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:{lw19277w, dd50506n, pm07433n, sn07217n, gsamuels, aleider, ctappert}@pace.edu

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]. Consider the electron in a hydrogen atom. It can be in its 1 + 1 1  0  ground state (i.e. an s orbital) or in an excited state. If this Φ = √ (|00i + |11i) = √   were a classical system, we could store a bit of information 2 2  0  in the state of the electron: ground = 0, excited = 1. The 1 qubits are usually thought to be spatially separated. The greater   the distance apart each are, the more each exhibit perfect 1 correlation even though there is no way to tell which state − 1  0  Φ = √ (|00i − |11i) =   2  0  Thanks to the IBM Faculty Award that made this research possible. −1 Once we created the virtual environment, installing the   0 Qiskit libraries with the Python package management system + 1  1  (PIP) followed. We now had our programming environment Ψ = √ (|01i + |10i) =   2  1  setup with the necessary libraries installed. 0 The next step was to clone the open source code project   from Github, a code sourcing repository (repo). This code 0 is a combination of efforts from IBM and other research − 1  1  Ψ = √ (|01i − |10i) =   institutions. Microsoft Windows systems required the instal- 2  −1  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 |1i to√ (|0i − |1i) was built through laser fields. The qubit-qubit interactions 2 decay with distance and entanglement is generated early , which means that a measurement will have equal prob- between neighboring qubits. abilities to become 1 or 0 (i.e. creates a superposition).√ According to Hu, Lamata, Sanz et all [12], the search for the It represents a rotation of π about the (xˆ + zˆ)/ 2. n-variable Boolean functions fulfilling global cryptographic Equivalently, it is the combination of two rotations, π constraints is computationally hard due to the sheer exponen- about the X-axis tial size O(22n) of the space. So a codification of the relevant 1  1 1  constraints in the ground state of an Ising Hamiltonian, has H = √ been introduced which provided a quantum speedup. Adding 2 1 −1 to this, small n cases in a D-Wave machine has been set as a • Controlled NOT gate (CNOT)- The CNOT is a quantum point of reference which demonstrates its capacity of devising gate that is an essential component in the construction bent functions, the most relevant set of cryptographic Boolean of a quantum computer. The CNOT gate operates on a functions. Belte, Hacker [13] suggests that neutral atoms quantum register consisting of 2 qubits. The CNOT gate trapped inside an optical cavity provide an ideal platform flips the second qubit (the target qubit) if and only if the for the implementation of quantum networks. A quantum first qubit (the control qubit) is |1i . The CNOT gate can network has nodes containing multiple atomic qubits, which be represented by the matrix (permutation matrix form): are essential for the construction of a quantum repeater as  1 0 0 0  they allow for entanglement swapping and thus the generation  0 1 0 0  of entanglement between qubits over long distances. The  0 0 0 1  realization of such a multi-qubit network node containing two   0 0 1 0 Rubidium (atomic number- 87) atoms in an optical cavity. Local entanglement between the two atoms is created with • The IBM Q Experience is a cloud based quantum com- an experimental technique called quantum state carving. puting platform, that gives users in the general public The entanglement properties of the grid states form a access to a set of IBMs prototype quantum processors discrete set of mixed quantum states. These states have been via the Cloud. There are two variations of the IBM Q 5 graphically represented by Lockhart, Ghne and Severini [14]. available - More precisely the entire entanglement properties have been – ibmqx2 in Yorktown defined, and evaluation methods for the entanglement criteria – ibmqx4 in Tenerife for the grid states have been computed graphically. The experiment shows that entanglement theory for grid states; although being a discrete set, grid states have a complexity similar to that for general states. Krenn, Gu, and Zeilinger [15] demonstrate a link between high-dimensional multipartite quantum states and graph theory. The paths of photons are identified in such a fashion that the photon-source information is never created. In fact, each specific setup corresponds to an undirected graph on its own, which in turn points to a separate experimental setup. In order to correlate graph theory and quantum states, Krenn et all have rephrased theorems from graph theory like Hall’s marriage problem in the language of pair creation. The issue, however is in calculating the final quantum state which is in the P-complete complexity class. Hence the evaluation could not be done efficiently. Fig. 2. Quantum Gates IV. METHODOLOGY V. PRELIMINARY RESULTS Certain terms need to explained before delving into the implementation of the bell states- Our goal was to look at each gate through IBM Q and become familiar with the graphical user interface(GUI). While • Hadamard (H) gate- The Hadamard gate acts on a single testing in this environment we would need to know exactly qubit. It is one qubit version of the quantum fourier how to create each Bell state using the quantum gates and also transform. It maps the basis state how to measure what was being produced after simulation. 1 First we explored the first Bell state which is the most |0i = √ (|0i + |1i) 2 documented bell state out of the four. Creating the first Bell state involved using two quantum gates and placing each on gate (phase-flip) from q[0] allowing the Bell state to remain qubits, q[0] and q[1]. The first gate being an CNOT, which positive(Ψ+). was placed on q[0] with a measure gate. During the process it was found that if each line must have a measuring gate along with the conventional flipping gates in order to see the results of the test. Previously we only measured one line with the CNOT gate and received graphs and other data based on one measurement. Placing both an Hadamard gate and an measurement gate to the next line q[1], we were able to see the amount of bit flips (0 to 1) would take place in each gate, the blue line from q[0] to q[1] represents entanglement of the two qubits as shown in figure 3. Next was to explore Fig. 5. Results of the third Bell state(Ψ+)

