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Bell States in Quantum Computing Bell States in Quantum Computing George Samuels, Paul Mahon, Debarshi Dutta, Sheetal Vasant Nikam, Lewis Westfall, Sukun Li, Avery Leider and Charles C. Tappert Pace University Pleasantville, NY 10570, USA Email:flw19277w, dd50506n, pm07433n, sn07217n, gsamuels, aleider, [email protected] Abstract—The rise of Quantum Computing in the industry begets a closer analysis of the topic and its methods. Quantum computers are not limited to the two states 0 and 1; rather, they encode information as quantum bits, or qubits, which can exist in superposition and can be entangled. Understanding the entanglement of qubits requires an understanding of the Bell states. The rst Bell state(Φ +), has been widely studied and is prominent in quantum computing literature. However, the other three (Φ-, Ψ+, Ψ-), are relatively untouched by researchers Fig. 1. Qubit verse Classical Bit since Quantum Computers have only recently been available for research that could test them. Years of theoretical research has not been backed by experiments due to the lack of available each qubit is in a situation that cannot be explained without technology. Now with its availability from companies like IBM we have the resources to test and work with all four Bell states, quantum mechanics. Qubits represent atoms, ions, photons further developing Quantum Computing as a strong pillar in or electrons and their respective control devices that are computing. working together to act as computer memory and a processor. Index Terms—quantum, qubit, Bell state, IBM Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times I. INTRODUCTION more powerful than today’s most powerful supercomputers [3]. Quantum Computing is a method that not many people Understanding what qubits are and how they are represented can articulate or even imagine. What is Quantum Comput- now helps us, as we look to understand Bell states. One Bell ing? What is it used for? How do developers interact with state can be defined as a maximally entangled quantum state these computers? During the course of our studies these of two qubits. The qubits are seen as a spatially separated questions and more will be addressed. Quantum Computing . Quantum entanglement can be understood by breaking it is known as the process of assembling instructions called down in simply terms. It can be imagined as if a person’s quantum programs which are capable of running on a quantum hair is tangled, the two strains of hair are now connected computer. These instructions can be written in either Python as one; however, these strains can be separated once more or JavaScript,both have a variety of Quantum Computation creating two parts. In the quantum space these strains are libraries [1]. For this project in particular we will use Python not simply taken apart once they are entangled. Due to the to write programs that will be complied on a local machine, entanglement, measurement of one qubit will assign one of two and ran on IBMs cloud quantum computer which is for public possible values to the other qubit instantly, where the values use. A vital asset in understanding Quantum Computing is are assigned depends on which Bell state the two qubits are knowing about classical computation. Classical computation in. are illustrated by classical bits, which are seen as singular Bell states can also be measured, the Bell measurement is strings of 1 or 0. The bits of 1 or 0 are representing Boolean an important concept in quantum information science: It is values of true or false. a joint quantum-mechanical measurement of two qubits that The basic entity of quantum information is a qubit or a determines which of the four Bell states the two qubits are in. quantum bit. Qubits can represent a 1, a 0 or both at once, The four Bell states are in contrast to the binary digits used in classical computing 2 3 [2]. Consider the electron in a hydrogen atom. It can be in its 1 + 1 1 6 0 7 ground state (i.e. an s orbital) or in an excited state. If this Φ = p (j00i + j11i) = p 6 7 were a classical system, we could store a bit of information 2 2 4 0 5 in the state of the electron: ground = 0, excited = 1. The 1 qubits are usually thought to be spatially separated. The greater 2 3 the distance apart each are, the more each exhibit perfect 1 correlation even though there is no way to tell which state − 1 6 0 7 Φ = p (j00i − j11i) = 6 7 2 4 0 5 Thanks to the IBM Faculty Award that made this research possible. −1 Once we created the virtual environment, installing the 2 3 0 Qiskit libraries with the Python package management system + 1 6 1 7 (PIP) followed. We now had our programming environment Ψ = p (j01i + j10i) = 6 7 2 4 1 5 setup with the necessary libraries installed. 