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Journal of Applied Mathematics and Physics

Three-Qutrit Topological SWAP Logic Gate for ISK (I = 1, S = 1, K = 1) Spin System

Özge Şahin, Azmi Gençten

Published Jan 1, 2017• 2 citations
Publisher

Abstract

Three Zeeman levels of spin-1 electron or nucleus are called as qutrits in quantum computation. Then, ISK (I = 1, S = 1, K = 1) spin system can be represented as three-qutrit states. Quantum circuits and algorithms consist of quantum logic gates. By using SWAP logic gate, two quantum states are exchanged. Topological quantum computing can be applied in quantum error correction. In this study, first, Yang-Baxter equation is modified for ISK (I = 1, S = 1, K = 1) spin system. Then three-qutrit topological SWAP logic gate is obtained. This SWAP logic gate is applied for three-qutrit states of ISK (I = 1, S = 1, K = 1) spin system. Three-qutrit SWAP logic gate is also applied to the product operators of ISK (I = 1, S = 1, K = 1) spin system. For these two applications, expected exchange results are found.

DOI
10.4236/jamp.2017.512189

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Three Zeeman levels of spin-1 electron or nucleus are called as qutrits in quantum computation. Then, ISK (I = 1, S = 1, K = 1) spin system can be represented as three-qutrit states. Quantum circuits and algorithms consist of quantum logic gates. By using SWAP logic gate, two quantum states are exchanged. Topological quantum computing can be applied in quantum error correction. In this study, first, Yang-Baxter equation is modified for ISK (I = 1, S = 1, K = 1) spin system. Then three-qutrit topological SWAP logic gate is obtained. This SWAP logic gate is applied for three-qutrit states of ISK (I = 1, S = 1, K = 1) spin system. Three-qutrit SWAP logic gate is also applied to the product operators of ISK (I = 1, S = 1, K = 1) spin system. For these two applications, expected exchange results are found.
Quantum information processing has become a promising research area. In quantum information processing qubits and qutrits will be used instead of bits. A qubit is a two level quantum system and a qutrit is a three level quantum system. Zeeman levels of spin-1 electron or nucleus can be considered as qutrit states. In this study, first, two and three qutrit SWAP logic gates are developed. For two qutrit states, there exist different types of CNOT quantum logic gates. The relationships between these different types of CNOT gates are obtained in this study.
In NMR quantum computing, spin states of spin-1/2 nuclei are called qubits. Quantum logic gates are represented by unitary matrices. As a universal gate, controlled-NOT (CNOT) is a two-qubit gate. For the IS (I = 1/2 and S = 1/2) spin system, two-qubit CNOT gate is represented by a 4 × 4 matrix. SWAP logic gate, which exchanges two quantum states, is constructed by CNOT gates. In this study, first, four-qubit CNOT gates are constructed for the IS (I = 3/2, S = 3/2) spin system. Then, by using these CNOT gates, a four-qubit SWAP logic gate is found. As an application and verification, an obtained SWAP logic gate is applied to the matrix representation of product operators for the IS (I = 3/2, S = 3/2) spin system. SWAP logic gate can also be presented by an NMR pulse sequence. By using the product operator theory, the pulse sequence of the SWAP logic gate is applied to product operators of the IS (I = 3/2, S = 3/2) spin system. The expected exchange results are obtained for both the matrix representation and the pulse sequence of SWAP logic gate.
Ternary quantum systems are being studied because they provide more computational state space per unit of information, known as qutrit. A qutrit has three basis states, thus a qubit may be considered as a special case of a qutrit where the coefficient of one of the basis states is zero. Hence both $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional and $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors can occur on qutrits. In this paper, we (i) explore the possible $(2\ifmmode\times\else\texttimes\fi{}2)$-dimensional as well as $(3\ifmmode\times\else\texttimes\fi{}3)$-dimensional Pauli errors in qutrits and show that any pairwise bit swap error can be expressed as a linear combination of shift errors and phase errors, (ii) propose a special type of error called a quantum superposition error and show its equivalence to arbitrary rotation, (iii) formulate a nine-qutrit code which can correct a single error in a qutrit, and (iv) provide its stabilizer and circuit realization.
One of the current trends in quantum computing is the application of algebraic topological methods in the design of new algorithms and quantum computers, giving rise to topological quantum computing. One of the tools used in it is the Yang-Baxter equation whose solutions are interpreted as universal quantum gates. Lately, more general Yang-Baxter equations have been investigated, making progress as two-spectral equations and Yang-Baxter systems. This paper intends to apply these new findings to the field of topological quantum computation, more specifically, the proposition of the two-spectral Yang-Baxter operators as universal quantum gates for 2 qubits and 2 qutrits systems, obtaining 4x4 and 9x9 matrices respectively, and further elaboration of the corresponding Hamiltonian by the use of computer algebra software Mathematica® and its Qucalc package. In addition, possible physical systems to which the Yang-Baxter operators obtained can be applied are considered. In the present work it is demonstrated the utility of the Yang-Baxter equation to generate universal quantum gates and the power of computer algebra to design them; it is expected that these mathematical studies contribute to the further development of quantum computers
As classical computers struggle to keep up with Moore's law, quantum computing may represent a big step in technology and yield significant improvement over classical computing for many important tasks. Building a quantum computer, however, is a daunting challenge since it requires good control but also good isolation from the environment to minimize decoherence. It is therefore important to realize quantum gates efficiently, using as few operations as possible, to reduce the amount of required control and operation time and thus improve the quantum state coherence. Here we propose a superconducting circuit for implementing a tunable spin chain consisting of a qutrit (three-level system analogous to spin-1) coupled to two qubits (spin-1/2). Our system can efficiently accomplish various quantum information tasks, including generation of entanglement of the two qubits and conditional three-qubit quantum gates, such as the Toffoli and Fredkin gates, which are universal for reversible classical computations. Furthermore, our system realizes a conditional geometric gate which may be used for holonomic (non-adiabatic) quantum computing. The efficiency, robustness and universality of our circuit makes it a promising candidate to serve as a building block for larger spin networks capable of performing involved quantum computational tasks.
Abstract In recent years, quantum computing has emerged as a leading approach for accelerating complex computations, leveraging quantum parallelism and entanglement to surpass classical limits. Although numerous quantum circuits based on reversible gates have been proposed, optimizing circuit efficiency remains a critical challenge, particularly in the noisy intermediate-scale quantum era, where error rates and coherence times constrain performance. Ternary quantum computing, which operates on three-level quantum systems (qutrits), offers a promising alternative by reducing circuit complexity and resource overhead. In particular, balanced ternary logic presents advantages over conventional ternary logic by enhancing resource efficiency and potentially mitigating quantum gate error rates. This paper focuses on two key objectives: first, the synthesis of a reversible 1-qutrit comparator circuit using balanced ternary logic, and second, the extension to a generalized n-qutrit comparator. The proposed 1-qutrit comparator achieves notable improvements over existing unbalanced designs, demonstrating a 65% reduction in quantum cost, 50% fewer constant inputs, and a 33% decrease in garbage outputs. These results highlight the potential of balanced ternary logic to enhance the efficiency of quantum circuits, making it a valuable contribution to the development of scalable and error-resilient quantum hardware.