Efficient simulatability of continuous-variable circuits with large Wigner negativity

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 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.

Classical simulation of Gaussian quantum circuits with non-Gaussian input states

“…Our results are complementary to those in a recently appeared work [7], where non-Gaussian states with unbounded Wigner negativity supplemented to Gaussian circuits are also shown to be classically efficiently simulable. In that work, the simulatability with input unbounded non-Gaussianity, namely with states characterized by infinite stellar rank, is possible due to the fact that the input states are discrete-variable stabiliser states encoded in CV by means of some bosonic encoding, such as for instance the Gottesman-Kitaev and Preskill one [9].…”

“…Since Gaussian states and processes have positive Wigner functions, this necessarily corresponds to the use of non-Gaussian resources. However, establishing under which conditions non-Gaussianity is also sufficient for quantum advantage [6], and when instead non-Gaussian circuits are classically efficiently simulable [7], is still an open question.…”

Witnessing Wigner Negativity

“…In addition to its fundamental relevance as a non-classical property of physical systems [18], Wigner negativity is also essential for quantum computing, since continuous-variable quantum computations described by positive Wigner functions can be simulated efficiently classically [19]. Wigner negativity is thus a necessary resource, though not sufficient [20], for quantum computational speedup with continuous variables.…”

Conditional non-Gaussian quantum state preparation

“…They have been shown to be easily simulated on classical devices [33], and in particular Wigner negativity is known to be a necessary resource for reaching a quantum computation advantage [34]. However, it should be stressed that recent work has found large classes of Wigner negative states that can also be simulated easily [35]. In other words, Wigner negativity is necessary but not sufficient to reach a quantum computation advantage [36].…”