The generation of non-Gaussian states of bosonic systems described by continuous quantum degrees of freedom is of interest in a variety of contexts, ranging from quantum foundations to computation and communication. A crucial feature of genuine non-Gaussian states is that they cannot be described in general with a non-negative Wignerquasi-probability distribution. In particular, this feature is equivalent to contextuality and it underpins all attempts at attaining quantum computational advantage in a universal setting. In this thesis, the generation of Wigner-negative states is addressed, pursuing two approaches. First, in an optomechanical-like setting, we develop schemes to generate Wigner-negative states of two massive mechanical systems, in turn correlated such that they also feature entanglement, another genuine quantum hallmark. These schemes are based on driven-dissipative systems where dissipation is properly engineered by taking advantage of linear-and-quadratic optomechanical interactions. We address the robustness of these schemes when uncontrollable noisy channels are included in the dynamics, proving that their resilience is comparable to Gaussian schemes that have already been implemented experimentally. Second, considering in particular circuit QED systems and trapped ions, we propose a method to engineer a set of Hamiltonian evolutions that are universal in terms of single-mode state preparation. This method is based on a driven quantum Rabi model, where the desired evolution is effectively imprinted on a continuous-variable degree of freedom by its interaction with a properly driven two-level system. We demonstrate the universality of such approach by simulating the generation of multi-squeezed and other states, including the celebrated cubic-phase state. We also consider the effect of noise in realistic settings, showing in particular that in systems involving superconducting qubits, the generation of tri-squeezed states --- a universal resource for continuous-variable quantum computation which we prove to display larger Wigner negativities with respect to Fock and cubic-phase states for the same energy --- is within experimental reach.
The generation of non-Gaussian states of bosonic systems described by continuous quantum degrees of freedom is of interest in a variety of contexts, ranging from quantum foundations to computation and communication. A crucial feature of genuine non-Gaussian states is that they cannot be described in general with a non-negative Wignerquasi-probability distribution. In particular, this feature is equivalent to contextuality and it underpins all attempts at attaining quantum computational advantage in a universal setting. In this thesis, the generation of Wigner-negative states is addressed, pursuing two approaches. First, in an optomechanical-like setting, we develop schemes to generate Wigner-negative states of two massive mechanical systems, in turn correlated such that they also feature entanglement, another genuine quantum hallmark. These schemes are based on driven-dissipative systems where dissipation is properly engineered by taking advantage of linear-and-quadratic optomechanical interactions. We address the robustness of these schemes when uncontrollable noisy channels are included in the dynamics, proving that their resilience is comparable to Gaussian schemes that have already been implemented experimentally. Second, considering in particular circuit QED systems and trapped ions, we propose a method to engineer a set of Hamiltonian evolutions that are universal in terms of single-mode state preparation. This method is based on a driven quantum Rabi model, where the desired evolution is effectively imprinted on a continuous-variable degree of freedom by its interaction with a properly driven two-level system. We demonstrate the universality of such approach by simulating the generation of multi-squeezed and other states, including the celebrated cubic-phase state. We also consider the effect of noise in realistic settings, showing in particular that in systems involving superconducting qubits, the generation of tri-squeezed states --- a universal resource for continuous-variable quantum computation which we prove to display larger Wigner negativities with respect to Fock and cubic-phase states for the same energy --- is within experimental reach.
