Discriminating between quantum computing architectures that can provide quantum advantage from those that cannot is of crucial importance.From the fundamental point of view, establishing such a boundary is akin to pinpointing the resources for quantum advantage; from the technological point of view, it is essential for the design of nontrivial quantum computing architectures.Wigner negativity is known to be a necessary resource for computational advantage in several quantum-computing architectures, including those based on continuous variables (CVs).However, it is not a sufficient resource, and it is an open question under which conditions CV circuits displaying Wigner negativity offer the potential for quantum advantage.In this work we identify vast families of circuits that display large, possibly unbounded, Wigner negativity, and yet are classically efficiently simulatable, although they are not recognized as such by previously available theorems.These families of circuits employ bosonic codes based on either translational or rotational symmetries (e.g., Gottesman-Kitaev-Preskill or cat codes) and can include both Gaussian and non-Gaussian gates and measurements.Crucially, within these encodings, the computational basis states are described by intrinsically negative Wigner functions, even though they are stabilizer states if considered as codewords belonging to a finite-dimensional Hilbert space.We derive our results by establishing a link between the simulatability of high-dimensional discrete-variable quantum circuits and bosonic codes.
Discriminating between quantum computing architectures that can provide quantum advantage from those that cannot is of crucial importance.From the fundamental point of view, establishing such a boundary is akin to pinpointing the resources for quantum advantage; from the technological point of view, it is essential for the design of nontrivial quantum computing architectures.Wigner negativity is known to be a necessary resource for computational advantage in several quantum-computing architectures, including those based on continuous variables (CVs).However, it is not a sufficient resource, and it is an open question under which conditions CV circuits displaying Wigner negativity offer the potential for quantum advantage.In this work we identify vast families of circuits that display large, possibly unbounded, Wigner negativity, and yet are classically efficiently simulatable, although they are not recognized as such by previously available theorems.These families of circuits employ bosonic codes based on either translational or rotational symmetries (e.g., Gottesman-Kitaev-Preskill or cat codes) and can include both Gaussian and non-Gaussian gates and measurements.Crucially, within these encodings, the computational basis states are described by intrinsically negative Wigner functions, even though they are stabilizer states if considered as codewords belonging to a finite-dimensional Hilbert space.We derive our results by establishing a link between the simulatability of high-dimensional discrete-variable quantum circuits and bosonic codes.
Abstract In this work, we study the Wigner phase-space representation of qubit states encoded in continuous variables (CV) by using the Gottesman–Kitaev–Preskill (GKP) mapping. We explore a possible connection between resources for universal quantum computation in discrete-variable (DV) systems, i.e. non-stabilizer states, and negativity of the Wigner function in CV architectures, which is a necessary requirement for quantum advantage. In particular, we show that the lowest Wigner logarithmic negativity corresponds to encoded stabilizer states, while the maximum negativity is associated with the most non-stabilizer states, H-type and T-type quantum states.
(Abridged.) Quantum computers promise to solve some problems exponentially faster than traditional computers, but we still do not fully understand why this is the case. While the most studied model of quantum computation uses qubits, which are the quantum equivalent of a classical bit, an alternative method for building quantum computers is gaining traction. Continuous-variable devices, with their infinite range of measurement outcomes, use systems such as electromagnetic fields. Given this infinite-dimensional structure, combined with the complexities of quantum physics, we are left with a natural question: when are continuous-variable quantum computers more powerful than classical devices? This thesis investigates this question by exploring the boundary of which circuits are classically simulatable and which unlock a quantum advantage over classical computers. A series of proofs are presented demonstrating the efficient simulatability of progressively more complex circuits, even those with high amounts of Wigner negativity. Specifically, circuits initiated with highly Wigner-negative Gottesman-Kitaev-Preskill states, which form a grid-like structure in phase space, can be simulated in polynomial time.
