**Guy Kindler and I identified a very primitive complexity class LDP that describes noisy quantum systems in the small scales (few bosons, qubits etc.) **

Today I would like to discuss, with the help of illustrations, a central issue in the debate about quantum computing: In view of a putative impossibility (and obvious difficulty) of quantum computing, why is classical computing possible (and so common)? This was a major challenge that Aram Harrow raised in our 2012 debate (in this post), and it goes back to Dave Bacon and others. My 2014 work with Guy Kindler on noisy BosonSampling (taking them as an analogy for noisy quantum circuits) leads to a fairly detailed answer that I will explain below, mainly with illustrations.

**A common view: in principle, there is no difference between achieving quantum computing via quantum error-correcting codes and achieving classical computing via classical error-correcting codes. **

**Encoding by repetition, and decoding by “majority” are both supported by low degree polynomials**

Encoding by repetition refers to a logical ‘one’ encoded by many physical ones and the same with zero. Approximate version of majority enables decoding. Both encoding and decoding are supported by low degree polynomials. (A variant which is more realistic is that one is encoded by 70% ones (say) and zero by 30% ones.)

**It is commonly expected that creating good-quality quantum error correcting codes is harder than demonstrating quantum supremacy.**

Unlike the repetition/majority mechanism which is supported by very primitive computational power, creating a quantum error correcting code and the easier task of demonstrating quantum supremacy are not likely to be achieved by devices which are very low-level in terms of computational complexity.

This is the difference!

(Quantum supremacy refers to the ability of quantum computers to perform computations which are infeasible for classical computers. )

Bounded depth computation () is an extremely primitive computational complexity class. Sampling tasks that can be performed approximately by low degree polynomials represent an even more primitive computational task. (The proof that LPD is in is quite clever but seems to Guy and me quite irrelevant to quantum computing )

LDP is so low-level that it allows efficient learning!

(This leads to the following prediction: distributions coming from pure quantum states that can be approached by noisy quantum devices allows efficient approximate learning.)

**Very simple, isn’t it?**

The notion of “primitive computational devices” in both principles applies to the asymptotic behavior. The second principle extends to quantum devices that do not involve quantum error-correction.

Let me try. (Trying to understand how insights from the theory of computation relate to real life computation is quite important.)

I don’t know how general is this insight. (I asked about it in TCSoverflow.) Note that this insight gave the rationale for the threshold theorem to start with.

My earlier work (Section 6 of the Notices paper, and earlier works cited there) proposes other goals that appear to be easier than creating quantum error correcting codes and I expect them to be impossible.

The remarkable IBM machine applies two-qubit CNOT gates. One can expect that errors for the gated qubits have large positive correlation. This is not controversial but it would be nice to check it.

You can implement CNOT gate indirectly by a chain of other gates. I expect that it will not be possible to reduce the positive correlation by a process that reaches the CNOT indirectly. (You can experience the IBM quantum computer here.)

**It is a positive and not terribly small constant! I am not sure what “fundamental” means.**

(Sometimes, I get corrected when I pronounce my name correctly.)

Q: But why can’t you simply create good enough qubits to allow universal quantum circuits with 50 qubits?

A: This will allow very primitive devices (in terms of the asymptotic behavior of computational complexity) to perform superior computation.

Q: This is an indirect explanation. Can you give an explanation on the level of a single qubit?

A: Yes, if the qubit is of too high quality, the microscopic process leading to it also demonstrates a primitive device leading to a superior computation.

**What about topological quantum computing?**

**My model of errors is precisely the ordinary model.**

**Also the Commonwealth Bank**

(Inspiration: math with bad drawing)

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Sergiu Hart raises a very interesting idea regarding elections. Consider the Brexit referendum. Sergiu proposes to have two rounds two weeks apart. Every voter can vote in each, and the votes of both rounds add up! The outcomes of the first round are made public well before the second round.

In the paper Repeat Voting: Two-Vote May Lead More People To Vote Segiu Hart suggests the following method:

**A. Voting is carried out in two rounds.**

**B. Every eligible voter is entitled (and encouraged) to vote in each one of the two rounds.**

**C. All the votes of the two rounds are added up, and the final election result is obtained by applying the current election rules to these two-round totals.**

**D. The results of the first round are officially counted and published; the second round takes place, say, two weeks after the first round, but no less than one week after the official publication of the first round’s results**

(The pun in the title is taken from an early version by Sergiu.)

