标题： 量子计算在离散和连续变量架构中的应用 显示英文标题

标题： Quantum Computing in Discrete- and Continuous-Variable Architectures

摘要： 本论文提出了一个混合连续变量（CV）和离散变量（DV）量子系统的理论框架，重点在于量子控制、状态制备和错误校正。 核心贡献是非阿贝尔量子信号处理（NA-QSP），这是将量子信号处理推广到控制参数为非对易算子的情况。 在该框架内，我们引入了高斯控制旋转（GCR）协议，该协议利用DV辅助系统实现了对CV状态的高保真控制。 这种方法允许在不进行数值优化的情况下确定性地制备压缩态、猫态和Gottesman-Kitaev-Preskill（GKP）态。 包含两个之前未发表的贡献：(i) 第2.3章介绍了高斯层次结构，这是一种类似于 Clifford 层次结构的CV操作分类，为理解CV门集提供了新的视角；(ii) 第5章提出了一个分析框架，用于纠正有限能量GKP码中的光子丢失，引入了概率错误校正的概念，为近期超越阈值的GKP实验提供了见解。 总体而言，这项工作为混合CV-DV架构中可扩展的容错量子计算奠定了基础，应用于使用辅助振子的逻辑门合成、读出和混合算法设计。 显示英文摘要

摘要： This thesis develops a theoretical framework for hybrid continuous-variable (CV) and discrete-variable (DV) quantum systems, with emphasis on quantum control, state preparation, and error correction. A central contribution is non-abelian quantum signal processing (NA-QSP), a generalization of quantum signal processing to settings where control parameters are non-commuting operators. Within this framework, we introduce the Gaussian-Controlled-Rotation (GCR) protocol, which enables high-fidelity control of CV states using DV ancillae. This approach allows for deterministic preparation of squeezed, cat, and Gottesman-Kitaev-Preskill (GKP) states without numerical optimization. Two previously unpublished contributions are included: (i) Chapter 2.3 introduces the Gaussian hierarchy, a classification of CV operations analogous to the Clifford hierarchy, offering a new lens for understanding CV gate sets; (ii) Chapter 5 presents an analytical framework for correcting photon loss in finite-energy GKP codes, introducing the notion of probabilistic error correction, providing insight into recent GKP experiments surpassing break-even thresholds. Overall, this work lays the groundwork for scalable, fault-tolerant quantum computation in hybrid CV-DV architectures, with applications to logical gate synthesis, readout, and hybrid algorithm design using ancilla oscillators.