This thesis investigates two complementary computational approaches to obtaining the outcome statistics of bosonic quantum experiments without directly performing the experiments themselves. In particular, we consider how continuous-variable quantum computations can be approximated using discrete-variable quantum systems and how bosonic outcome probabilities can be estimated using classical computational methods.The first part of the thesis focuses on the relation between continuous-variable (CV) and discrete-variable (DV) quantum computation. A framework is presented for approximating realistic CV quantum computations, consisting of polynomially bounded-energy states and a polynomial number of elementary gates, using DV quantum hardware. The proposed approximation error scales polynomially with the number of modes and can be made arbitrarily small by increasing the dimension of the simulating qudits. However, some technical aspects of the derivation remain to be rigorously justified. If these can be resolved, the result would establish a computational equivalence between the two paradigms. The remaining potential issues and possible approaches to resolving them are discussed.In the second part, we develop a framework for classically estimating individual outcome probabilities of general bosonic experiments using Monte Carlo methods based on (s)-ordered phase-space quasi-probability distributions. The resulting sampling complexity depends on properties of the phase-space quasi-probability distributions, in particular, their negativity and range, showing that negativity alone does not prevent efficient classical estimation. The outcome probability can be decomposed in several ways, giving rise to distinct trade-offs between these quantities. We discuss how experimentally relevant error channels can be incorporated into the framework and how they affect the classical estimation complexity, using pure loss as an illustrative example.Together, these results contribute to understanding the resources required to reproduce the outcome statistics of bosonic quantum experiments using both DV quantum and classical computation.
This thesis investigates two complementary computational approaches to obtaining the outcome statistics of bosonic quantum experiments without directly performing the experiments themselves. In particular, we consider how continuous-variable quantum computations can be approximated using discrete-variable quantum systems and how bosonic outcome probabilities can be estimated using classical computational methods.The first part of the thesis focuses on the relation between continuous-variable (CV) and discrete-variable (DV) quantum computation. A framework is presented for approximating realistic CV quantum computations, consisting of polynomially bounded-energy states and a polynomial number of elementary gates, using DV quantum hardware. The proposed approximation error scales polynomially with the number of modes and can be made arbitrarily small by increasing the dimension of the simulating qudits. However, some technical aspects of the derivation remain to be rigorously justified. If these can be resolved, the result would establish a computational equivalence between the two paradigms. The remaining potential issues and possible approaches to resolving them are discussed.In the second part, we develop a framework for classically estimating individual outcome probabilities of general bosonic experiments using Monte Carlo methods based on (s)-ordered phase-space quasi-probability distributions. The resulting sampling complexity depends on properties of the phase-space quasi-probability distributions, in particular, their negativity and range, showing that negativity alone does not prevent efficient classical estimation. The outcome probability can be decomposed in several ways, giving rise to distinct trade-offs between these quantities. We discuss how experimentally relevant error channels can be incorporated into the framework and how they affect the classical estimation complexity, using pure loss as an illustrative example.Together, these results contribute to understanding the resources required to reproduce the outcome statistics of bosonic quantum experiments using both DV quantum and classical computation.
This tutorial offers a pedagogical guide to hybrid quantum processors that integrate discrete-variable (DV) qubits and continuous-variable (CV) oscillators. Aimed at computer scientists, engineers, and physicists, it provides an overview of the experimental, algorithmic, and architectural aspects of this novel and rapidly developing hardware model. Experimental realizations of this model include superconducting, trapped-ion, and neutral-atom platforms. By combining DV and CV components, hybrid oscillator-qubit processors enable a powerful new paradigm that offers complementary strengths for quantum control, error correction, computation, and simulation. Working toward the goal of a full-stack system connecting applications to CV-DV hardware, we define and formulate abstract machine models and instruction set architectures. These essential abstractions enable codesign of hardware and software, and resource estimation for exploring the potential of current and future hardware for computational and simulation tasks. Using these abstractions, we present both new and existing examples that illustrate the benefits of hybrid CV-DV processors relative to traditional DV-only hardware in computation as well as quantum simulation of physical models. Examples include algorithms for transferring states between DV and CV systems, performing the quantum Fourier transform, and simulation of lattice gauge theories. Relative to qubit-only hardware, the bosonic degrees of freedom natively available in hybrid architectures can substantially reduce the circuit complexity of simulations for physical models containing bosons. A key technique is the extension of quantum signal processing ideas to CV-DV systems. This work is intended to serve as a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and to provide a road map to guide future development.