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**Alistair and the Simons Institure friendly and helpful staff **

Luca Trevisan invited me to give a 3-minute (vidotaped or live) toast for Alistair Sinclair to celebrate that Alistair much deservedly received the SIGACT service award and to mourn that he also just retired from his role of associate director of the Simons Institute. These toasts will be part of FOCS 2017 Saturday evening reception (October 14) which will be hosted by the SI and turn into a party for Alistair. It is great that leading scientists excel also in the all so important academic administration tasks. I can think in this context about Michael Rabin who was the rector (provost) of my university and many others.

But I thought it could be a good idea to mainly mention in my toast two of Alistair’s great and mind-boggling scientific contributions. Miraculously, I had a videotaped which was lying on the editing floor for three years. It was part of the first ever (and maybe the only ever) Simons Institute videotaped production. (The toast video was done by me using my smartphone and it took many takes to do.) So you can hear for one minute and a half quick stories about rapid mixing and approximate permanents and inspiring persistence and volume.

—Click here for the toast video for Alistair!—

This brief video was edited out from my videotaped lecture on quantum computers (see this earlier post). When we prepared the videos, I was quite excited by the fact that we do not need to shoot the video in the right order. (I even use this fact to outline a major difference between classical computation and quantum computation.) We first shot the pictorial + entertaining ending of Video II. Then we moved to shoot Video I and the plan was to start it with Alistair Sinclair introducing me. In the beginning of the unedited video, I say to Tselil in Hebrew that we do not need to bring Alistair up to the production room since he can record his introduction later. (At the end we decided not to have an introduction at all.) As seen from my smile, when I thanked Alistair for his yet-to-be-given introduction I felt as sophisticated as Niccolò Machiavelli.

And here are links to the papers mentioned: Approximating the Permanent by Jerrum and Sinclair, A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries by Jerrum, Sinclair and Vigoda (and here is the Journal version), and A random polynomial time algorithm for approximating the volume of convex bodies by Dyer, Frieze and Kannan.

Indeed these amazing algorithmic applications of rapidly mixing random walks came as great surprises and are extremely important!

Talking about my old Simon Institute lectures videos, let me first recommend to you the two minutes ending of the second video – lovely pictures and a wonderful song in the background. Can you identify the people there? (There are 49 altogether!)

The second video includes (toward the end, click here to jump to this part) a presentation of some pictures +funny things related (more or less) to the debate on quantum-computers. It contains (click to jump) a forgotten picture of Avi Wigderson and Oded Goldreich, and a minute later you can see a picture with Terry Tao and hear my clumsy editing effort to change the audio from “propaganda” to “public relation.” This follows with a “kililish” (ululations) (Click to jump) for my daughter’s wedding, and then my response (click here to jump to it) to Scott Aaronson’s famous $100,000 challenge for an argument he will find convincing that quantum computers cannot work, and a discussion in the context of QC of Jesus’s statement regarding rich people, camels and needles.

Also left on the editing floor was a thorough comparison of Scott’s challenge to a “ketubah” where one guarantees to pay a certain amount of money when he divorces some idea he fell in love with in his youth, and a thorough discussion on if and how Jesus’s statement that it is harder for rich people to enter heaven than for a camel to pass through the eye of a needle refers also to Jim Simons the founder of the Simons Institute at Berkeley.

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Ropemaker (source)

Rados Radoicic wrote me:

“Several years back, I heard the following puzzle that turns out to be rather ‘classical’:

“There are N ropes in a bag. In each step, two rope ends are picked uniformly at random, tied together and put back into a bag. The process is repeated until there are no free ends. What is the expected number of loops at the end of the process?”

Before moving on, please try to answer this question.

Let me go on with Rados’ email:

“Upon hearing this puzzle, I came up with and have been wondering (for years now) about the following “natural” variation:

“What if at each step, each end is picked with the probability proportional to the length of its rope?”

I made no epsilon-worthy progress on this problem since then. A properly-trained probabilist (unlike me) might find it easy, but somehow my gut feeling tells me it may be very interesting and not simple at all.”

The challenges are to test yourself on the first question and try to answer the second. No polls this time; comments and thoughts on both versions are welcome.

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(From right): Udo Pachner, Peter Kleinschmidt, and Günter Ewald. (Oberwolfach photo collection)

It is possible that the link of a vertex in a simplicial sphere is no longer a sphere. This is not good if we look for a proof by induction. A subfamily which is closed under links is the piecewise linear spheres. Another advantage of this family for inductive proof is given by Pachner: any PL--sphere can be obtained from the boundary of the -simplex by a finite sequence of bistellar moves aka Pachner moves. This is a family of possible local moves, defined combinatorially.

