Phase space quasi-probability functions provide powerful representations of quantum states and operators, as well as criteria for assessing quantum computational resources. In discrete, odd-dimensional systems (qudits), protocols involving only non-negative phase space distributions can be efficiently classically simulated. For bosonic systems, defined in continuous variables, phase space negativities are likewise necessary to prevent efficient classical simulation of the underlying physical processes. However, when quantum information is encoded in bosonic systems, this connection becomes subtler: as negativity is only a necessary property for potential quantum advantage, encoding (i.e., physical) states may exhibit large negativities while still corresponding to architectures that remain classically simulable. Several frameworks have attempted to relate non-negativity of states and gates in the computational phase space to non-negativity of processes in the physical bosonic phase space, but a consistent correspondence remains elusive. Here, we introduce a general framework that connects the physical phase space structure of bosonic systems to their encoded computational representations across arbitrary dimensions and encodings. This framework highlights the key role of the reference frame-equivalently, the choice of vacuum-in defining the computational basis and linking its phase space simulability properties to those of the physical system. Finally, we provide computational and physical interpretations of the planar (quadrature-like) phase space limit, where genuinely quantum features may gradually vanish, yielding classically simulable behavior.
Phase space quasi-probability functions provide powerful representations of quantum states and operators, as well as criteria for assessing quantum computational resources. In discrete, odd-dimensional systems (qudits), protocols involving only non-negative phase space distributions can be efficiently classically simulated. For bosonic systems, defined in continuous variables, phase space negativities are likewise necessary to prevent efficient classical simulation of the underlying physical processes. However, when quantum information is encoded in bosonic systems, this connection becomes subtler: as negativity is only a necessary property for potential quantum advantage, encoding (i.e., physical) states may exhibit large negativities while still corresponding to architectures that remain classically simulable. Several frameworks have attempted to relate non-negativity of states and gates in the computational phase space to non-negativity of processes in the physical bosonic phase space, but a consistent correspondence remains elusive. Here, we introduce a general framework that connects the physical phase space structure of bosonic systems to their encoded computational representations across arbitrary dimensions and encodings. This framework highlights the key role of the reference frame-equivalently, the choice of vacuum-in defining the computational basis and linking its phase space simulability properties to those of the physical system. Finally, we provide computational and physical interpretations of the planar (quadrature-like) phase space limit, where genuinely quantum features may gradually vanish, yielding classically simulable behavior.
Quantum kernel methods are a proposal for achieving quantum computational advantage in machine learning. They are based on a hybrid classical-quantum computation where a function called the quantum kernel is estimated by a quantum device while the rest of computation is performed classically. Quantum advantages may be achieved through this method only if the quantum kernel function cannot be estimated efficiently on a classical computer. In this paper, we provide sufficient conditions for the efficient classical estimation of quantum kernel functions for bosonic systems. These conditions are based on phase-space properties of data-encoding quantum states associated with the quantum kernels: negative volume, non-classical depth, and excess range, which are shown to be three signatures of phase-space negativity. We consider quantum optical examples involving linear-optical networks with and without adaptive non-Gaussian measurements, and investigate the effects of loss on the efficiency of the classical simulation. Our results underpin the role of the negativity in phase-space quasi-probability distributions as an essential resource in quantum machine learning based on kernel methods.
An important problem in quantum computation is to characterize the resources required for a computational speedup over classical computation. Veitch et al. showed that one necessary condition for a computational speedup in the model of quantum computation with magic states is that the discrete Wigner function representing the input state of the quantum circuit must take negative values. The amount of negativity in the discrete Wigner function quantifies the complexity of classical simulation of a quantum computation with simulation being efficient if the Wigner function is nonnegative everywhere. In this sense, negativity of the Wigner quasiprobability representation serves as an indicator of quantumness from a computational perspective. However, this result only holds for systems of qudits where the local Hilbert space dimension is odd. The first main result discussed in this thesis relates to a discrete Wigner function suitable for describing quantum computation with magic states defined for any local Hilbert space dimension. When the local Hilbert space dimension is odd it subsumes the standard discrete Wigner function. When the local Hilbert space dimension is even, as a result of state-independent proofs of contextuality among multiqubit (or multi-even-dimensional-qudit) Pauli observables, the phase space over which the Wigner function is defined becomes much larger. However, for systems of qubits, the properties required for simulation of quantum computation with magic states remain. This simulation method effectively extends the result described above for odd-dimensional qudits to qubits. Although in the even-dimensional case the phase space is much larger, points in multiqubit phase space can be characterized and classified and some structure can be imparted on the phase space. The second main result discussed here is a hidden variable model for quantum computation with magic states on qubits. This model is similar in structure to quasiprobability representations like the discrete Wigner function, but unlike those representations the model is capable of representing all elements of any quantum computation---states, operations, and measurements---using only classical probabilities. No negativity is required. This calls into question the role of negativity in quasiprobability representations as an indicator of quantumness for models of quantum computation.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
Since the early days of quantum information science, diverse strategies have been proposed to encode qubits and implement quantum gates across various physical platforms. Modes and bosonic states offer a flexible framework for quantum information processing, from single-photon optics to superconducting circuits, combining massless and massive particles. Universal gate sets have been established for specific encodings—such as those based on Fock states or the field’s quadratures. However, a general framework clarifying how different bosonic resources—such as mode structure, particle-number statistics, Gaussian and non-Gaussian states and operations—combine to enable universality and potential quantum advantage irrespectively of the encoding remains lacking. A main difficulty lies in the coexistence of distinct paradigms that appear to combine loosely in bosonic quantum information: discrete-variable schemes—where information is encoded in qubits or effective finite-dimensional systems—and continuous-variable schemes—which rely on infinite-dimensional Hilbert spaces associated with the field’s quadratures. Despite the success of each approach, they remain seemingly incompatible in many aspects—in particular in terms of physical resource analysis for universality—due to their underlying mathematical differences. In this work, we apply a superselection-rule-compliant framework for bosonic quantum information that unifies these paradigms. This representation explicitly incorporates the phase reference and enforces total particle number conservation, allowing for a consistent treatment of bosonic quantum information protocols and resources across both discrete and continuous regimes. We derive necessary and sufficient conditions expressed in terms of Gaussian and non-Gaussian resources for universality that apply to any encoding, without resorting to specific mappings. Our results show that continuous variable encodings are not fundamentally distinct from discrete ones, but rather emerge as subspaces within a general bosonic structure. Importantly, in such a widely studied framework, non-Gaussian properties of bosonic states and operations can be washed out, providing evidence that the continuous variables regime is not the most suitable to fully assess the quantum properties of bosons.
Non-Gaussian bosonic states are ubiquitous in interacting light--matter systems, many-body platforms, and relativistic quantum field settings, but their quantitative characterization is hindered by the infinite-dimensional Hilbert space and by the poor scalability of Fock-space truncation methods. We introduce an exact finite-manifold encoding for states supported on a finite span of Gaussian branches, enabling the use of standard finite-dimensional quantum-information tools directly on an effective density matrix whose entries are determined by Gaussian overlaps. As demonstrations, we obtain closed-form and numerically stable evaluations of entropies and relative-entropy non-Gaussianity, and derive an analytic expression for the bipartite entanglement negativity of arbitrary multimode two-branch Gaussian superpositions, including a minimal which-branch dephasing model. Our framework provides a practical bridge between experimentally accessible continuous-variable resources (e.g., cat-like and measurement-conditioned states) and discrete-variable information measures, with immediate applications to benchmarking non-Gaussian resources in several quantum technology platforms.
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.