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The superconducting quantum supremacy demonstrations serve as crucial milestones on the path to full-scale quantum computation. On the other hand, the choice of technologies used in the photonic machines1,2, and their consequential lack of programmability and scalability, places them outside any current proposed roadmap for fault-tolerant photonic quantum computing9,10,11 or any GBS application12,13,14,15,16,17,18. A demonstration of photonic quantum computational advantage incorporating hardware capabilities required for the platform to progress along the road to fault-tolerance is still lacking.
For Gaussian states undergoing only common loss (including the special case of lossless GBS), it is straightforward to show that the third-order photon-number cumulants involving any three distinct modes are all strictly zero. Thus, the fact that significant third- and fourth-order cumulants are observed in refs. 1,2 is simply a reflection of the fact that most of their inputs are vacuum and that their experiment lacks photon-number resolution. The latter observation could in principle be exploited by a classical adversary to speed up the simulation of GBS with mostly vacuum inputs because strategies exist to speed up the simulation of GBS when the number of input squeezed states is fixed and is a small fraction of the total number of photons observed. These strategies used the fact that hafnians of low-rank matrices32,33 can be calculated faster than hafnians of full rank matrices of equal size. For our system, the matrices needed for simulation are all full rank as every input is illuminated with squeezed light.
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