However, so far no proof of the -conjecture for PL-spheres is known.

A superfamily of simplicial spheres, closed under links, is that of *homology spheres*, where we just require that the *homology* of all face links are as of the sphere of the appropriate dimension. The hard Lefschetz property is conjectured for this family too.

What we can show is that the hard Lefschetz property is preserved under some combinatorial constructions on spheres, namely: connected sum, join, stellar subdivision – or more generally the inverse moves of topology preserving edge contractions, which are exactly the admissible contractions defined above for *minors*. Let us just mention two remarks:

1. These last moves played a role in Murai’s refinement of Billera-Lee theorem. He showed that any squeezed sphere can be obtained from the boundary of the simplex in this way, hence squeezed spheres have the hard Lefschetz property.

2. Any PL-sphere can be obtained from the boundary of a simplex by a finite sequence of stellar and inverse-stellar moves, so it remains to show that inverse-stellar moves on PL-spheres preserve the hard Lefschetz property.

is *doubly Cohen-Macaulay* (2-CM) if is CM and deleting any vertex of , the induced complex on the rest of the vertices is also CM of the same dimension as . Examples include all homology spheres, as well as other examples where the vector is known to be an -vector. This led Björner and Swartz to conjecture that the -vector of any 2-CM complex is an -vector. Note that the -vector need not be symmetric anymore. Algebraically, the *weak Lefschetz* property may hold, i.e. injections

for .

What we can show is only that is an -vector for 2-CM complexes. This is done using *rigidity theory for graphs*, to be discussed next time.

Another case were is an -vector is when is the barycentric subdivision of a simplicial sphere, or more generally of any CM complex!

This was shown recently in a work with Martina Kubitzke, by proving a hard Lefschetz type result.

Let a finite triangulation of a connected orientalble -manifold without boundary. We have seen that and depends on the Euler characteristic of , so may not be symmetric. Neither it is an -vector.

Can we fix that?

Kalai defined a new vector which is a function of and of the Betti numbers of

where is the dimension of the reduced j-th homology of with coefficients.

The vector is symmetric and an -vector, and recently Isabella Novik and Ed Swartz figured out its algebraic meaning:

The *socle* of a graded standard ring is the ideal of elements in annihilated by the maximal ideal (generated by the variables). Let be generic elements on , so they make as small as possible (it is a finite vector. Such , that makes this quotient ring of zero Krull-dimension is called a linear system of parameters).

Let be the part of the degree at most in the socle of this ring. Then

Now define and for as for the -vector.

Kalai conjectured that is an -vector, and it is possible that has the hard Lefschetz property. Novik and Swartz showed further that the hard Lefschetz property for the vertex links of implies that is an -vector.

So we are back to the algebraic -conjecture.

To end, even is not known for say PL-5-spheres.

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This remarkable 3D geometric object tiles space! It is related to a theory of “spacial networks” extensively studied by Michael Burt and a few of his students. The network associated to this object is described in the picture below.

And below you can see the first step in tiling the whole space: How two tiles fit together.

Michael Burt started finding and classifying related objects in his 1967 Master thesis. My academic grandfather Branko Grunbaum (below left) helped him in his early works. The thesis was so impressive that Michael (below, right) was awarded a doctorate for it.

You can find more about it in Michael’s site. The site includes slides for various lectures like the one on UNIFORM NETWORKS, SPONGE SURFACES AND UNIFORM SPONGE POLYHEDRA IN 3-D SPACE.

Below are some more pictures from Michel Burt’s home. In one of them you can see Michael, his wife Tamara, and my friend since university days Yoav Moriah who initiated the visit.

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———–

It was high time to raise the level of the discussion, I thought. Princeton, Fall 1995. We were a group of mathematicians at the IAS lunch table, and discussing a really profound idea was called for. But I was missing a word and I asked Avi Wigderson what is the English word for a lion’s hair. Avi replied “layish”.

This was precisely what I was missing. It was time to voice my new thought:

” Isn’t it the case and isn’t it amazing that Avi’s hair looks just like a layish?”, I asked

Pierre Deligne looked busy thinking about his own business, but Enrico Bombieri looked interested and he was trying to understand what I was saying. Robert Langlands seemed to have got it – he was nodding his head in agreement – or so I thought. I also sensed a positive reaction from Bob MacPherson. Well, it takes time for great ideas, however simple, to get through.

But as is often the case, one great idea led to another.

