Symmetric ternary quantum Fourier transform and its application
Authors: Dong Hao , Lu Dayong , Sun Xiaoyun 
Published: May 2022

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.