Abstract
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 non-trivial 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.
Highlights
With the advent of Noisy Intermediate-Scale Quantum Computing (NISQ) devices [1] and quantum computational advantage [2,3], it becomes of paramount importance to identify resourceful architectures—i.e., those that are capable of yielding quantum speed-up for computation [4]— and to distinguish them from those that cannot
III we introduce the basic formalism for DV systems of different dimensions, and we review how to encode qubits into qudits and more generally systems of dimension d1 into systems of dimension d2, following schemes developed in the context of quantum error correction
VI we develop our framework for assessing the simulatability of continuous variables (CVs) architecture based on embedding the logical quantum information of lower-dimensional systems into higher-dimensional systems and demonstrate our main results
Summary
With the advent of Noisy Intermediate-Scale Quantum Computing (NISQ) devices [1] and quantum computational advantage [2,3], it becomes of paramount importance to identify resourceful architectures—i.e., those that are capable of yielding quantum speed-up for computation [4]— and to distinguish them from those that cannot. The most used criteria for setting a boundary to the power of CV quantum computing architectures, in analogy to the Gottesman-Knill theorem, are based on the positivity of quasi-probability distributions [24], and in particular of the Wigner function: It always exists a classical algorithm that can simulate efficiently the output of a quantum circuit with input states, unitary operations, and measurements described by non-negative Wigner functions [25,26]. No CV circuit that uses bosonic codes to process information can be assessed using the aforementioned criteria, regardless of whether it can provide quantum computational advantage or not To exemplify this impasse, consider a CV architecture composed of initial stabilizer GKP-encoded states, over which encoded Clifford operations and computational basis measurements act.
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