模擬導向設計於超導量子位元單量子閘控制脈衝之實現

Abstract

本論文研究了超導量子位元 Transmon 的數學模型，並將其應用於數值模擬中，探討了不同激發態（excited state）數量下的模擬優化結果、不同波型條件下的優化效果，並進行了與確定性測試(Deterministic Benchmarking)的實驗比對，最後測試優化前後隨機性測試(Randomized Benchmarking)結果的變化。模擬結果顯示，隨著考慮的激發態數量增加，優化所需時間也隨之增長，模擬所得的閘保真度（fidelity）會逐漸偏離理論上的上限。而在使用不同波型進行優化的情況下，可以觀察到參數數量的增加雖然會導致更長的優化時間，但同時也能提升優化效果，讓結果更接近理論值。在確定性測試的實驗比對中，模擬所設定的 T1、T2 以及旋轉誤差（rotation error）與實驗結果有良好的符合程度；然而在相位誤差（phase error）方面，模擬與實驗結果有明顯差異。推測原因可能是實驗中 −X 與 X 操作在實際實現上存在細微差異，導致實驗中的衰減速度明顯快於模擬結果。此外，優化後相位誤差的平衡狀態亦未能與實驗一致，可能是因為模型中僅考慮了 T1 和 T2 的衰減機制，未將其他實驗上可能存在的雜訊來源（例如 1/f 誤差或準粒子雜訊）納入考量，導致模擬結果傾向維持在 |0⟩ 態，而實驗結果則趨近於無資訊的混合態。最後將模擬所得到的最佳參數應用於隨機性測試實驗，可以觀察到保真度有小幅提升；然而此結果亦可能受到其他因素影響，例如實驗時間長短、擬合誤差，或是隨機性測試量測次數不足等。因此，若欲進一步驗證模擬與優化方法的有效性，未來仍需透過更多次的實驗比對與數據收集來進行佐證。

This thesis investigates the mathematical modeling of the superconducting qubit transmon and its application in numerical simulations. Various optimization scenarios are explored, including different numbers of excited states and distinct pulse shapes. Deterministic Benchmarking (DB) is employed to validate the simulations against experimental data, and Randomized Benchmarking (RB) is used to evaluate fidelity before and after pulse optimization. In the simulations, increasing the number of excited states results in longer optimization time and a greater deviation of the fidelity from its theoretical upper bound. For different pulse shapes, increasing the number of parameters similarly increases the optimization time, but can lead to improved performance that approaches the theoretical limit. In DB comparisons, simulated results for T1, T2, and rotation errors align well with experimental data. However, phase error shows a significant discrepancy: the experimental decay is faster than simulated results, likely due to subtle differences between the X and −X gates in practice. Furthermore, the steady-state behavior of the optimized phase error does not match experimental trends, possibly because the model accounts only for T1 and T2 decay and neglects other noise sources. As a result, the experimental data tend to converge to a fully mixed state, whereas simulations remain closer to the |0⟩ state. Finally, applying the optimized parameters in RB measurements reveals a modest improvement in gate fidelity. However, this enhancement may also arise from other factors, such as the total experiment duration, fitting inaccuracies, or insufficient measurement repetitions. Therefore, further verification would require a larger number of experimental trials to establish more conclusive comparisons.