Second, apparently the joke about “Bosons having harder time” was first made by Tishby in response to a lecture by Wigderson on the computational complexity of permanents in the mid 90s, and Tishby;s joke have led to his 1996 paper with Troyansky. Of course, finding the precise historical origins of jokes and giving correct credit for them is extremely important. ]]>

Also I want to stress the conclusion: all the hardness results for exact quantum sampling applies for fault-tolerant quantum computers.

Finally, there are good theoretical evidence (similar to the argument in Scott and Alex’ paper) that scenario b) is correct. quantum sampling is considerably stronger computational class than BQP. (E.g. evidence that a classical computer with BQP oracle cannot perform quantum sampling.)

]]>Of course, we do not know to implement quantum fault tolerance within the very limited classes of quantum sampling (boson sampling, bounded depth polynomial size quantum circuits, …) for which PH collapse results are proved.

Let me also mention a nice essay by Scott on quantum sampling https://www.scottaaronson.com/blog/?p=3427 with references (and links) also to earlier relevant results by Fenner et al. and Terhal and DiVincenzo.

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