Quantum computers have many desirable features but are physically challenging to build. They use quantum physics to solve practically motivated yet classically intractable problems, and because the experimental technology is still in its infancy, quantum mechanical devices are susceptible to errors that compromise data integrity. As a result, quantum error correction is necessary to protect important information from such undesirable influences, which inevitably increases the resource overhead to ensure a reliable quantum computation. In this thesis, we develop methods that are relevant to reducing the utilization of physical resources in a quantum computer. The core operations considered here are the so-called stabilizer operations, which have fault-tolerant constructions that are vital to achieving an error-resistant quantum computation. By applying our practices, we achieve small optimizations that have considerable value when implementing quantum algorithms in the near-term, when small quantum systems are much easier to manage than a single large quantum system. We cover two techniques to improve the efficiency of stabilizer operations applied to qubit states.

First, we introduce protocols that can probabilistically recreate an initial input qubit from the output qubit of specific quantum processes. These protocols are ideally suited for recovery purposes, and are designed with potentially many nested layers of stabilizer operations. We subsequently give a precise analysis on the effectiveness of the nested recovery. By integrating recovery at the optimal nesting depth, the resource usage of the relevant quantum processes can be reduced by up to half in expectation. Second, we define a new special arrangement of elementary stabilizer operations for realizing certain quantum computations, which we call a binary in-tree decomposition. We show that such implementations lead to a better process for lowering resource consumption. We then propose an efficient classical algorithm to assemble stabilizer operation sequences with such binary in-tree form. Finally, we demonstrate the merits of the binary in-tree structure on several examples.

Quantum computing seeks to use the powers of quantum mechanics to accomplish tasks that classical computers cannot easily accomplish. Adiabatic quantum computing is one flavor of quantum computing that slowly changes a system with time in such a way that the state remains close to the ground state for the entire evolution. An easy way to implement and study adiabatic computing is using stoquastic Hamiltonians with no-sign problem. These Hamiltonians can be readily simulated on classical computers using quantum Monte Carlo schemes; though, it is unknown whether such simulation captures the speed and power of the quantum computation.

In this work, I study stoquastic Hamiltonians, focusing on quantum mechanical tunneling through classically inaccessible regions and barriers. I study multi-qubit tunneling problems to determine what properties of the barrier make this problem quantum mechanically or classically hard or inefficient. I compare quantum adiabatic computing with a path-integral Quantum Monte Carlo algorithm, which is a classical algorithm designed to simulate the quantum mechanical dynamics of the adiabatic evolution. Limited numerical data shows strong correlation between adiabatic and Monte Carlo runtimes, but due to computational limits, only small system sizes could be sampled.

Additionally, I study the properties of the quantum adiabatic algorithm in both the asymptotic limit of a large number of qubits and long runtimes and more realistic finite system sizes and non-adiabatic conditions. I develop a modification of the existing Villain transformation that allows it to find the asymptotic running time of the quantum evolution for different barrier sizes. Furthermore, I compare this to finite system sizes and discover the extremely large numbers of qubits are needed for these asymptotics to be useful.

When the quantum dynamics are run faster than adiabatically, new behavior arises that could potentially lead to enhancements in the quantum runtimes. I explore these enhancements, discovering when they do and do not occur. I also develop a broad framework for determining the nature of the near-adiabatic behavior of any quantum evolution, using only information about the energy gap between the ground state and the first excited state.

There exist numerous proofs of Bell's theorem, stating that quantum mechanics is incompatible with local realistic theories of nature. Here the strength of such nonlocality proofs is defined in terms of the amount of evidence against local realism provided by the corresponding experiments. Statistical considerations show that the amount of evidence should be measured by the Kull-back-Leibler (KL) or relative entropy divergence. The statistical strength of the following proofs is determined: Bell's original proof and Peres' optimized variant of it, and proofs by Clauser, Horne, Shimony, and Holt (CHSH), Hardy, Mermin, and Greenberger, Horne, and Zeilinger (GHZ). The GHZ proof. is at least four and a half times stronger than all other proofs, while of the two-party proofs, the one of CHSH is the strongest.