The Quantum Debate is Over! (and other Updates)

Quid est noster computationis mundus?

Nine months after is started, (much longer than expected,) and after eight posts on GLL, (much more than planned,)  and almost a thousand comments of overall good quality,   from quite a few participants, my scientific debate with Aram Harrow regarding quantum fault tolerance is essentially over. Some good food for thought, I hope. As always, there is more to be said on the matter itself, on the debate, and on other”meta” matters, but it is also useful to put it now in the background for a while, to continue to think about quantum fault tolerance in the slow pace and solitude, as I am used to, and also to move on in other fronts, which were perhaps neglected a little.

Here are the links to the eight posts: My initial post “Perpetual Motion of The 21st Century?” was followed by three posts by Aram. The first “Flying Machines of the 21st Century?” the second “Nature does not conspire” and the third “The Quantum super-PAC.” We had then two additional posts “Quantum refutations and reproofs” and “Can you hear the shape of a quantum computer?.” Finally we had  two conclusion posts: “Quantum repetition” and “Quantum supremacy or classical control?

EDP 22-27

We had six new posts on the Erdos Discrepancy Problem over Gowers’s blog (Here is the link to the last one EDP27). Tim contributed a large number of comments and it was interesting to follow his line of thought.  Other participants also contributed a few comments. One nice surprise for me was that the behavior of the hereditary discrepancy for homogeneous arithmetic progression in {1,2,…,n} was  found by Alexander Nikolov and  Kunal Talwar. See this post From discrepancy to privacy and back and the paper. Noga Alon and I showed that it is ${\tilde{\Omega}(\sqrt{\log n})}$ and at most ${n^{O(\frac{1}{\log\log n})}}$, and to my surprise Alexander and Kunal showed that the upper bound is the correct behavior. The argument relies on connection with differential privacy.

Möbius randomness and computation

After the $AC^0$-prime number theorem was proved by Ben Green, and the Mobius randomness of all Walsh functions and monotone Boolean function was proved by Jean Bourgain, (See this MO question for details) the next logical step are low degree polynomials over Z/2Z . (The Walsh functions are degree 1 polynomials.) The simplest case offered to me by Bourgain is the case of the Rudin-Shapiro sequence. (But for an ACC(2) result via Razborov-Smolensky theorem we will need to go all the way to polynomial of degree polylog.) I asked it over MathOverflaw. After three months of no activity I offered a bounty of 300 of my own MO-reputations. Subsequently, Terry Tao and Ben Green discussed some avenues and eventually Tao solved the problem (and earned the 300 reputation points). Here is a very recent post on Möbius randomness on Terry Tao’s blog.

Influences on large sets

In my post Nati’s Influence I mentioned two old conjectures (Conjecture 1 dues to Benny Chor and Conjecture 2) about influence of large sets on Boolean functions. During Jeff Kahn’s visit to Israel we managed to disprove them both. The disproof is inspired by an old construction of Ajtai and Linial.

Tel Aviv, New Haven, Jerusalem

Last year we lived for a year in Tel Aviv which was a wonderful experience: especially walking on the beach every day and feeling the different atmosphere of the city. It is so different from my Jerusalem and still the people speak fluent Hebrew. I am now in New Haven. It is getting cold and the fall colors are getting beautiful. And it also feels at home after all these years. And next week I return to my difficult and beautiful  (and beloved) Jerusalem.

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3 Responses to The Quantum Debate is Over! (and other Updates)

1. Shmuel Weinberger says:

Gil, although Chicago is not Jerusalem, you are always welcome here!

2. valuevar says:

MathOverflaw? That’s very nice, but what did it do to deserve it?