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Quantum Information and Computation

Symmetric ternary quantum Fourier transform and its application

Hao Dong, Dayong Lu, Xiaoyun Sun

Published May 30, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.26421/qic22.9-10-2

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The research of ternary quantum system has gradually come into the attention of scholars in recent years. In 201820182018, 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 201920192019, 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 nnn 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 201820182018, 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 201920192019, 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 nnn 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.
Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of O(n)O(n)O(n) and uses only 111 ancilla, making it more efficient in both, circuit depth and width than previous constructions. Our ternary carry lookahead circuit has a circuit depth of only O(log⁡ n)O(\log\,n)O(logn), while using with O(n)O(n)O(n) ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate C(X):∣i,j⟩↦∣i,j+δi,2mod  3⟩C(X): |i,j\rangle \mapsto |i, j + δ_{i,2} \mod 3\rangleC(X):∣i,j⟩↦∣i,j+δi,2​mod3⟩ or from the two-qutrit Clifford gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 π\, i/9},1,e^{2 π\, i/9})$. The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomial over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.
Qutrit (or ternary) structures arise naturally in many quantum systems, notably in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders. The main distinction from the binary adders is a richer ternary carry which leads potentially to higher resource counts in universal ternary bases. Our ternary ripple adder circuit has a circuit depth of O(n) and uses only 1 ancilla, making it more efficient in both, circuit depth and width, when compared with previous constructions. Our ternary carry lookahead circuit has a circuit depth of only O(log n), while using O(n) ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate C(X) : |i, ji 7→ |i, j + δi,2 mod 3i or from the two-qutrit Clifford gates and the non-Clifford gate P9 = diag(e −2π i/9 , 1, e 2π i/9 ). The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomials over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.
Synthesis of ternary quantum circuits involves basic ternary gates and logic operations in the ternary quantum domain. Works that define ternary algebra and their applications for ternary quantum logic realization, are very few. In this paper, we express a ternary logic function in terms of projection operations including a new one. We demonstrate how to realize a few new multi-qutrit ternary gates in terms of generalized ternary gates and projection operations. We then employ our synthesis method to design ternary adder circuits which have better cost than that obtained by earlier method.
Since the difficulty in preparing the equal superposition state of amplitude is 1/√ N , we construct a quantile transform of quantum Fourier transform (QFT) over Z N based on the elementary transforms, such as Hadamard transform and Pauli transform. The QFT over Z N can then be realized by the quantile transform, and used to further design its quantum circuit and analyze the requirements for the quantum register and quantum gates. However, the transform needs considerable quantum computational resources and it is difficult to construct a high-dimensional quantum register. Hence, we investigate the design of t -bit quantile transform, and introduce the definition of t -bit semiclassical QFT over Z N . According to probability amplitude, we prove that the transform can be used to realize QFT over Z N and further design its quantum circuit. For this transform, the requirements for the quantum register, the one-qubit gate, and two-qubit gate reduce obviously when compared with those for the QFT over Z N .
The designs using ternary logic exploit its logarithmic reduction in the number of qudits compared with the binary circuits. In this paper, we propose quantum ternary multiplication gate. We term it as QTMG. Then we present the symbol and the realization of QTMG. Researchers will be able to use this gate as well as its symbol and realizations in their future studies. We also present a new realization of ternary Toffoli gate in specific state. Moreover, in this paper, we propose 1-qutrit multiplier circuit. The symbol and the realization of the proposed 1-qutrit multiplier circuit are also provided. Afterward, we proposed ternary partial products generation circuit (TPPG) and summation network circuit in order to design quantum ternary 2-qutrit multiplier circuit. Overall, the proposed design of QTMG in this paper is suggested for the first time. In addition, the proposed realization of ternary Toffoli gate, TPPG, summation network and 2-qutrit multiplier circuits are compared with the existing designs and the improvements are reported. The proposed gate and circuits are realized using macro-level ternary gates which are built on the top of the ion-trap realizable 1-qutrit gates and 2-qutrit Muthukrishnan–Stroud gates.
Ternary quantum circuits play a significant role in future quantum computing technology because they have many advantages over binary quantum circuits. Subtraction is considered as being one of the key arithmetic operations; hence, subtractors are very essential for the construction of various computational units of quantum computers and other complex computational systems. In this paper, we have proposed the realization of a quantum reversible ternary half-subtractor circuit using a generalized ternary gate, a ternary Toffoli gate, and a ternary C2NOT gate. Based on the realization of the ternary half-subtractor, we proposed the realization of a ternary full-subtractor.