量子電腦透過位元獨特的「量子疊加」和「量子糾纏」特性，對比傳統電腦的運算步驟被位元數所限制，量子位元能夠同時產生0和1的疊加態，跳脫線性關係，往指數型運算邁進，傳統電腦需要上萬年才能解決的運算，量子電腦只要短時間即能完成。
小波轉換解決了傅立葉轉換在處理非平穩訊號上的缺陷，對於非平穩訊號的處理，傅立葉轉換只能分析訊號中內含有哪些頻率的成分，無法加入時頻進行分析，這也造成了在不同時域訊號中可能會對應好幾個類似頻譜圖的問題，因此小波轉換透過改變傅立葉轉換基底，進而能藉由小波函數的伸縮、平移對應到非平穩訊號在時域上每個位置的頻率，且得到一張精準的＂時頻譜圖＂。
本文目的是將小波轉換與量子電腦做結合，成為新的「量子小波理論」。我們選用了具有高消失動量且小波函數波形近似對稱的Coiflet小波作為研究的主題，利用矩陣分解、量子排列、以及量子邏輯閘設計方法，以基礎一位元邏輯閘和CNOT邏輯閘組合出Coiflet’C6小波轉換之完整量子電路，同時計算此電路設計需要的時間與空間複雜度。
最後加入NASS在量子電路中表達多維度的圖像，並實際地將量子C6小波轉換應用於圖像壓縮上，且加上量子容錯理論，估算容錯前後所需代價之差異。

Quantum computers have the unique characteristics of "quantum superposition" and "quantum entanglement" of qubits. Qubits can produce superposition states of 0 and 1 at the same time, escaping the linear relationship and moving towards exponential calculations. Therefore, quantum computers have better performance than classical computing.

Wavelet transform solves the defects of Fourier transform in processing non-stationary signals. For the processing of non-stationary signals, Fourier transform only analyze which frequency components are contained in the signal, and doesn’t analyze time-frequency. Therefore, the wavelet transform changes the Fourier transform base. It can correspond to the frequency of each position of the non-stationary signal in the time domain through the expansion and translation of the wavelet function. Moreover we can obtain an accurate "time-spectrogram".

The purpose of this article is to combine wavelet transformation with quantum computers to become a new "quantum wavelet theory. I chose the Coiflet wavelets that have high vanishing momentum and approximately symmetric wavelet function, and use matrix decomposition, quantum permutation, and quantum logic gate to combine the basic one-qubit logic gate and CNOT logic gate to form Coiflet' C6 wavelet transform. Moreover I also calculate the time complexity and space complexity required by the circuit design.

Finally, I add NASS method to express multi-dimensional images in quantum circuits, and the quantum C6 wavelet transform is actually applied to image compression. I also add quantum fault tolerance theory to estimate the difference in cost before and after fault tolerance.

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