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.
Abstract 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.
For discussing the solving methods of NP problems in classical computers and studying the circuit model which is the most representative in quantum computing to simulate the compute processes, this article discussed how to simulate the quantum Fourier transform by using the quantum logic gates, implemented the quantum Fourier transform and constructed the quantum information and compute simulation platform. The experiment introduced the quantum register structure to be the storage medium, which is better than the form of matrix in space. The operation processes adopted the bit manipulation to avoid the mass time for matrix multiplications. The results presented the changes of quantum amplitudes and probabilities of quantum states according to the quantum effect and compared with the approximate quantum Fourier transform. The experimental platform provided a groundwork for the further simulations of other important quantum algorithms and quantum circuits.
Hardware emulation based on field programmable gate array (FPGA) platform is vital to harness the power of quantum parallelism. As resource requirement grows exponentially in classical modeling quantum system, an optimum hardware architecture is crucial to emulate practical quantum circuits. Quantum Fourier transform (QFT) finds application in several critical quantum algorithms. In this work, experiments are conducted based on QFT to identify suitable qubit representation and hardware design technique. Experimental results show that 24-bit fixed point representation and serial architecture achieve optimal computation accuracy and resource utilization in QFT circuit emulation.
The research of ternary quantum system has gradually come into the attention of scholars in recent years. In
2018, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In
2019, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit flipping is realized by Not gate. Therefore, we need to extend Not gate to symmetric ternary system to realize qutrit flipping, which is called M-S gate. And then, by decomposing single-qutrit unitary gate in symmetric ternary quantum system, the universal gates are given. It means that any unitary operation on
n qutrits can be accurately implemented by single-qutrit symmetric ternary quantum gates and two-qutrit symmetric ternary M-S gates. By extending the QFT to the symmetric ternary quantum system, the paper successfully use some symmetric ternary quantum gates to construct the circuit which can realize symmetric ternary quantum Fourier transform (STQFT). Finally, the circuit of adder in symmetric ternary quantum system are designed based on the STQFT and the universal quantum gates. ternary quantum Fourier transform and its application (pp733-754) Hao Dong, Dayong Lu, and Xiaoyun Sun doi: https://doi.org/10.26421/QIC22.9-10-2 Abstracts: The research of ternary quantum system has gradually come into the attention of scholars in recent years. In
2018, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In
2019, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit flipping is realized by Not gate. Therefore, we need to extend Not gate to symmetric ternary system to realize qutrit flipping, which is called M-S gate. And then, by decomposing single-qutrit unitary gate in symmetric ternary quantum system, the universal gates are given. It means that any unitary operation on
n qutrits can be accurately implemented by single-qutrit symmetric ternary quantum gates and two-qutrit symmetric ternary M-S gates. By extending the QFT to the symmetric ternary quantum system, the paper successfully use some symmetric ternary quantum gates to construct the circuit which can realize symmetric ternary quantum Fourier transform (STQFT). Finally, the circuit of adder in symmetric ternary quantum system are designed based on the STQFT and the universal quantum gates.