So p(n)=(1/4)*(1-(1/3)^n) which tends to 1/4 for large n.

The expected value Sum(n*p(n)) is infinite. ]]>

In my opinion, the general context to think about Formula (77) is not necessarily “experimental findings in computational complexity” but rather in theory and practice of reliability theory. A crucial question is also if other implementations of quantum computers (like IBM’s) exhibit a similar behavior.

(A small correction to my earlier remark. There is also some possibility that the computational task exhibited by the Google experiment is much easier computationally than what we expect at present. But, my intuition is that this is not the issue with the supremacy claim.)

]]>“Finally it seems to indicate that your hypothesis about there being unavoidable noise at a rate too high for fault tolerance to work is wrong.”

Right, as I said in the post, if the Google claims are correct then my argument that there being unavoidable noise at a rate too high for fault tolerance to work is wrong. Even by formula (77) you will need better (even substantially better) gate performances for good quality quantum error-correcting codes and certainly for quantum fault-tolerance. But my argument that this is impossible is based on the claim that even achieving the noise-rate for quantum supremacy is impossible.

Nice poem!

]]>The real question remains whether quantum fault tolerance works, and how well, which I believe is one of the next items on the experimentalists’ list.

]]>.

Here, is the number of qubits, is the number of 1-gates and is the number of 2-gates. For the full circuit with layers and . (For the simplified circuits “elided” and “patch” circuits the number of gates is somewhat different.)

The formula expresses independence of the fidelity from computational complexity and from entanglement. (This assertion actually agrees with my own intuition: for example, I don’t expect a difference between noisy boson sampling and noisy fermion sampling which in the noiseless case are very different computationally. I expect strong effect of entanglement on error correlation but not on the fidelity.)

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