Continuous-variable bosonic systems stand as prominent candidates for implementing quantum computational tasks. While various necessary criteria have been established to assess their resourcefulness, sufficient conditions have remained elusive. We address this gap by focusing on promoting circuits that are otherwise simulatable to computational universality. The class of simulatable, albeit non-Gaussian, circuits that we consider is composed of Gottesman-Kitaev-Preskill (GKP) states, Gaussian operations, and homodyne measurements. Based on these circuits, we first introduce a general framework for mapping a continuous-variable state into a qubit state. Subsequently, we cast existing maps into this framework, including the modular and stabilizer subsystem decompositions. By combining these findings with established results for discrete-variable systems, we formulate a sufficient condition for achieving universal quantum computation. Leveraging this, we evaluate the computational resourcefulness of a variety of states, including Gaussian states, finite-squeezing GKP states, and cat states. Furthermore, our framework reveals that both the stabilizer subsystem decomposition and the modular subsystem decomposition (of position-symmetric states) can be constructed in terms of simulatable operations. This establishes a robust resource-theoretical foundation for employing these techniques to evaluate the logical content of a generic continuous-variable state, which can be of independent interest. Published by the American Physical Society 2024
Characterizing the origin of quantum computational advantage amounts to identifying the key properties, or quantum resources, that distinguish quantum computers from their classical counterparts. Fault-tolerant computation, however, requires error-correcting codes to encode logical information into a larger, physical quantum system. This makes the task of resource identification more difficult, as what constitutes a resource from the logical and physical points of view can differ significantly. Here, we introduce a framework to correctly identify quantum resources in general encoded computations. For a given quantum code, our construction provides a Wigner function that accounts for how the symmetry of the code space is contained within the symmetry of the physical space, resulting in an object capable of describing the logical content of any physical state, both within and outside the code space. We illustrate our general construction with the Gottesman–Kitaev–Preskill encoding of qudits with odd dimension. The resulting Wigner function, which we call the Zak–Gross Wigner function, is shown to correctly identify quantum resources through its phase-space negativity. For instance, it is positive for encoded stabilizer states and negative for the bosonic vacuum. We further show that its negativity provides a measure of magic for the logical content of a state and that its marginals are modular measurement distributions associated with conjugate Zak patches.
The use of quantum information in technology promises to supersede the so-called classical devices used nowadays. Understanding what features are inherently non-classical is crucial for reaching better-than-classical performance. This thesis focuses on two nonclassical behaviours: quantum contextuality and Wigner negativity. The former is a notion superseding nonlocality that can be exhibited by quantum systems. To date, it has mostly been studied in discrete-variable scenarios. In those scenarios, contextuality has been shown to be necessary and sufficient for advantages in some cases. On the other hand, negativity of the Wigner function is another unsettling non-classical feature of quantum states that originates from phase-space formulation in continuous-variable quantum optics. Continuous-variable scenarios offer promising candidates for implementing quantum computations. Wigner negativity is known to be a necessary resource for quantum speedup with continuous variables. However contextuality has been little understood and studied in continuous-variable scenarios. We first set out a robust framework for properly treating contextuality in continuous variables. We also quantify contextuality in such scenarios by using tools from infinite-dimensional optimisation theory. Building upon this, we show that Wigner negativity is equivalent to contextuality in continuous variables with respect to Pauli measurements thus establishing a continuous-variable analogue of a celebrated result by Howard et al. We then introduce experimentally-friendly witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, using again tools from infinite-dimensional optimisation theory. We further extend the range of previously known discrete-variable results linking contextuality and advantage into a new territory of information retrieval.
We show that quantum circuits where the initial state and all the following quantum operations can be represented by positive Wigner functions can be classically efficiently simulated. This is true both for continuous-variable as well as discrete variable systems in odd prime dimensions, two cases which will be treated on entirely the same footing. Noting the fact that Clifford and Gaussian operations preserve the positivity of the Wigner function, our result generalizes the Gottesman-Knill theorem. Our algorithm provides a way of sampling from the output distribution of a computation or a simulation, including the efficient sampling from an approximate output distribution in the case of sampling imperfections for initial states, gates, or measurements. In this sense, this work highlights the role of the positive Wigner function as separating classically efficiently simulable systems from those that are potentially universal for quantum computing and simulation, and it emphasizes the role of negativity of the Wigner function as a computational resource.