American University in Cairo AUC Knowledge Fountain Capstone and Graduation Projects Student Research Winter 12-30-2020 Quantum Simulation of the Schrodinger Equation using IBM's Quantum Computers Mohamed Abouelela [email protected] Follow this and additional works at: https://fount.aucegypt.edu/capstone Part of the Quantum Physics Commons Recommended Citation Abouelela, Mohamed, "Quantum Simulation of the Schrodinger Equation using IBM's Quantum Computers" (2020). Capstone and Graduation Projects. 20. https://fount.aucegypt.edu/capstone/20 This Dissertation/Thesis is brought to you for free and open access by the Student Research at AUC Knowledge Fountain. It has been accepted for inclusion in Capstone and Graduation Projects by an authorized administrator of AUC Knowledge Fountain. For more information, please contact [email protected]. Quantum Simulation of the Schr¨odingerEquation using IBM's Quantum Computers by Mohamed Abouelela A Thesis Presented for the Bachelor of Science in Physics Thesis Advisor: Dr. Tarek El Sayed Department of Physics American University in Cairo Egypt, December 2020 Contents List of Figures 3 List of Tables 3 Abstract 5 1 Introduction 6 2 Quantum Computation 7 2.1 Qubits . 7 2.2 Quantum Gates and Quantum Logic . 8 2.3 Bernstein-Vazirani Algorithm . 11 2.3.1 The Algorithm . 11 2.3.2 Running on QASM and IBMq . 13 3 Quantum Simulation 15 3.1 Initialisation . 15 3.2 Time-Dependant Schr¨odingerEquation . 15 3.2.1 Quantum Fourier Transform . 16 3.2.2 Momentum Operator . 19 3.2.3 Algorithm Overview . 21 4 Results 22 4.1 Infinite Square Well . 22 4.2 Free Particle . 24 4.3 Particle in a Step Potential . 25 4.4 Quantum Tunneling . 27 4.4.1 Barrier Through the jx111xxi Bases . 29 4.4.2 Barrier Through the jx11xxxi Bases . 30 5 Running 4-qubit Simulations on ibm vigo 31 5.1 Infinite Square Well . 31 5.1.1 QASM . 31 5.1.2 ibmq vigo . 31 5.2 Free Particle . 32 5.2.1 QASM . 32 5.2.2 ibmq vigo . 32 5.3 Particle in a Step Potential . 33 5.3.1 QASM . 33 5.3.2 ibmq vigo . 33 5.4 Quantum Tunneling . 34 5.4.1 QASM . 34 5.4.2 ibmq vigo . 34 6 Discussion & Conclusion 35 References 36 Appendix 1 38 List of Figures 1 The state of a qubit can be represented in the image of a Bloch Sphere. This can be easily θ iφ θ interpreted by the more general form of the wavefunction: j i = cos 2 j0i + e sin 2 j1i, where θ is the polar angle that determines the probabilities of the measurement, and φ is the az- imuthal angle known as the phase, which has no effect on the probabilities of the measurement. Figure courtesy of Nielsen. 1 ................................ 7 2 A quantum circuit schematic of the CNOT Gate. The control is represented by a dot, while the target is box with the '+' ................................ 10 3 The quantum circuit of Bernstein-Vazirani's Algorithm. 11 4 A quantum circuit representation of the unitary property of the Hadamard gate. 12 5 The oracle acting on an input jxi ............................. 13 6 Histogram produced by QASM and IBM's quantum computer [20 pt] . 14 Quantum circuit implementation of the QFT, where φ = 2πi . Figure 1 7 N 2N . 18 8 Quantum Fourier Transform followed by its inverse. Notice how φ is negative in the inverse QFT. Figure courtesy of Nielsen and Chang 1 ........................ 19 1 9 Quantum circuit simulating the Hamiltonian H = Z1 ⊗ Z2 ⊗ Z3 as shown by Nielsen . 20 10 The momentum operator for a 3-qubit system. ....................... 20 11 Quantum circuit representation of the time evolution operator for a single time step. 21 12 A visualisation of the 6-qubit initialisation of the half sine with the potential at half of the simulation space ...................................... 23 13 Results from QASM: Left-hand side showing the initial state, right-hand side showing state after 15 iterations ..................................... 23 14 The Gaussian curve that will be initialised onto the qubits, with µ = 0 and σ = 0:4 . 24 15 Three different states of the free particle as measured by QASM with a ∆t = 0:3 . 24 16 Three different states of the particle in a step potential of V = 2 simulation as measured by QASM with ∆t = 0:1 ................................... 25 17 Three different states of the particle in a step potential of V = 3 simulation as measured by QASM with ∆t = 0:1 ................................... 25 18 Three different states of the particle in a step potential of V = 5 simulation as measured by QASM with ∆t = 0:1 ................................... 26 19 A potential barrier with the incident particle energy below the potential barrier height 2 [10 pt] 27 20 An incident sinusoidal wavefunciton quantum tunneling through a barrier. The wavefunciton exponentially decays within the potential barrier. 2 [10 pt] . 28 21 Three different states of the quantum particle tunneling through a potential barrier of height V = 1:75. ........................................ 