My doctoral thesis details my research on two major projects. Both projects investigate the correspondence between the discrete (photon number) and the continuous (quadrature) representation of optical states. Understanding the relationships between the two representations is very important for the investigation of quantum physics phenomena. Moreover it has significant applications in quantum computing where it is found that discrete-variable bases are easier to error-correct but some continuous-variable architectures are more scalable. Research in the connection between the two representation might help develop new architectures for scalable fault-tolerant quantum computing.
My doctoral thesis details my research on two major projects. Both projects investigate the correspondence between the discrete (photon number) and the continuous (quadrature) representation of optical states. Understanding the relationships between the two representations is very important for the investigation of quantum physics phenomena. Moreover it has significant applications in quantum computing where it is found that discrete-variable bases are easier to error-correct but some continuous-variable architectures are more scalable. Research in the connection between the two representation might help develop new architectures for scalable fault-tolerant quantum computing.
The thesis focuses on the experimental investigation of the optical hybrid approach forquantum information processing. Specifically, two traditionally separated approaches, i.e.the discrete and the continuous-variable ones, are combined in the same experiment with twodistinct quantum measurements based on photon counting (photon number) and homodynedetection (quadrature components).The optical hybrid approach is first applied to generate high-fidelity non-Gaussian states,e.g. Fock states and Schrödinger cat states, which correspond to two types of qubit encodingsused in optical quantum information. The use of high-efficiency superconducting nanowiresingle-photon detectors leads to an unprecedented preparation rate, which facilitates thesubsequent use of these states. For instance, the two types of states are combined to generatefor the first time a hybrid entanglement between particle-like and wave-like optical qubits, aswell as the extension to hybrid qutrit entanglement. These novel resources may pave the wayto implement heterogeneous networks where discrete and continuous-variable operations andtechniques can be efficiently combined. Additionally, we also experimentally demonstratefor the first time the so-called squeezing-induced micro-macro entangled states.During this PhD, efforts have also been dedicated to implement a high-efficiency andlow-noise frequency up-conversion system based on two synchronized fiber lasers. Suchquantum frequency converter not only permits to extend the spectra of quantum statesto difficultly accessible wavelengths with current technology, but also constitutes a highperformancephoton detector especially in the infrared regime. Based on the conversionsystem, several applications are demonstrated, such as infrared photon-number-resolvingdetection, and few-photon-level infrared imaging.
My research has three main foci, one which is of interest to the fields of quantum information and quantum optics is to characterize the effects of coupling between quantum systems. Advances in technology made since the conception of quantum mechanics spurred the development of quantum technologies like quantum optics and quantum computing. These applications require a good understanding of the quantum systems that compose them, and the clever ways in which the information in these systems is stored and manipulated. By understanding the ways which quantum systems interact with their environment, their usefulness can be further utilized. The second is to develop methods of calculation which simplify the calculational and computational side of quantum dynamics. Understanding how quantum systems evolve in time is very important, but traditional methods of solving for the dynamics of quantum systems struggle to deal with time dependent parameters. Using results from the theory of Lie algebras, I will show how quantum dynamics problems can be made less computationally intensive by avoiding the partial differential equations typically associated with such problems and reducing the problem to a system of coupled ordinary differential equations. The third focus is to develop methods of visualizing properties of quantum systems. I think this is important as an educational tool and as a way to develop intuition about how quantum systems evolve considering how few opportunities there are to visualize quantum effects on a macroscopic scale. This problem is particularly challenging because quantum mechanics is non-deterministic, meaning that the only nontrivial definitive statements that can be made regarding quantum systems deal with probabilities. This coupled with the fact that many of the systems of interest depend on upwards of six variables and also evolve in time make creating useful visual representations challenging. Despite the challenges, I feel that useful insights can be gained from the visualizations, and will illustrate a few examples.
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 quantum systems are characterized by physical quantities that can take continuous values.It appears in many physical systems, such as the position and momentum of oscillator modes of trapped ions and the complex amplitude of an optical mode.Especially, the quantum optical system is one of the most important continuous-variable systems for the quantum information application since light is an information carrier for telecommunication and behaves quantum mechanically even in the room-temperature environment.Furthermore, the complex amplitude of the optical mode can easily be manipulated.However, using a continuous variable, as it is, in quantum information processing does not work since a continuous variable is subject to continuous noises in the physical world, and such noises cannot completely be corrected.In order to perform reliable and fault-tolerant quantum information processing, information should be encoded in a digitized degree of freedom.In this thesis, digital quantum information processing with continuous-variables systems is studied in the quantum key distribution (QKD) and the quantum computation (QC), two major fields of quantum information processing.For the QKD, the so-called continuous-variable QKD protocols, in which information is encoded on the complex amplitude of an optical pulse and the homodyne or heterodyne detector is used to measure it, are studied.More specifically, a continuousvariable QKD protocol that uses discretely modulated signals and the classical digital information processing is developed, and its security proof against general attacks in the finite-key case is established.This is achieved by virtually introducing the discretevariable quantum system into the continuous-variable system and thereby reducing its security argument to that of discrete-variable QKDs, which is more mature.This is in sharp contrast to the previous security analyses on this kind of QKD protocol, where they directly use continuous variables in the security analyses and thus have difficulty in extending the analyses to the finite-key case.Another key to our security proof is the development of the method of estimating a lower bound on the fidelity to the coherent states, which eventually leads to the evaluation of how well the discrete-variable system can be approximated by the continuous-variable one.These security analyses are further elaborated, and a protocol whose key rate against the loss noise achieves almost optimal scaling is obtained.For the QC with the continuous-variable system, the Gottesman-Kitaev-Preskill (GKP) code, which encodes discrete quantum information into a continuous-variable system, is studied.This code has many desirable properties for the optical implementation of QC; e.g., the universal QC is possible using only the Gaussian operations, which are relatively easy to implement in the quantum optical system, along with the GKP-encoded states.Because the ideal GKP-encoded state is unphysical, only i I have received much support and advice during my Ph.D. studies.First, I would like to express my sincere thanks to my supervisor Masato Koashi for the instruction and all the constructive suggestions.He was always ready to listen to my questions and concerns for both academic and non-academic matters.From him, I learned the approaches to challenging problems and the importance of thorough consideration.