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Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS)

Generalization of Common Gates for Quantum Computation

Anca Ralescu, Logan Mayfield

Published Dec 10, 2005• 1 citation
Publisher

Abstract

As problems in computer science become increasingly more computationally intense, researchers have begun to examine other methods of computation. One of the largest new models of computation is the quantum computer. Based on the concepts of quantum mechanics, quantum computers process bits of quantum information, or qubits, using quantum gates. So far some impressive results have been obtained using quantum computers on current problems (e.g. search problems). Many quantum algorithms make use of the Hadamard gate, and a few use a more general form of that gate. In this paper we present a generalized quantum gate that can be shown to encapsulate the standard gates used in quantum algorithms and quantum information processing: Pauli X, Y, and Z, Hadamard, T phase shift, S phase shift, and identity. This generalized gate may also lead to new, interesting, and useful gates for quantum computation.

DOI
10.1109/nafips.2005.1548607

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As problems in computer science become increasingly more computationally intense, researchers have begun to examine other methods of computation. One of the largest new models of computation is the quantum computer. Based on the concepts of quantum mechanics, quantum computers process bits of quantum information, or qubits, using quantum gates. So far some impressive results have been obtained using quantum computers on current problems (e.g. search problems). Many quantum algorithms make use of the Hadamard gate, and a few use a more general form of that gate. In this paper we present a generalized quantum gate that can be shown to encapsulate the standard gates used in quantum algorithms and quantum information processing: Pauli X, Y, and Z, Hadamard, T phase shift, S phase shift, and identity. This generalized gate may also lead to new, interesting, and useful gates for quantum computation.
Although classical computing has brought great convenience to mankind’s lives, classical computing is facing some problems. Combining computer science, information theory, and quantum mechanics, quantum information has been developed in many fields. Quantum gates can operate quantum bits, the carrier of quantum information, while quantum circuits are composed of quantum gates and quantum bits, showing the logic of operations. This study summarizes the fundamental concepts of quantum informatics: qubits, quantum circuits, and quantum gates where it is proved that single-bit quantum gates and CNOT gates are universal quantum gates. In addition, this research also demonstrates the current progress of quantum gates in the examination of fidelity of quantum gates, improvement of fidelity of quantum gates, and teleportation of quantum gates. Meanwhile, according to the current development status, the limits and prospects of quantum information are discussed. These results shed light on helping readers understand the basic concepts of quantum information, thus understand the future applications and developments in quantum information.
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Quantum mechanics explains the behavior of matter and its movement with energy in the scale of atoms and subatomic particles. In quantum circuits there are many gates such as Hadamard Gate, Pauli Gates, many more gates for half turns, quarter turns, eighth turns, sixteenth turns and so on, rest for spinning, parametrized etc. Linear operators in general and quantum mechanics can be represented in the form of vectors and in turn can be viewed as matrices format because linear operators are totally equivalent with the matrices view point. This paper discloses creation of various quantum matrices using Quantum bits. Square, Identity and Transposition of matrices are performed considering whole process in entanglement. Angle, phase, coordinates, magnitude, complex numbers and amplitude has been noted and documented in this paper for further research.
Quantum computing applies quantum physics ideas to problems that traditional computers cannot address. The qubit, or quantum equivalent of the classical bit, is fundamental to this paradigm shift. Unlike its classical equivalent, a qubit can exist in a superposition of states, representing both 0 and 1. This superposition is defined not only by magnitudes, but also by important phase variables. These phases have a significant impact on qubit behavior and quantum computation outputs. This work conducts a thorough investigation of qubit phases, exploring their tremendous impact on the efficacy and capabilities of quantum algorithms. We investigate how constructive and destructive interference caused by phase interactions provides the foundation of quantum algorithms. Furthermore, we look into the intricate role of phases in establishing and managing entanglement, a unique quantum phenomenon that allows tremendous interactions between qubits. Our investigation includes the effects of numerous quantum operations on qubit phases. We present a thorough mathematical framework for describing how typical quantum gates, such as Hadamard, Pauli, and phase-shift gates, change the phase and thus the overall state of a qubit. We show these concepts through actual implementations of the Qiskit library. Finally, we discuss the intrinsic difficulty of managing and monitoring qubit phases, particularly the negative impacts of decoherence, which disrupts the delicate phase relationships. We describe tactics for mitigating these obstacles and investigate techniques for extracting phase information indirectly, such as quantum state tomography and interferometry. This comprehensive study seeks to provide a better understanding of the critical role phases play in quantum computing, paving the way for advances in algorithm design, quantum control, and the development of fault-tolerant quantum computers.
To this point only quantum gates derived from invertible digital counterparts or general invertible Boolean functions have been considered. Here we introduce a number of quantum gates without digital equivalents. In particular, we consider the Pauli gates, the https://www.w3.org/1998/Math/MathML" display="inline"> n o t https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math21_1.tif "/> gate, the phase-shift gate and the Hadamard gate, some of which occur in quantum algorithms introduced later. These gates impart additional power to quantum computation.