29 22 Three different states of the quantum particle tunneling through a potential barrier of height V = 1:75. ........................................ 30 23 Two states of the eigenstate in an infinite well simulation as measured by QASM with ∆t = 0:1 31 24 Two states of the eigenstate in an infinite well simulation as measured by ibmq vigo with ∆t = 0:1 ......................................... 31 25 Three states of the free particle simulation as measured by QASM with ∆t = 0:1 . 32 26 Three states of the free particle simulation as measured by ibmq vigo with ∆t = 0:1 . 32 27 Three states of the step potential simulation at V = 2 as measured by the QASM simulator with ∆t = 0:1 ....................................... 33 28 Three states of the step potential simulation with V = 2 as measured by ibmq vigo with ∆t = 0:1 33 2 29 Three states of the quantum tunneling simulation with V = 1:75 as measured by the QASM simulator with ∆t = 0:1 .................................. 34 30 Three states of the quantum tunneling simulation with V = 1:75 as measured by ibmq vigo with ∆t = 0:1 ......................................... 34 31 A 6-qubit implementation of the Quantum Fourier Transform. Note that the q0 is the ancillary qubit. ........................................... 38 32 A 6-qubit implementation of the Inverse Quantum Fourier Transform. Note that the q0 is the ancillary qubit. ...................................... 38 33 A 6-qubit implementation of the momentum operator with φ = π. Note that the q0 is the ancillary qubit ....................................... 39 34 The potential used for the 6-qubit simulation, to apply a potential V on the highest order qubit. 40 35 A filter for the states jx111xxi. The Toffoli gate's controls are two of the 2nd, 3rd, or 4th highest order qubits, and its target is the ancillary qubit. The controlled phase gate's control is the ancillary qubit, while the target is the remaining high order qubit from the stated three. 40 36 A filter for the states jx11xxxi. The CNOT gates' control is on the 3rd highest order qubit, and its target is the ancillary qubit. The controlled phase gate's control is the ancillary qubit, while the target is the 5th highest order qubit. ........................ 41 List of Tables 1 List of commonly used quantum gates. Row 1: Hadamard Gate - NOT Gate. Row 2: Phase Gate - Controlled Phase Gate. Row 3: Controlled NOT Gate - Controlled Controlled NOT (Toffoli) Gate ....................................... 10 3 Acknowledgements I am profoundly grateful to have had Dr. Tarek Elsayed as my thesis advisor. I would like to thank him for his patience, and for putting so much effort in teaching me something new, giving me the opportunity to research and unlock a new passion I never knew I had. I would also like to thank my colleague, Mohamed Eltohfa, for helping me throughout this period as my thesis partner. I would like to thank for teaching me to look at problems through other perspectives. I would like to thank the every professor in the physics department at AUC for having a strong impact on me as a student and continue to push me forward to reach my goals: Dr. Mohamed Swillam, Dr. Mohammad Alfiky, Dr. Ashraf Alfiky, Dr. Ahmed Hamed, Dr. Nageh Allam, Dr. Amr Shaarawi, Dr. Salah Elsheikh, Dr. Hosny Omar, Dr. Karim Addas, Dr. Mo- hamed Orabi, and Dr. Ezzeldin Soliman. I thank Miss Lobna Abdelrehim and Farah Lotfy for guiding throughout my years at uni- versity. Their impact was truly great. I would like to thank my family for their continued emotional support and their belief in following my dreams and passion. Finally I would like to thank my friends that I have made along the away. Their support is truly appreciated. 4 Abstract This thesis explores the capabilities of a quantum computer to simulate quantum systems. We give an introduction to the basics of quantum computing with the Bernstein-Vazirani algorithm as a demonstration. Four quantum systems are then simulated using IBM's QASM simulator using 6 qubits: the free particle, eigenstate of an infinite-well, particle in a step potential, and quantum tunneling. Because of the high number of gates, a 6-qubit simulation will not be feasible on current quantum computers. The number of qubits was, thus, reduced to 4 qubits, and was simulated on IBM's 5 qubit quantum computers (ibmq 5 vigo). We conclude that quantum simulations on quantum computers are theoretically achievable, as shown by the QASM simulator; however, no useful information can be extracted using the real quantum computers, due to high noise and high errors. 5 1 Introduction In his 1982 paper Simulating Physics with Computers, Richard Feynman popularised the idea of a non-Turing machine that would simulate quantum systems.