It remains questionable whether all physically realizable computational model can be simulated by currently available digital computer in an efficient manner. The theory of Quantum Mechanics offers an answer to this question through one of its branches that intersects with Computer Science, known as Quantum Computing. It claims that current computers, which is operating based on classical mechanics cannot efficiently simulate quantum mechanical process. Under this claim, a computer that is operating based on quantum mechanics holds more computational power than classical computers, promising a possibility to computationally solve problems that are classically intractable. To show that this claim holds, numerous quantum computational models have been proposed and analyzed through the lens of Computational Complexity Theory, providing a rigorous notion of efficiency. Models that are more feasible experimentally, although not necessarily universal, have also been of a particular interests given the numerous complicated challenges surrounding the experimental implementation of a universal quantum computer. One of this non-universal models is called Gaussian Boson Sampling (GBS), which embraces the methods of Quantum Optics by sampling the number of photon outputs from a linear optical network as a means to process quantum information. Compared to its prototype model, Boson Sampling, GBS is more experimentally feasible due to its method of generating photon that is based on the Continuous Variable (CV) paradigm of Quantum Information. Nevertheless as with Boson Sampling, it was also shown that classical simulation of GBS cannot be performed efficiently under some highly plausible conjectures. In the work presented in this thesis, we probe the complexity of classically constructing the probability distribution induced by GBS using the method of Quantum Homodyne Tomography. Instead of photon number measurement, this method further embraces the CV paradigm in the measurement of GBS by using Homodyne measurements that outputs real numbers. Sample data obtained from the measurement outcomes is then classically processed using Pattern Functions, which we use to approximate the probability distribution of GBS. A series of computer simulations that approximates some probability values of GBS are presented here, which shows that the number of samples needed to decrease the error of the approximation increases rapidly for more accurate approximation. We have derived a bound for the approximation error which can be achieved given that the sampled data satisfies a certain statistical property. This statistical property can eventually be satisfied as the number of Homodyne samples increases. Having shown another evidence for the complexity of approximating GBS using Homodyne Tomography, we further investigate the nature of the complexity of GBS by analyzing its classical-quantum computational boundary in the spectrum of amount of noise present in a GBS setup. In doing this, we built our discussions on the measure of non-classicality using the presence of negative values in the Quasi-probability distribution (QPD) representation of states of a quantum system. We then discuss existing results that shows the possibility of efficient classical simulation of a quantum computational model given a sufficient amount of noise. These results impose a necessary criteria of the minimum amount of noise that an experiment has to surpass in order for it to be impossible to simulate classically in an efficient manner. We extend this idea for GBS, where we propose a modified GBS where we consider noise in its measurements that opens a way for a possible formulation of a noise condition that a GBS experiment has to fulfill to rule out possible efficient classical simulation. To further probe the classical-quantum computational boundary, we also propose a different quantum computational model called the Photon-added Thermal State CV-Sampling (PATS CVS), which can be formulated similarly to our noisy GBS model. We then discuss existing results that analyzes the negativity of the QPD representation of a noisy Photon-added Thermal state (PATS), which constitutes the inputs for PATS CVS. Using the result that shows the quantum-classical transition in terms of the QPD negativity for a PATS state when the amount of noise changes, we discuss the potential of PATS CVS model as a possible avenue to further probe the transition between efficiently simulable and non-efficiently simulable quantum mechanical experiment.
(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.
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.
Hybrid quantum systems combining discrete-variable (DV) and continuous-variable (CV) subsystems arise across a broad range of physical platforms and enable quantum information processing beyond purely qubit-based designs. Their classical simulation, however, is challenging since computational costs of both CV and DV systems exponentially increase. In this paper, we introduce a simulation framework for CV--DV hybrid systems based on linear combination of Gaussian (LCoG) decomposition. We decompose the hybrid density matrix into blocks in a DV basis and represent each block as an LCoG expansion. Within the representation, Gaussian operations on the bosonic subsystem independently act on each Gaussian function. We derive analytic formulas describing how complex Gaussian functions transform under controlled Gaussian operations. We further extend the framework to controlled Gaussian channels with state-dependent Gaussian noise. The method avoids a Fock-space truncation for the CV subsystem, and the cost of updating each Gaussian term scales polynomially with the number of modes. The framework provides a general computational tool for analyzing hybrid quantum algorithms, quantum simulation protocols, and non-Gaussian state generation schemes. As an example, we simulate Gottesman--Kitaev--Preskill (GKP) state generation based on a cavity-QED system.
Continuous systems are studied in many branches of modern physics, such as high-energy physics, cosmology, condensed matter physics, quantum chemistry, and field theories. Such systems are expected to benefit from the substantial advantages in computational power of quantum computers. The continuous-variable paradigm of quantum computation provides the most natural computational formalism for these tasks. However, most existing quantum hardware is based on discrete-variable systems. We address this fundamental discrepancy by providing a rigorous framework for translating native continuous-variable algorithms onto qubit-based quantum processors. This mapping is constructed from a gate-based model of continuous-variable quantum computers, consisting of states and operations built from a polynomial sequence of elementary gates in a finite set, with total energy polynomial in the number of modes. We prove that, under realistic constraints, a gate-based model of continuous-variable quantum computers can be efficiently simulated using discrete-variable devices, thereby establishing a computational equivalence between these paradigms.