Isn’t it amazing that the word “layish” means in Hebrew “a lion,” a meaning so close to the English meaning? I asked

While trying to further repeat, promote and discuss these two ideas a conflicting piece of information came to my mind. Another word for a lion’s hair in English was something like “main” or “mane”, I started to vaguely remember. When I asked Avi about it, he could not hold his laughter any longer, and the sad reality had emerged.

The second great idea was just an artifact of Avi’s juvenile behavior, a behavior that also had a devastating effect on the presentation of the first idea. The first idea, as great as it might be, failed to come through due to problematic notation. It should have waited for another opportunity. And now, 22 years later this opportunity has come!

The observation regarding Avi’s hair is not isolated. Nati Linial already noted in AviFest, Avi’s resemblance to the famous Georgian writer Shota Rustaveli and mention also the resemblance to Bin Laden – a connection which was first made when Avi, shortly after 9/11, entered a stand up joint in NYC, and the comedian immediately stopped everything and said: “Ladies and gentlemen, Osama Bin Laden”! The shape of Rustaveli’s layish remains however, a mystery.

Shota Rustaveli

From Nati’s videotaped lecture

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Puzzles on trees, high dimensions, elections, computation and noise.

A videotaped lecture is here. An earlier Hebrew version is

**חידות על עצים , ממדים גבוהים, בחירות, חישוב ורעש **

(It deals only with the first four puzzles.) A short videotaped Hebrew lecture (where I only discuss the first two puzzles and touch on the third) is here.

A drawing by my daughter Neta for puzzle number 3. (Based on a famous Florida recount picture.)

**Summary:** I will talk about some mathematical puzzles that have preoccupied me over the years, and I will also reveal to you some of the secrets of our trade. The first puzzle we shall discuss is about high-dimensional trees: what they are and how to count them. The second puzzle deals with high-dimensional geometric bodies, and a question of Borsuk. The third puzzle is about errors made when counting votes during elections, and the fourth puzzle raises the question: are quantum computers possible? I will conclude with a puzzle that I am currently thinking about: random RNA trees.

I am very curious about how accessible the paper is, and comments both on the presentation and content are most welcome. I plan to devote soon one post to a fresh look on each puzzle, and I also have posted about them in the past- Puzzle 1 (1,2,3,4,5), Puzzle2 (1,2,3), Puzzle 3 (1,2), and Puzzle 4. The fifth puzzle is closely related to TYI 29 about the diameter of various random trees. (Since writing the paper, I have learnt more about earlier relevant research.) I am also planning to write a similar but more technical version for mathematicians (perhaps for the ICM2018 Proceedings).

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SESAME (**S**ynchrotron-light for **E**xperimental **S**cience and** A**pplications in the **M**iddle **E**ast) is a “third-generation” synchrotron light source that was officially opened in Allan (Jordan) on 16 May 2017. It is the Middle East’s first major international research center. It is very nice to see scientific cooperation between countries which otherwise do have some difficulties in their relationships.

Narrowing on Israel and Iran, there are many personal scientific friendships and collaborations between Israeli and Iranian scientists and it will certainly be nice to see in the future visits of Israeli scientists to Iran and of Iranian scientists to Israel.

Last July, Marina Ratner and Maryam Mirzakhani, two eminent mathematicians working in the area of dynamics passed away. Maryam was the first woman to win the Fields Medal, and Marina was, for a short time, a faculty member at my department and a close friend of the Hebrew University ever since. Many of my colleagues knew both Marina and Maryam very well, and were great admirers of their personalities as much as their mathematics. Amie Wilkinson wrote a beautiful NYT article about Marina and Maryam’s mathematics: With Snowflakes and Unicorns, Marina Ratner and Maryam Mirzakhani Explored a Universe in Motion. (See also this HUJI memorial page, this post on Terry Tao’s blog, and this post on GLL.)

Amir Asghari had a nice idea to open a site dedicated to maths friendships in memory of Maryam Mirzakhani. Let me also mention that there is a well known Israeli Facebook group for “digging into mathematics” that dedicated its profile picture to Maryam.

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TYI 30 asked Elchanan Mossel’s Amazing Dice Paradox (that I heard from Yuval Peres yesterday)

You throw a die until you get 6. What is the expected number of throws (including the throw giving 6)** conditioned** on the event that all throws gave even numbers?

Most people answered 3.

Is it the right answer?

Please use now the comments thread to offer your answers, explanations, insights, intuition, thoughts and after-thoughts. I am especially eager to hear your take, James Martin! For a nice explanation by Paul Cuffis, see this comment by Yuval.

Comments on the English dilemma between “a die” or “a dice” are also welcome.

(Let me also draw your attention to TYI 29 about exciting models of random trees.)

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