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.
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.
Abstract Quantum computing leverages the power of superposition, entanglement, and interference to surpass classical systems in certain computational tasks. The first studies revolved around two-state quantum systems (qubits), which are a superposition of the |0⟩ and |1⟩ states. These qubits are physically realized using subatomic particles (e.g., electrons or photons) with the two states corresponding to different energy levels or polarization directions. However, some of these particles can exist in more than two discrete states. For example, an electron may occupy several energy levels, or a photon may have more than two polarization states. As a result, there has been an interest in qudits, quantum systems with d different states. A fascinating subcase of this is a qutrit, a three-level quantum system with states |0⟩, |1⟩, and |2⟩ as its basis. It is understood that qutrits with their larger state space can improve the storage and processing of information, and Qutrit-based quantum computing can be realized by defining suitable quantum gates to concatenate the three-level systems. Researchers have proposed qutrit gates, some of which are direct generalizations of qubit gates, such as the Pauli-X gate, and others are entirely new gate constructs. This paper summarizes recent advances in qutrit-based quantum computing, where the definitions of various qutrit gates are presented, and their significance in the development of qutrit-based quantum computing, particularly in implementing Deutsch and Shor’s algorithms, is explored.
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.
In quantum information processing, spin-3/2 electron or nuclear spin states are known as two-qubit states. For SI (S = 3/2, I = 1/2) spin system, there are eight three-qubit states. In this study, first, three-qubit CNOT logic gates are obtained. Then three-qubit entangled states are obtained by using the matrix representation of Hadamard and three-qubit CNOT logic gates. By considering single 31P@C60 molecule as SI (S = 3/2, I = 1/2) spin system, three-qubit entangled states are also obtained using the magnetic resonance pulse sequences of Hadamard and CNOT logic gates.
Quantum entanglement is essential for some applications of quantum information processing such as quantum cryptography, quantum teleportation and superdence coding. A qubit is a two level quantum system and four two-qubit entangled states called Bell states can be easily obtained for two-qubit states. A qutrit is a three level quantum system and Zeeman levels of spin-1 electron or nucleus can be referred as qutrit. For SI (S=1, I=1) spin system there exist nine two-qutrit states. So nine two-qutrit entangled states can be obtained by using the Hadamard and CNOT logic gates. In this study by considering N+@C60 molecule as SI (S=1, I=1) spin system, two-qutrit entangled states are also obtained by using the magnetic resonance selective pulse sequences of Hadamard and CNOT logic gates. Then it is shown that these entangled states can be transformed into each other by the suggested transformation operators.
In this paper, the synthesis and implementation of three-quhit SWAP gate is discussed. The three-quhit SWAP gate can be decomposed into product of 2 two-qubit SWAP gates, and it can be realized by 6 CNOT gates. Research illustrated that although the result is very simple, the current methods of matrix decomposition for multi-qubit gate can not get that. Then the implementation of three-qubit SWAP gate in the three spin system with Ising interaction is investigated and the sequence of control pulse and drift process to implement the gate is given. It needs 23 control pulses and 12 drift processes. Since the interaction can not be switched on and off at will, the realization of three-qubit SWAP gate in specific quantum system also can not simply come down to 2 two-qubit SWAP gates.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.