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.2 Two tone spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.3 Rabi oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.4 Power and time Rabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.5 T1 relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.6 Ramsey fringe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.7 Spin echo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.8 Quantum process tomography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.9 Randomized Benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Conclusion and future work 121
bibliography 123
                                

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