Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Physics

First Advisor

Girvin, Steven

Abstract

This dissertation develops a theoretical framework for hybrid discrete-variable (DV) and continuous-variable (CV) quantum systems, focusing on control, state preparation, and error correction. Quantum computing holds the potential to surpass classical computation in tasks such as factorization, secure communication, and quantum simulation. Hybrid CV-DV systems offer a promising path by combining the stability and long coherence times of oscillators with the fast gate operations of qubits.

A central contribution of this work is the development of “non-abelian quantum signal processing” (NA-QSP), a generalization of quantum signal processing (QSP) [1] where the control parameters are non-commuting quantum operators, such as oscillator position and momentum. We introduce the “Gaussian-Controlled-Rotation” (GCR) technique, the first non-abelian composite pulse sequence that enables precise control of CV states using DV ancillae. GCR outperforms traditional composite pulse sequences in terms of gate fidelity and robustness to control errors. This framework can be extended to quantum singular value transformation (QSVT). In light of understanding the CV instruction set, we also propose the Gaussian hierarchy for CV operations, a classification of CV operations, analogous to the Clifford hierarchy for qubits, and raise open questions about the comparison and mapping between the two hierarchies.

The dissertation also addresses deterministic state preparation in oscillators, including squeezed states, two-legged and four-legged cat states, and Gottesman-Kitaev-Preskill (GKP) states. The GCR technique enables high-fidelity preparation of these states, which are essential for quantum simulation and error correction, without the need for numerical optimizers.

A key challenge in oscillator-based architectures is photon loss, which degrades state coherence. This work analyzes probabilistic error correction for photon loss in finite-energy GKP codes, introducing the concept of ‘probabilistic distance’ to quantify error correction performance of the recent GKP experiments showing promising realizations of beyond break-even error correction for qudits.

The dissertation further explores high-fidelity universal control of error-corrected qubits encoded in oscillators. It introduces protocols for high-fidelity logical readout in the presence of residual errors and a pieceable error-corrected gate teleportation. A key finding is that logical operations on GKP qubits using our scheme can achieve high fidelity using GCR, even in the presence of errors, with a biased-noise ancilla. The extension of GCR to multi-mode systems enables efficient entangling gates and error-corrected two-qubit rotations. Our schemes are generalizable to arbitrary qubit as well as qudit GKP lattices.

We also explore how oscillator codes can reduce resource overheads in fault-tolerant quantum computing, alongside potential applications of a hybrid CV-DV architecture. To this end, we also present a quantum phase estimation compilation using an ancillary oscillator and a non-abelian QSP-based circuit, demonstrating the utility of the thesis framework for hybrid CV-DV algorithms. The dissertation establishes NA-QSP as a foundation for hybrid CV-DV quantum control, state preparation, and GKP-based error correction, laying the groundwork for scalable fault-tolerant quantum computation in CV-DV architectures.

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