MS Thesis Defense: Tian Xia

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"Algorithms and Numerical Simulations for Quantum Platforms"

Abstract

Quantum computing has the potential to solve certain problems exponentially faster than classical computers. However, significant challenges lie ahead in realizing this potential. On the algorithmic side, few quantum algorithms have demonstrated verifiable quantum-classical separations for practical problems. On the hardware side, experimental quantum platforms suffer from noise, limited connectivity, and decoherence, making it difficult to scale beyond small problem instances.  This thesis offers a modest investigation on both fronts, developing quantum and classical algorithms for graph problems, and theoretically and numerically modeling novel quantum platforms to better understand and explore hardware-level limitations.

In the first part of this thesis, we explore the relative power of quantum and classical algorithms in combinatorial optimization. Specifically, we design a class of quantum algorithms that approximates the hardcore partition function on neutral atom platforms by exploiting the native Rydberg blockade mechanism. We complement this with a classical distribution testing protocol that certifies the hardcore distribution in polynomial number of samples in the high temperature regime. We also present a set of polynomial-time reductions from other #P-complete problems, extending the algorithmic framework to a broader family of combinatorial counting problems. Our results offer an algorithmic bridge at the intersection of quantum algorithms, statistical mechanics and computational complexity of neutral atom systems.

In the second part, we theoretically model and numerically simulate two novel quantum platforms. The first platform integrates a traveling-wave parametric amplifier (TWPA) in a feedback loop with a 3D microwave cavity, realizing a parametrically driven nonlinear oscillator. We model the feedback architecture using the SLH framework for networked quantum systems and numerically explore regimes where exotic quantum states emerge. The second platform builds on our previously developed Broadband Cryogenic Transient Dielectric Spectroscopy (BCTDS) technique, which enables coherent control and time-resolved probing of two-level system (TLS) defect ensembles. We employ computational electrodynamics and Floquet theory to qualitatively capture the observed experimental behaviors. Our results establish BCTDS as a versatile platform for defect spectroscopy and exploring non-equilibrium phenomena in disordered quantum systems.

Thesis Committee

• Mattias W. Fitzpatrick (Chair)
• Anthony J. Rizzo
• Deeparnab Chakrabarty
• James D. Whitfield

Contact

For more information, contact Thayer Registrar at thayer.registrar@dartmouth.edu .