由於觀察及記錄一個隨時間演化的連續分布函數需要耗費相當龐大的資源，因此要去研究真實的量子系統並不容易，所以在資源有限的情況下，我們希望能利用離散的機率分布去近似連續函數帶來的消相位效果。而此研究將連續頻譜的真實量子系統離散化，會得到一組機率分布，接下來最重要的一個步驟就是必須重建出量子系統的量子電路。

重建量子電路需要利用Qiskit所提供的數據編碼套件中的Amplitude encoding來拆解量子電路，並且將其電路放入至IBM Q的量子電腦中求得其隨時間演化期望值的變化，最後比較真實量子系統跟模擬量子系統的差異，同時找到模擬效果最佳的參數。需要注意的是當構成環境的量子位元數越多時，整個系統會越來越龐大，所需要的計算時間跟資源就會隨之增加，因此目前最多只模擬到以5個量子位元構成環境的情況。

從實驗結果可以得知當構成環境的位元數太少導致機率數據不夠多，會使模擬的量子系統與真實量子系統幾乎不吻合，但若是構成環境的位元數足夠，頻率卻沒有平均且完整分布在頻譜上，也會使擬合更加失真，因此提升環境位元數的同時也要控制機率分布的間距大小。也就是說，當間距變小的同時，需要提高離散機率的數量，這樣擬合的效果才會好，但缺點就是將會更耗費計算資源。另外實驗結果也顯示，當我們增加構成環境的的量子位元數時，會使得動力學行為的震盪震幅變小且頻率變高; 而增加取樣間距也會使動力學行為的震盪頻率變高。由於古典電腦要模擬量子電腦很不容易，並且只要是取樣間距相同的情形，其動力學行為就會在固定週期上有反覆震盪的情形，因此還無法模擬出完全一致的量子系統。

Due to the substantial resources required for measuring a continuously evolving distribution function over time, we aim to approximate the dephasing effects caused by continuous functions using discrete probability distributions, particularly in scenarios with limited resources.
This research involves discretizing real quantum systems to obtain a set of probability distributions. The next step is to reconstruct the quantum circuit of the quantum system, which requires utilizing the Amplitude encoding provided by Qiskit. The circuit is then run on IBM Q's quantum computer to determine the variations in the expected values over time. Finally, a comparison is made between the real quantum system and the simulated system. However, as the number of quantum bits in the environment increases, the system becomes more extensive, requiring more resources. Therefore, the current simulations are limited to environments consisting of up to 5 quantum bits.
The experimental results show that having too few quantum bits in the environment leads to insufficient probability data, resulting in suboptimal simulation results. On the other hand, if the number of environment qubits is sufficient, but the frequencies are not evenly and thoroughly distributed in the spectrum, the fitting becomes distorted. Therefore, increasing the number of environment qubits must be accompanied by controlling the spacing of the probability distribution. The experimental results also demonstrate that increasing the number of quantum bits in the environment reduces the amplitude of oscillations and increases the frequency of the dynamical behavior.

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