Study And Implementation of Unitary Gates in Quantum Computation Using Schrodinger Dynamics

TL;DR

This thesis investigates how to realize quantum gates through Schrodinger dynamics by perturbing physical systems with time-dependent Hamiltonians, enabling the implementation of various quantum gates for quantum computation.

Contribution

It presents a method to implement a wide class of quantum gates using time-varying Hamiltonians and explores controllability conditions for such gate realizations.

Findings

Controlled unitary gates can be realized via perturbations of harmonic oscillators.

Controllability depends on the existence of suitable time-dependent functions.

Infinite-dimensional gates are achievable through electromagnetic field modulation.

Abstract

This thesis explores the concept of realizing quantum gates using physical systems like atoms and oscillators perturbed by electric and magnetic fields. The basic idea is that if a time-independent Hamiltonian $H_0$ is perturbed by a time-varying Hamiltonian of the form $f(t)V$, where $f(t)$ is a scalar function of time and $V$ is a Hermitian operator that does not commute with $H_0$, then a large class of unitary operators can be realized via the Schrodinger evolution corresponding to the time-varying Hamiltonian $H_0+f(t)V$. This is a consequence of the Baker-Campbell-Hausdorff formula in Lie groups and Lie algebras. The thesis addresses two problems based on this idea: first, taking a Harmonic oscillator and perturbing it with a time-independent anharmonic term, and then computing $U_g=e^{-\iota T H_1}$. Then, perturbing the harmonic Hamiltonian with a linear time-dependent term, and calculating the unitary evolution corresponding to $H(t)$ at time $T$. This gate can be expressed as $U(T)=U(T,\epsilon,f)=T\{e^{-\iota\int_0^TH(t)dt}\}$.The anharmonic gate $U_g$ is replaced by a host of commonly used gates in quantum computation, such as controlled unitary gates and quantum Fourier transform gates. The control electric field is selected appropriately. The thesis also addresses the controllability issue, determining under what conditions there exists a scalar real valued function of time $f(t), 0\leq t\leq T$ such that if $|\psi_\iota\rangle$ is any initial wave function and $|\psi_f\rangle$ is any final wave function, then $U(T,f)|\psi_i\rangle=|\psi_f\rangle$. A partial solution was obtained by replacing the unitary evolution kernel by its Dyson series truncated version. In all design procedures, the gates that appear are infinite-dimensional, with an interaction between the atom and the electromagnetic field modulated by a controllable function of time.