Testing the final Bell state does follow the rule in which we discover while on the third Bell state, that being similar to it’s predecessor. Entering this state required the addition of the Z gate(phase-flip),and also leaving the X gate for the bit-flipping which took place in the third Bell state. The last Bell state helped us as we were able to compare what makes each Bell state so different and how these states are achieved through bit manipulation using the bit-flipping gate of X and the phase flipping gate of Z. Fig. 3. Results of the first Bell state(Φ+) The more interesting part of our findings began when testing different qubits by placing the quantum gates on q[2], q[3] the second Bell state through simulation, this would be done and so on then, analyzing the results. There is no one or two similar to the first Bell state by placing a Hadamard gate on methods of entering the 4 Bell states, there are several other q[1]; however,q[0] does not receive a gate. Instead a CNOT implementations not explained in this section. This was great gate is place on the same line as the Hadamard gate, and new for us to see, prior to these results the thought was the contrary. gate would then be needed. The Pauli Z gate as shown in Quantum Computing as we explore it continues to deepen our figure 2, is a gate that has a property of X to -X , Z to Z. understanding of the growing technology as well as widen the The Z gate is known as the phase-flip gate and will do such possibilities of to what it may be used for in years to come. while being place on q[1] to flip the outcome from CNOT and the Hadamard gate creating (Φ-). The results are as shown in figure 4.

Fig. 6. Results of the fouth Bell state(Ψ-)

Fig. 4. Results of the second Bell state(Φ-)

VI.CONCLUSION Simulations continue with the third Bell state, at this point it is a lucid notion that each Bell state will be the same with q[1] Using the resources given to the public by IBM, we were having both the CNOT and the Hadamard gates respectively. able to test the 4 Bell states by actually creating them Similar to the gate before it, have some newly introduced gate and simulating each, producing graphed results of the qubit to flip the bits on each qubit to receive the outcome needed to flipping. Reviewing what has already been achieved through enter each Bell state. As we seen in the last example the Z gate the first Bell state will helped us look past and explore the was introduced at q[1] to flip the result of the Hadamard gate; other three Bell states; writing code for quantum machines for the third Bell state another flip gate is needed, the Pauli has only helped in learning about each Bell state and it’s X gate. The X gate is known as the bit-flip and will flip bits polarization. Through our research it is our goal to learn more based on the outcome of the CNOT gate and the Hadamard about what we can use this technology for not being limited gate, while being entangled.The important difference to notice to one Bell state. We hope that our research educates and from the second Bell state to the third is the removal of the Z intrigues many to help grow this very new technology further. REFERENCES [1] X-Team, Quantum Computation Python JavaScript, 2018. Available at https://x-team.com/blog/quantum-computation-python-javascript/. [2] M. Maheswaran, QP-difference, 2018. https://www.quora.com/What-is- the-difference-between-a-regular-computer-and-a-quantum-computer. [3] K. BONSOR, Qbits, 2018. https://computer.howstuffworks.com/quantum- computer1.htm. [4] V. T, youtube-video, 2018. https://www.youtube.com/watch?v=lLbMvHRQ6IY. [5] P. H. Vasconcellos, python-ide, 2018. https://www.datacamp.com/community/tutorials/data-science-python- ide. [6] Anaconda, ana-ide, 2018. https://www.anaconda.com/what-is- anaconda/. [7] IBM, ibm-ide, 2018. https://www.research.ibm.com/ibm-q/. [8] IBM, ibm-quantum, 2018. https://quantumexperience.ng.bluemix.net/qx/experience. [9] V. Thanasekaran, Quantum Spin, Entanglement and Teleportation, 2014. Available at https://youtu.be/-Tw-GzAWvNI. [10] Y.-Y. Liao, “Bell states and entanglement of two-dimensional polar molecules in electric fields,” The European Physical Journal D, vol. 71, p. 277, Nov 2017. [11] N. Friis, O. Marty, C. Maier, C. Hempel, M. Holzapfel,¨ P. Jurcevic, M. B. Plenio, M. Huber, C. Roos, R. Blatt, et al., “Observation of entangled states of a fully controlled 20-qubit system,” Physical Review X, vol. 8, no. 2, p. 021012, 2018. [12] F. Hu, L. Lamata, M. Sanz, X. Chen, X. Chen, C. Wang, and E. Solano, “Quantum computing cryptography: Unveiling cryptographic boolean functions with quantum annealing,” arXiv preprint arXiv:1806.08706, 2018. [13] S. Welte, B. Hacker, S. Daiss, L. Li, S. Ritter, and G. Rempe, “Quantum state carving of two atomic qubits in an optical cavity,” in 2017 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC), pp. 1–1, 2017. [14] J. Lockhart, O. Guhne,¨ and S. Severini, “Entanglement properties of quantum grid states,” Phys. Rev. A, vol. 97, p. 062340, Jun 2018. [15] M. Krenn, X. Gu, and A. Zeilinger, “Quantum experiments and graphs: Multiparty states as coherent superpositions of perfect matchings,” Phys. Rev. Lett., vol. 119, p. 240403, Dec 2017.