Resources for Bosonic Quantum Computational Advantage

Quantum computers promise to dramatically outperform their classical counterparts. However, the nonclassical resources enabling such computational advantages are challenging to pinpoint, as it is not a single resource but the subtle interplay of many that can be held responsible for these potential advantages. In this Letter, we show that every bosonic quantum computation can be recast into a continuous-variable sampling computation where all computational resources are contained in the input state. Using this reduction, we derive a general classical algorithm for the strong simulation of bosonic computations, whose complexity scales with the non-Gaussian stellar rank of both the input state and the measurement setup. We further study the conditions for an efficient classical simulation of the associated continuous-variable sampling computations and identify an operational notion of non-Gaussian entanglement based on the lack of passive separability, thus clarifying the interplay of bosonic quantum computational resources such as squeezing, non-Gaussianity, and entanglement.

Figure 1

To any bosonic computation (left, in blue) is associated a coherent state sampling setup (right, in orange), which takes as input the same state $\widehat{\rho }$, together with auxiliary single-photon Fock states, and whose output probability density approximates to arbitrary precision the output probability of a given outcome up to normalization, i.e., $P({\alpha }_1,\dots ,{\alpha }_m,0,\dots ,0)\approx (1/{\mathcal{N}}^{\prime })P(y_1,\dots ,y_m)$. The number of auxiliary Fock states $n$ is the sum of the stellar ranks of the projectors associated with the outcomes $y_1,\dots ,y_m$.