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Tikrit Journal of Pure Science

Quantum Teleportation and Entanglement of the 5-Qubit State Over a GHZ-Like Channel

Aseel Hamad, Shareef Faiq Al-tikrity, Abdullah. H. Mohammed

Published Oct 25, 2023• 1 citation

Abstract

Quantum information processing has a wide range of uses, including the solution of complex algorithms, cryptography, dense coding, quantum computation, and quantum teleportation. Quantum entanglement, a phenomenon that is crucial to many different applications of quantum computation, is an essential resource for quantum computation, and the need for quantum circuits that generate entangled states is one of the ongoing needs for constructing quantum computers. In this study, we present a general approach to the design of quantum circuits that produce entangled states. The approach can be applied to the design of various entanglement circuits with any number of qubits (n-qubit systems), and it makes use of a set of CNOT and Hadamard gates, where the number of CNOT gates should always be (n-1) and each gate connects the two adjacent qubits. In this paper, a three-entangled teleportation scheme of a GHZ-like state (named after its inventor Greenberger-Horne-Zeilinger) through three particles as a quantum channel is presented. The probability of successful teleportation depends on the degree of entanglement of GHZ-like states. A 5-qubit quantum teleportation over a GHZ-like channel has also been used. And single-qubit gates are defined, which are Pauli gates. These gates are represented by arrays I, X, Y, and Z. The results in this paper are good and promising theoretical results in the field of quantum entanglement and quantum teleportation using the Mathematica program.

DOI: 10.25130/tjps.v28i5.1389

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It is well-known that maximally entangled states such as the Greenberger-Horne-Zeilinger (GHZ) states, with the Bell states as the simplest examples, are widely exploited in quantum information and computation. We study the application of such maximally entangled states from the viewpoint of the GHZ transform, which is a unitary basis transformation from the product states to the GHZ states. The algebraic structure of the GHZ transform is made clear and representative examples for it are verified as multi-qubit Clifford gates. In this paper, we focus on the Bell transform as the simplest example of the GHZ transform and apply it to the reformulation of quantum circuit model of teleportation and the reformulation of the fault-tolerant construction of single-qubit gates and two-qubit gates in teleportation-based quantum computation. We clearly show that there exists a natural algebraic structure called the teleportation operator in terms of the Bell transform to catch essential points of quantum teleportation, and hence we expect that there would also exist interesting algebraic structures in terms of the GHZ transform to play important roles in quantum information and computation.

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Abstract It is crucial to design quantum circuits as small as possible and as shallow as possible for quantum information processing tasks. We design quantum circuits with simplified gate-count, cost, and depth for implementing quantum teleportation among various entangled channels. Here the gate-count/cost/depth of the Greenberger-Horne-Zeilinger-based quantum teleportation is reduced from 10/6/8 to 9/4/6, the two-qubit-cluster-based quantum teleportation is reduced from 9/4/5 to 6/3/5, the three-qubit-cluster-based quantum teleportation is reduced from 12/6/7 to 8/4/5, the Brown-based quantum teleportation is reduced from 25/15/17 to 18/8/7, the Borras-based quantum teleportation is reduced from 36/25/20 to 15/8/11, and the entanglement-swapping-based quantum teleportation is reduced from 13/8/8 to 10/5/5. Note that, no feed-forward recover operation is required in the simplified schemes. Moreover, the experimentally demonstrations on IBM quantum computer indicate that our simplified and compressed schemes can be realized with good fidelity.

We proposed an N-qubit GHZ-like state quantum teleportation protocol using a two-qubit entangled quantum channel state assisted with only one classical bit. In this protocol, we use CNOT gates and a reset procedure, assuming the sender has many copies of the state that will be teleported. The CNOT gates can reduce the number of N-qubit to only one qubit, while the reset procedure may reduce the number of classical bits required to only one classical bit. Lastly, we show that the teleportation success probability depends on the state of the N-qubit state’s coefficients and is generally more than one-half.

The declaration of 2025 as the International Year of Quantum Science and Technology has galvanized numerous researchers working on innovative quantum applications. Recently, Quantum Computing and Quantum Information Science have gained significant attention as key areas of study within the broader field of Quantum Science and Technology. This is due to their potential to address complex computational challenges, their interdisciplinary applications across diverse fields such as telecommunications and biological sciences, and the intriguing theoretical framework they offer alongside practical implementation challenges. One notable application is Quantum Teleportation (QT), which involves transferring an arbitrary quantum state across spacelike distances using entangled resources. This paper explores the QT of a multi-qubit superposition state, utilizing maximally entangled Greenberger–Horne–Zeilinger (GHZ) states as both the entanglement resource and the basis for quantum measurements. The entire QT procedure is outlined with a notation system that is particularly suited for handling multi-qubit scenarios.