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中国科学通报：英文版

t-bit semiclassical quantum Fourier transform

 Fu, Xiang-Qun, Bao, Wansu, Zhou Zhou, Chun, Song, Zhen

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

因为造一个高度维的量寄存器的困难，这份报纸基于一个低维的量寄存器论述高度维的量 Fourier 变换(QFT ) 的实现。首先，我们定义 t 小点半古典作品量 Fourier 变换。以概率振幅，我们证明变换能认识到量 Fourier 转变，说明为 two-qubit 门的要求显然，并且进一步减少设计变换的一个量电路。联合古典改正窗户方法和 Shors 量因式分解算法的实现，我们然后为 Shors 算法重新设计一个电路，谁的要求计算资源近似等于 Parkers 的。为为 Parkers 算法的基本的量门的要求是 O (日志 N3，和为我们的电路的量寄存器要求 t1 更多的尺寸比 Parkers。然而，我们的电路是 2 倍快于 Parkers 的 t，在 t 是窗户的宽度的地方。

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因为造一个高度维的量寄存器的困难，这份报纸基于一个低维的量寄存器论述高度维的量 Fourier 变换(QFT ) 的实现。首先，我们定义 t 小点半古典作品量 Fourier 变换。以概率振幅，我们证明变换能认识到量 Fourier 转变，说明为 two-qubit 门的要求显然，并且进一步减少设计变换的一个量电路。联合古典改正窗户方法和 Shors 量因式分解算法的实现，我们然后为 Shors 算法重新设计一个电路，谁的要求计算资源近似等于 Parkers 的。为为 Parkers 算法的基本的量门的要求是 O (日志 N3，和为我们的电路的量寄存器要求 t1 更多的尺寸比 Parkers。然而，我们的电路是 2 倍快于 Parkers 的 t，在 t 是窗户的宽度的地方。
针对目前大维数量子寄存器生成的困难性, 研究了基于小维数量子寄存器实现大维数量子Fourier 变换的方法. 首先, 定义了t 比特半经典量子Fourier 变换, 从几率幅的角度证明该变换可以实现量子Fourier 变换, 且所需2 位量子门的规模显著降低, 并设计了该变换的量子实现线路. 然后基于t 比特半经典量子Fourier 变换, 将经典固定窗口法与Shor 算法实现方法相融合, 重新设计了Shor 整数分解量子算法的实现线路, 与Parker 等人的实现线路相比, 计算资源大体相同(所需的基本量子门均为所需量子寄存器的维数前者较后者多t -1维), 而实现速度提高了t2 倍, t 是窗口宽度.
Because of the difficulty of building a high-dimensional quantum register, this paper presents an implementation of the high-dimensional quantum Fourier transform (QFT) based on a low-dimensional quantum register. First, we define the t-bit semiclassical quantum Fourier transform. In terms of probability amplitude, we prove that the transform can realize quantum Fourier transformation, illustrate that the requirement for the two-qubit gate reduces obviously, and further design a quantum circuit of the transform. Combining the classical fixed-window method and the implementation of Shor’s quantum factorization algorithm, we then redesign a circuit for Shor’s algorithm, whose required computation resource is approximately equal to that of Parker’s. The requirement for elementary quantum gates for Parker’s algorithm is O(⌈ log N⌉3, and the quantum register for our circuit requires t−1 more dimensions than Parker’s. However, our circuit is t 2 times as fast as Parker’s, where t is the width of the window.
To overcome the difficulty of realizing large-scale quantum Fourier transform (QFT) within existing technology, this paper presents a resource-saving method, namely t-bit semiclassical QFT over (Z_(2^n)), which could realize large-scale QFT using arbitrary-scale quantum register. Using our method, the scale of quantum register can be determined flexibility according to the scale of quantum system, enabling the quantum resource and speed of realizing QFT to be optimal. By developing a feasible method to realize the control quantum gate R_k, we experimentally demonstrate the 2-bit semiclassical QFT over (Z_(2^3)) on IBM's quantum cloud computer, showing the feasibility of our proposed method. Then, we compare the actual performance of 2-bit semiclassical QFT and standard QFT in the experiments. Experimental data show that the fidelity of the result of 2-bit semiclassical QFT is higher than that of standard QFT, which is mainly due to less two-qubit controlled gates are required in the semiclassical QFT. Furthermore, based on the proposed method, we successfully implement the Shor's algorithm to factorize N=15.