Dissertation Defense

Quantum computing promises to transform our approach to solving significant computational challenges, such as factorization and quantum system simulation. Harnessing this quantum power in real life necessitates software stack support. This talk focuses on the critical challenges encountered in the software for quantum computing, aiming to shape high-assurance software stacks for controlling quantum computing devices in the immediate future and beyond.

Continuous optimization problems arise in virtually all disciplines of quantitative research, including applied mathematics, computer science, and operations research. While convex optimization has been well studied in the past decades, nonconvex optimization generally remains intractable in theory and practice. Quantum computers, an emerging technology that exploits quantum physics for information processing, could pave an unprecedented path toward nonconvex optimization.

Optomechanical systems have enabled a variety of novel sensors that transduce an external force on a mechanical sensor to an optical signal which can be read out through different measurement techniques. Based on recent advances in these sensing technologies, we suggest that heavy dark matter candidates around the Planck mass range could be detected solely through their gravitational interaction.

Random Circuits have emerged as an invaluable tool in the quantum mechanics’ toolkit. On one hand, the task of sampling outputs from a random circuit has established itself as a leading contender to experimentally demonstrate the intrinsic superiority of quantum computers using near-term, noisy platforms. On the other hand, random circuits have also been used to deduce far-reaching conclusions about the theoretical foundations of quantum information and communication.

Significant advancements in controlling and manipulating individual quantum degrees of freedom have paved the way for the development of programmable strongly-interacting quantum many-body systems. Quantum simulation emerges as one of the most promising applications of these systems, offering insights into complex natural phenomena that would otherwise be difficult to explore.

The difficulty of efficiently simulating quantum many-body systems was one of the first motivations for developing quantum computers and may also be one of the first applications to find practical computational advantage on real quantum hardware. With the relatively recent advent of publicly available quantum technologies, we have now entered the era of noisy intermediate-scale quantum (NISQ) computing.

The variational algorithm is a paradigm for designing quantum procedures implementable on noisy intermediate-scale quantum (NISQ) machines. It is viewed as a promising candidate for demonstrating practical quantum advantage.

In this dissertation, we look into the optimization aspect of the variational quantum algorithms as an attempt to answer when and why a variational quantum algorithm works. We mainly focus on two instantiations of the family of variational algorithms, the Variational Quantum Eigensolvers (VQEs) and the Quantum Neural Networks (QNNs).

A quantum speedup occurs when a computational problem can be solved faster on a quantum computer than on a classical computer. This thesis studies the circumstances under which quantum speedups can arise from three perspectives. First, we study the structure of the problem. We show how a problem’s symmetries relate to whether it can admit a polynomial or superpolynomial quantum speedup. In particular, we show that the computation of any partial function of a hypergraph’s adjacency matrix can only admit a polynomial speedup.

Quantum many-body systems with long-range interactions—such as those that decay as a power-law in the distance between particles—are promising candidates for quantum information processors. Due to their high degree of connectivity, they are potentially capable of generating entanglement more quickly than systems limited to local interactions, which may lead to faster computational speeds.