Truncating the Hilbert Space: Topics on Hardware-Efficient Quantum Control, Error Correction, and Tomography

Over the past century, people have observed the development of quantum mechanics from a conceptual mystery to a powerful framework for information processing. However, the quantum information remains fragile due to the ubiquitous noise. To address the challenge, device physicists continue to improve pulse-level control schemes for better performance in physical operations, while information scientists employ resource redundancy to protect the quantum states from direct damage caused by noise. Bridging these efforts, the hardware-efficient paradigm exploits the specific hardware structures to suppress the error or reshape the error pattern, which facilitates the error correction or characterization in the next step. On the other hand, the Hilbert space is usually intractable due to its size. In this dissertation, I will provide several case studies to achieve hardware-efficiency through the truncation of Hilbert space. First, we truncate the Hilbert space of a resonator through a destructive interference, streamlining pulse design for high-fidelity operations. Then, we utilize driven-dissipative processes to autonomously stabilize subspaces in resonators or atomic systems, which provides an encoded qubit with a structured error channel. We also design operations that preserve such error structures, as they are essential for the next-level error correction. Finally, a collective truncation in a multipartite system or constraints on specific subsets of states in the Hilbert space can also provide efficiency in state characterization tasks. I will present several experimentally relevant examples to justify this claim.