0 The next step was to clone the open source code project 2 3 from Github, a code sourcing repository (repo). This code 0 is a combination of efforts from IBM and other research − 1 6 1 7 Ψ = p (j01i − j10i) = 6 7 institutions. Microsoft Windows systems required the instal- 2 4 −1 5 lation of the Github client prior to pulling the code from 0 GitHub. Github was already installed on our Mac systems. Each state can be achieved based on the polarization of a The team also cloned the Qiskit tutorial repo from Github single photon (spin up, spin down). The states (Φ) and (Ψ) that provided us with help files and configuration templates are represented by the polarization of the photon, either being (Qconfig.py.template). horizontal or vertical. The polarization being the same are IBM has taken the first initiative to build a quantum represented by the Greek symbol (Φ), while the polarization computer for research [7] and they have provided access to being opposites are shown as the Geek symbol (Ψ) [4]. this computer through an application programming interface These two divisions are subdivided by the rotational state. (API). For the Qiskit project, our team needed a token (code) The rotations include the positive horizontal state, negative to access this API for our code testing. To get the code, we horizontal state, positive vertical state, and negative vertical logged in to IBM Q experience site [8] and generated an API state. The public computers at IBM now allow full exploration token. This token was subsequently entered in to the Qiskit into all four Bell states, where previously researchers and configuration file for access to the IBM quantum computer via computer scientist were limited to one Bell state. the API. Finally, Jupyter Notebook was launched through the Ana- II. PROJECT REQUIREMENTS conda Navigator and loaded the Qiskit home page (in- To setup our Quantum Information System Kit (Qiskit) dex.ipynb). development environment we used both Microsoft Windows • IBM’s QISKit and Apple Mac configurations. • Q-Experience API key Our team selected Jupyter Notebook for the Integrated • Anaconda Development Environment (IDE) platform. Jupyter Notebook • Python is an cross-platform, open source application that is based on • Jupyter Notebook a server-client structure to enable an interactive programming • Slack for QISKit experience for Python [5].Jupyter Notebook is highly suited • Understanding of entangled states [9] to data science and also works well as a presentation tool. Our team used the Anaconda distribution of Python, which III. LITERATURE REVIEW includes the Jupyter Notebook IDE, for our programming A review of the literature available on quantum computing interpreter. Anaconda is a popular distribution of Python reveals that the focus of the research thus far has been on the for data science and suited this project well [6]. The main first bell state (Φ +). This can be attributed to the fact that the advantage of using the Anaconda distribution is that it comes technology which was used to analyze the Bell states has been with a package manager (conda) that seamlessly installs key made available only relatively recently. In fact, prior to 2017 packages out of the box that were needed for our Qiskit the research on Bell states was largely hypothetical. Liao [10] project. The libraries needed for this project are as shown: has investigated entanglement generated from polar molecules 1) IBM IBMQuantumExperience of two-dimensional rotation in a static electric field. The 2) Numpy concurrence is used to estimate the degree of entanglement. 3) Scipy Parallel and perpendicular application of the electric field to 4) Matplotlib the inter-molecular direction reveals two overlapping features, Anaconda also has a graphical user interface, Anaconda Navi- which corresponds to the existence of Bell-like states. The gator, that enables our team to launch applications and manage characteristics of Bell-like states and overlapping concurrences these conda packages. are kept independent of the modulation of dipolefield and To setup our Qiskit Development Environment (QDE) we dipoledipole interactions. The Bell-like states however do first downloaded the Anaconda distribution of Python at ana- not coexist in other field directions, which signifies non- conda.com. After verifying the installation, we created our overlapping concurrences. Dissimilar suppressed concurrences Qiskit virtual environment with the Anaconda tool (conda occur due to different energy structures for the two specific create). The virtual environment enabled us to isolate the field directions. Friis, Marty et all [11] has characterized Python libraries and package installations from the local entangled states of a registry of twenty individually controlled computer systems. qubits. Each qubit was encoded into the electronic state of a trapped atomic ion. Entanglement is generated during the out- and 1 of-equilibrium dynamics of an Ising-type Hamiltonian, which j1i top (j0i − j1i) was built through laser fields.
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