The generation of entangled states that display negative values of the Wigner function in the quantum phase space is a challenging task, particularly elusive for massive, and possibly macroscopic, systems such as mechanical resonators. In this work, we propose two schemes based on reservoir engineering for generating Wigner-negative entangled states unconditionally. We consider two noninteracting mechanical resonators that are radiation-pressure coupled to either one or two common cavity fields; the optomechanical coupling with the field(s) features both a linear and quadratic part in the mechanical displacement and the cavity is driven at multiple frequencies. We show analytically that both schemes stabilize a Wigner-negative entangled state that combines the entanglement of a two-mode squeezed vacuum with a cubic nonlinearity, which we dub cubic-phase entangled (CPE) state. We then perform extensive numerical simulations to test the robustness of Wigner-negative entanglement attained by approximate CPE states stabilized in the presence of thermal decoherence. Published by the American Physical Society 2024
Non-Gaussian interactions are a key ingredient for achieving universality in continuous-variable quantum computation, yet their experimental characterization and the validation of their correct implementation remain challenging tasks. In this work, we focus on a three-mode non-Gaussian trilinear Hamiltonian that has recently been realized in superconducting microwave platforms, and present a comprehensive theoretical analysis of the computational resources it generates, together with experimentally accessible protocols to validate their presence. We systematically investigate its ability to generate two key resources for quantum computation: multipartite entanglement and Wigner negativity. In particular, using displaced-parity Bell tests, we demonstrate the generation of nonlocal states and thereby provide an operational certification of multipartite entanglement in the non-Gaussian states produced by the dynamics. We further quantify the Wigner logarithmic negativity and benchmark it against that of established non-Gaussian resource states. Building on this resource-based characterization, we introduce a measurement-efficient protocol for the experimental validation of the Hamiltonian implementation without requiring full reconstruction of the Wigner function. The protocol combines the measurement of zero-variance observables (nullifiers and stabilizers) with a limited number of targeted phase-space measurements, leading to a drastic reduction of the experimental overhead.
Driven-dissipative Kerr cavities serve as a versatile platform for generating nonclassical states, such as quantum cat states and high-photon-number Fock states, which are essential for quantum information processing. Building on the quantized Lugiato-Lefever equation (QLLE), which predicts steady-state coherent state superpositions in multi-mode Kerr resonators, and the framework for engineering Kerr nonlinearity ratios to mitigate spectral crowding in coupled oscillators, we propose an integrated approach for precise quantum control. By engineering the Kerr ratio K1/K2 to approximate an incommensurate value using complex rational approximations, we eliminate parasitic degeneracies and enable selective addressing of transitions in the QLLE steady state. Employing a Magnus expansion, we derive an effective Hamiltonian that incorporates Stark-shift corrections, facilitating deterministic synthesis of entangled cat states and Fock states with fidelities exceeding 99.9%. Numerical simulations using QuTiP validate robustness against dissipation and thermal noise, demonstrating quantum hysteresis and Wigner negativity in multi-mode configurations. This synthesis bridges dissipative phase transitions with architectural control, offering a blueprint for scalable bosonic quantum processors in circuit QED. The Wigner function plays a pivotal role in characterizing these nonclassical states, providing a phase-space representation that reveals quantum interference effects. In our framework, the emergence of negative values in the Wigner function for the central mode confirms the non-Gaussian nature of the cat states, as dictated by Hudson’s theorem, which states that only Gaussian pure states have non-negative Wigner functions everywhere [21]. This negativity arises from the interference term in the cat-state Wigner expression, quantifying macroscopic quantum coherence and serving as a witness for entanglement in multi-mode systems.For instance, our simulations at F = 2.0 yield a minimum Wigner value of approximately -0.006, highlighting the system’s departure from classical behavior and its utility for bosonic error correction, where cat codes leverage phase-space separation to suppress bit-flip errors exponentially with cat size. Furthermore, the relation to Wigner functions extends to dynamical properties: the steady-state Wigner negativity is robust against thermal noise due to the dissipative stabilization mechanism, aligning with results from driven Bose-Hubbard models where two-mode cat states maintain negativity under loss [46]. In the QLLE context, the multi-mode extension introduces spatial localization, where the Wigner function’s negative regions correspond to soliton-like structures, as evidenced by photon population distributions [0.33, 0.90, 0.33]. This negativity is not merely a feature but a consequence of fundamental limits on classical simulability, relating to the Gottesman-Knill theorem’s extension to continuous variables, where non-Gaussian operations are necessary for universal quantum computation. Quantum hysteresis, observed in our hysteresis plots, is intimately linked to the Wigner representation through the bifurcation dynamics. Near the codimension-2 point, the system’s bistability manifests as distinct paths in phase space, with Wigner functions switching between Gaussian-like (low-photon) and cat-like (high-photon) forms, supported by spectral theory of Liouvillians that guarantees metastable states in dissipative phase transitions [30]. The forward and backward sweeps in our simulations demonstrate this path dependence, with the Wigner negativity peaking in the high-drive regime, underscoring the role of Kerr nonlinearity in engineering quantum resources. Overall, the Wigner function’s negativity serves as a diagnostic tool for the efficacy of our engineered ratios, directly tying to experimental verifiability in circuit QED setups, where homodyne measurements can reconstruct Wigner distributions to confirm cat-state fidelity. This work not only advances control protocols but also highlights the profound interplay between phase-space quasi-probabilities and dissipative quantum engineering, paving the way for fault-tolerant bosonic qubits.
