Quantum technologies commonly operate on finite-dimensional systems, such as qubits, also known as discrete-variable (DV) systems. Continuous-variable (CV) bosonic systems, which occupy an infinite-dimensional Hilbert space, present a promising alternative. That is the focus of this thesis, which has two main goals. Firstly, we look at probing bosonic systems, to gain information about their dynamics and interaction with their environment. We employ linear response theory, which was recently extended to encompass physical situations where an open quantum system evolves toward a non-equilibrium steady-state. We use the framework put forward by Konopik and Lutz to go beyond unitary perturbations of the dynamics. Considering an open system comprised of two coupled quantum harmonic oscillators, we study the system's response to unitary perturbations, as well as non-unitary perturbations, affecting the properties of the environment, e.g., its temperature and squeezing. We show that linear response, combined with a quantum probing approach, can provide valuable quantitative information about the perturbation and characteristics of the environment. Secondly, we investigate bosonic quantum computation, which involves encoding discrete quantum information into continuous-variable quantum systems. CV quantum computation was introduced by Braunstein and Lloyd, who put forward a notion of universality that is independent of the encoding of the CV system. However, a proof that the Braunstein-Lloyd (BL) model allows for fault-tolerant computation is still missing. Here, we provide evidence towards this objective, using the Gottesman-Kitaev-Preskill (GKP) encoding. We show how to generate GKP states from vacua by optimising a circuit comprised of a set of CV gates, deemed universal according to the BL notion. We compute the threshold value of the Glancy-Knill error probability for a fault-tolerant quantum memory from the known squeezing threshold for the concatenated GKP-surface code. We demonstrate that our generated GKP states have error probabilities below this threshold, further corroborating our claim.
Quantum technologies commonly operate on finite-dimensional systems, such as qubits, also known as discrete-variable (DV) systems. Continuous-variable (CV) bosonic systems, which occupy an infinite-dimensional Hilbert space, present a promising alternative. That is the focus of this thesis, which has two main goals. Firstly, we look at probing bosonic systems, to gain information about their dynamics and interaction with their environment. We employ linear response theory, which was recently extended to encompass physical situations where an open quantum system evolves toward a non-equilibrium steady-state. We use the framework put forward by Konopik and Lutz to go beyond unitary perturbations of the dynamics. Considering an open system comprised of two coupled quantum harmonic oscillators, we study the system's response to unitary perturbations, as well as non-unitary perturbations, affecting the properties of the environment, e.g., its temperature and squeezing. We show that linear response, combined with a quantum probing approach, can provide valuable quantitative information about the perturbation and characteristics of the environment. Secondly, we investigate bosonic quantum computation, which involves encoding discrete quantum information into continuous-variable quantum systems. CV quantum computation was introduced by Braunstein and Lloyd, who put forward a notion of universality that is independent of the encoding of the CV system. However, a proof that the Braunstein-Lloyd (BL) model allows for fault-tolerant computation is still missing. Here, we provide evidence towards this objective, using the Gottesman-Kitaev-Preskill (GKP) encoding. We show how to generate GKP states from vacua by optimising a circuit comprised of a set of CV gates, deemed universal according to the BL notion. We compute the threshold value of the Glancy-Knill error probability for a fault-tolerant quantum memory from the known squeezing threshold for the concatenated GKP-surface code. We demonstrate that our generated GKP states have error probabilities below this threshold, further corroborating our claim.
Continuous-variable (CV) systems have shown remarkable potential for quantum computation, particularly excelling in scalability and holding great promise for error correction through bosonic encoding. Within this framework, the foundational notion of computational universality was introduced by Lloyd and Braunstein [], and has proven especially successful since it allows for the identification of finite sets of universal CV gates independent of the encoding scheme. However, realizing the full power of this computational model requires achieving fault tolerance, which in turn demands discretization via some form of encoding—and, to date, no proof exists that universal CV gates can give rise to encoded fault tolerance. We present compelling evidence in this direction by utilizing the Gottesman-Kitaev-Preskill (GKP) encoding. Specifically, we numerically optimize the generation of GKP states from vacua using circuits comprised solely of universal CV gates, and demonstrate that the resulting states are of sufficient quality to exhibit error probabilities below the threshold required for fault-tolerant memory via concatenated GKP-stabilizer codes.
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.
