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Huadong Shifan Daxue xuebao. Ziran kexue ban

Quantum Logic Transformation Based on Spin-1/2 States

Ma Lei

Published Apr 15, 2003• 0 citations
OpenAlex

Abstract

In this paper, by using the linear quantum transformation theory to spin operator, we offer an effective method to study the quantum logic gates based on spin - 1/2 states. The forms of some basic quantum logic operations are also obtained.

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In quantum mechanics,the evolution of a quantum state is a unitary evolution,the evolution process can be described by an action of a unitary operator on the quantum state and the process is time reversible.Based on this fundamental fact,Gerard t Hooft has introduced the concept of equivalence classes of quantum states and described the unitary evolutions of quantum states by unitary transformation between two equivalence classes.In the present paper,we will use the concept of equivalence classes and their transformations to construct universal quantum logic gates in the quantum information theory and give their generalized forms.In the end of this paper we will show that these quantum logic gates can be realized by some intrinsic interaction Hamiltonian of the system.
A quantum gate is the quantum analog of the classical logic gate of digital computers. It is a basic structural unit of a quantum circuit which operates on a small number of qubits to deliver a predefined output. A quantum circuit is a graphical representation of the sequence of the quantum gates used and the measurements performed in a quantum computation. Various quantum logic gates, namely, the Hadamard, T phase, NOT, Pauli-X, controlled NOT (CNOT), controlled-Z, SWAP and controlled phase gates as well as higher-level Toffoli and Fredkin gates are described. Conventions of reading quantum circuit diagrams are explained. Operation of a quantum full adder consisting of registers and gates is discussed, elaborating the role of the NOT gate, the sum calculation with CNOTgates and the carry C-out calculation with Toffoli gates. A primary tool used for performing arithmetic operations on a quantum computer is the quantum Fourier transform (QFT). The QFT adder and multiplier circuits are presented.
Abstract Quantum gates, equivalent to logic gates, modify the state vectors. Linear algebra calls these gates linear transformations. In quantum language, they are “operators.” These gates include the NOT gate, the Pauli gates, and the Hadamard gate. The operations of the gates are represented both as abstract symbols and by matrices multiplying state vectors (expressed as column vectors). Quantum computers use a series of gates called a quantum circuit. The chapter explains how to read a quantum circuit diagram and translate it into linear algebra mathematics. The mathematical requirements on the matrices allowed to be used is delineated. Gates are reversible by applying the same gate twice, but measurements are not reversible.
Much of the current understanding of logic gates for quantum computation relies on the application of one- and two-qubit gates in order to implement a universal set of logic gates, as well as other gates. Such ideas stem from the notion that the Hamiltonians available for quantum computation are one or two-body processes. However, present day NMR implementations of quantum information processing (QIP) rely on the internal Hamiltonian of a liquid state system in which there are many two-body processes that occur simultaneously. Such use of the internal Hamiltonian allows for the creation of `multiqubit' logic gates, i.e. logic gates that operate on many qubits simultaneously and are more efficient than a sequence of one- and two-qubit rotations that effect the same operation. Such larger qubit operations offer a universal set of gates (even when not all couplings within the internal Hamiltonian are readily accessible) as well as more convenient and efficient implementations of the Hadamard transform, the controlled-NOT gate, and a quantum Fourier transform that scales linearly (assuming all couplings can be turned on and off at will).