Continuous-variable quantum information processing necessitates the generation and manipulation of non-Gaussian photonic states, such as higher-order generalized squeezed states and cubic phase states. These non-Gaussian states are distinguished from conventional Gaussian states by their pronounced Wigner negativity in phase space. However, generating these states with large Wigner negativity remains experimentally challenging, hindering their practical applications. Here, we experimentally generate non-Gaussian generalized microwave squeezed states (including tri- and quad-squeezed states) and cubic phase states within a high-quality superconducting cavity. This is achieved by engineering generalized photon-number filter operations in a programmable parameterized quantum circuit that efficiently projects cavity states onto target non-Gaussian states. These generalized squeezed photonic states are demonstrated to exhibit large Wigner negativity and quantum Fisher information as the squeezing order and squeezing parameter increase, thereby confirming their non-Gaussianity and highlighting potential applications in continuous-variable quantum information processing and quantum metrology.
We experimentally realize trigonometric continuous-variable gates on a trapped-ion processor, for which a motional mode acquires a phase proportional to the cosine of its position quadrature, and, for the first time, implement the two-mode generalization, coupling two modes through a single nonlinear phase. Such gates provide an experimentally accessible, nonpolynomial primitive for periodic interactions acting on both compact and noncompact degrees of freedom, including rotor models, sine-Gordon-type systems, and lattice gauge theories. Scanning gate strength, spatial frequency, and circuit depth, we resolve via blue-sideband spectroscopy the parity selection rule that fingerprints the exact cosine evolution, and find that an open-system model incorporating residual thermal occupation and motional dephasing reproduces the data. We then derive the asymptotics of the Wigner negativity generated by these gates and find three scaling regimes. The negativity is beyond all algebraic orders in the gate strength while the negative regions sit in far phase-space tails, becomes linear once they reach the bulk, where it saturates a first-order bound we establish, and logarithmic at strong gate strength. These results expose a general mechanism, first identified here through the cosine gate, by which every finite-order perturbative estimate of a non-Gaussian resource can vanish even though the resource itself remains nonzero. Together, our results establish trigonometric gates as controllable, experimentally realizable building blocks for bosonic quantum simulation, expanding the class of nonlinear dynamics accessible to continuous-variable quantum processors.
Quantum non-Gaussian states are crucial for the fundamental understanding of non-linear bosonic systems and simultaneously advanced applications in quantum technologies. In many bosonic experiments, the important non-Gaussian quantum feature is the negativity of the Wigner function, a cornerstone for quantum computation with bosons. Unfortunately, the negativities present in complex quantum states are extremely vulnerable to the effects of decoherence, such as energy loss, noise, and dephasing, caused by the coupling to the environment, which is an unavoidable part of any experimental implementation. An efficient way to mitigate its effects is by adapting quantum states into more resilient forms. We propose optimal protection of superpositions of coherent states against a sequence of asymmetric thermal lossy channels by suitable squeezing operations.