(Abridged.) This thesis investigates scalable fault-tolerant quantum computation through the development of bosonic quantum codes, quantum LDPC codes, and decoding protocols that connect continuous-variable and discrete-variable error correction. We investigate superconducting microwave implementations of continuous-variable quantum computing, including the deterministic generation of cubic phase states, and introduce the dissipatively stabilized squeezed cat qubit, a noise-biased bosonic encoding with enhanced error suppression and faster gates. The performance of rotation-symmetric and GKP codes is analyzed under realistic noise and measurement models, revealing key trade-offs in measurement-based schemes. To integrate bosonic codes into larger architectures, we develop decoding methods that exploit analog syndrome information, enabling quasi-single-shot decoding in concatenated systems. On the discrete-variable side, we introduce localized statistics decoding, a highly parallelizable decoder for quantum LDPC codes, and propose quantum radial codes, a new family of single-shot LDPC codes with low overhead and strong circuit-level performance. Finally, we present fault complexes, a homological framework for analyzing faults in dynamic quantum error correction protocols. Extending the role of homology in static CSS codes, fault complexes provide a general language for the design and analysis of fault-tolerant schemes.
Continuous-variable (CV) cluster states are a universal resource for fault-tolerant quantum computation when supplemented with the Gottesman-Kitaev-Preskill (GKP) bosonic code. We generalize the recently introduced subsystem decomposition of a bosonic mode [G. Pantaleoni et al., Phys. Rev. Lett. 125, 040501 (2020)], and we use it to analyze CV cluster-state quantum computing with GKP states. Specifically, we decompose squeezed-vacuum states and approximate GKP states to reveal their encoded logical information, and we decompose several gates crucial to CV cluster-state quantum computing. Then, we use the subsystem decomposition to quantify damage to the logical information in approximate GKP states teleported through noisy CV cluster states. Each of these studies uses the subsystem decomposition to circumvent complications arising from the full CV nature of the mode in order to focus on the encoded qubit information.
Bosonic quantum systems are among the leading architectures for quantum information processing, offering continuous-variable degrees of freedom with strong error-correction capabilities. However, standard bosonic quantum computation models such as the Lloyd-Braunstein [Lloyd and Braunstein, 1999] and hybrid oscillator-qubit models [Brenner, Dias, and Koenig, 2025; Liu et al., 2026] permit dramatic energy growth, leading to unphysical computational power and the breakdown of fundamental algorithmic tools such as universal and efficient compilation [Brenner et al., 2026; Rudolph et al., 2025]. To address this, we introduce a new model of bosonic quantum computation, Bosonic Energy-Preserving Quantum Computation (BEQC), in which energy is treated as a computational resource. Namely, energy is supplied solely through input coherent states, and all gates are generated by energy-preserving Hamiltonians. Thus, by construction, dramatic energy growth is impossible, making the model physically grounded.
We next show that BEQC is a computationally robust and universal model in many respects, including: (1. Computational power) BEQC efficiently simulates all polynomial-energy computations in existing models, and exactly recovers BQP in the polynomial energy setting. It further admits several complexity-theoretic upper bounds when varying the energy, precision, and space parameters of the model. (2. Universal gate sets and state synthesis) BEQC has natural universal gate sets based on linear optics and Kerr interactions. In particular, we obtain a Solovay-Kitaev theorem which circumvents previous no-go results. We give various applications, including (a) a protocol for engineering GKP states with rigorous preparation guarantees, (b) Fock state preparation to exponential precision, and (c) native Fock space simulation of any qubit-based unitary.
Robust continuous-variable (CV) quantum information processing requires correcting realistic errors in bosonic systems, but all existing schemes rely on auxiliary Gottesman-Kitaev-Preskill (GKP) states which the preparation and operation are demanding in many platforms. In this work, we propose a novel CV quantum error correction (QEC) scheme that utilizes a broadly accessible resource: discrete-variable (DV) ancilla. Our scheme extracts information about CV displacement to the DV ancilla, measuring that allows counteracting the unwanted displacement error. We show that a simple single-qubit ancilla can already suppress CV infidelity by more than 20%. By concatenating with DV QEC codes, our scheme is robust against the physical errors in hybrid CV-DV systems, and yields a new class of oscillator-in-oscillator code that does not involve GKP states. Our work facilitates the implementation of CV QEC on realistic platforms.