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Title Page: Variational Quantum Eigensolver Implemented on a 1D Quantum Harmonic oscillator, my photo, photo of SAND and photo of NIP.

Line 0: Good day. My name is John Samuel Suico and I'm here to show a brief presentation of my undergraduate thesis.

Line 1: Quantum computers offer a different paradigm of processing information using the principles of quantum mechanics.
Line 2: The smallest unit of information in this paradigm is known as a qubit, which exists in a superposition of binary states ket 0 and ket 1
Line 3: As opposed to a classical bit which can only be in either state 0 or state 1.

Line 4: This implies that $n$ qubits gives us access to $2^n$ bits of information, offering new methods to tackle computationally hard problems. (1 sec)

Line 5: We consider a particular hybrid algorithm known as the Variational Quantum Eigensolver (1 sec), which minimizes the energy of a system using the variational principle of quantum mechanics.

Line 6: We study a single particle 1D Harmonic Oscillator, which models the physics of a particle in a parabolic potential trap.

Line 7: To work on a quantum computer, we discretize the operators, representing them as matrices (1 sec), and set \hbar,\omega,m to 1. The position operator represents a lattice of points with arbitrary precision determining the size of the matrix and accuracy of the approximation.

Line 8: To measure this on a quantum computer, the Hamiltonian is written as a weighted sum of Pauli strings, or tensor products of Pauli and Identity matrices. (2 sec)

Line 9: The Pauli strings are then measured over a parameterized ansatz, which is a quantum circuit composed of rotational and entangling gates. 

Line 10: Each term in the Pauli string is measured separately over each qubit in the Ansatz, and the eigenvalues are multiplied together and averaged over many repetitions of this process.

Line 11: The summation is then carried out in a classical computer, and the parameters are optimized in each iteration so as to arrive at the ground state energy. (2 sec, middle)

Line 12: The results show that the VQE Pipeline successfully minimized the energy, except for the position basis Hamiltonian in the 4 Qubit Ansatz which struggled to converge.

Line 13: Studies like this lead towards better simulation of quantum materials in order to understand their properties. (add picture, 5 seconds))