We propose an implementation of the algorithm for the fast Fourier transform (FFT) as a quantum circuit consisting of a combination of some quantum gates. In our implementation, a data sequence is expressed by a tensor product of vector spaces. Namely, our FFT is defined as a transformation of the tensor product of quantum states. It is essentially different from the so-called quantum Fourier transform (QFT) defined to be a linear transformation of the amplitudes for the superposition of quantum states. The quantum circuit for the FFT consists of several circuits for elementary arithmetic operations such as a quantum adder, subtractor and shift operations, which are implemented as effectively as possible. Namely, our circuit does not generate any garbage bits. The advantages of our method compared to the QFT are its high versatility, and data storage efficiency in terms, for instance, of the quantum image processing.

中文翻译：

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快速傅立叶变换的量子电路

我们提出了一种用于快速傅立叶变换（FFT）的算法的实现，它是由一些量子门的组合组成的量子电路。在我们的实现中，数据序列由向量空间的张量积表示。即，我们的FFT被定义为量子态张量积的变换。它与所谓的量子傅里叶变换（QFT）本质上不同，量子傅里叶变换定义为对量子态进行叠加的幅度的线性变换。FFT的量子电路由几个用于基本算术运算的电路组成，例如量子加法器，减法器和移位运算，这些电路应尽可能有效地实现。即，我们的电路不会产生任何垃圾位。与QFT相比，我们的方法的优势在于其多功能性，