Analysis of Boolean functions – week 2

Post on week 1; home page of the course analysis of Boolean functions

Lecture II:

We discussed two important examples that were introduced by Ben-Or and Linial: Recursive majority and  tribes.

Recursive majority (RM): F_m is a Boolean function with 3^m variables and F_{m+1} (x,y,z) = F_1(F_m(x),F_m(y),F_m(z)). For the base case we use the majority function F_1(x,y,z)=MAJ(x,y,z).

Tribes: Divide your n variables into s pairwise disjoint sets (“tribes”) of cardinality t. f=1 if for some tribe all variables equal one, and thus T=0 if for every tribe there is a variable with value ‘0’.

We note that this is not an odd function i.e. it is not symmetric with respect to switching ‘0’ and ‘1’. To have \mu=1/2 we need to set t=\log_2n - \log_2\log_2n+c. We computed the influence of every variable to be C log n/n. The tribe function is a depth-two formula of linear size and we briefly discussed what are Boolean formulas and Boolean circuits (These notions can be found in many places and also in this post.).

I states several conjectures and questions that Ben-Or and Linial raised in their 85 paper:

Conjecture 1: For every balanced Boolean function with n variables there is a variable k whose influence is \Omega (\log n/n).

Conjecture 2: For every balanced Boolean function with n variables there is a set S of n/log n variables whose influence I_S(f) is 1-o(1).

Question 3: To what extent can the bound in Conjecture 2 be improved if the function f is odd. (Namely, f(1-x_1,1-x_2,\dots, 1-x_n)=1-f(x_1,x_2,\dots, x_n).)

Our next theme was discrete isoperimetric results.  I noted the connection between total influence and edge expansion and proved the basic isoperimetric inequality: If μ(f)=t then I(f) ≥ 4 t(1-t). The proof uses the canonical paths argument.

Lecture III:

We proved using “compression” that sharp bound on I(f) as a function of t=μ(f). We made the analogy between compression and Steiner symmetrization – a classic method for proving the classical isoperimetric theorem. We discussed similar results on vertex boundary and on Talagrand-Margulis boundary (to be elaborated later in the course).

Then We proved the Harris-Kleitman inequality and showed how to deduce the fact that intersecting family of subsets of [n] with the property that the family of complements is also intersecting has at most 2^{n-2} sets.

The next topic is spectral graph theory. We proved the Hoffman bound for the largest size of an independent set in a graph G.

I mentioned graph-Laplacians and the spectral bound for expansions (Alon-Milman, Tanner)..

The proofs mentioned above are so lovely that I will add them on this page, but sometime later.

Next week I will introduce harmonic analysis on the discrete cube and give a Fourier-theoretic explanation for  the additional log (1/t) factor in the edge isoperimetric inequality.

Important announcement: Real analysis boot camp in the Simons Institute for the Theory of Computing, is part of the program in Real Analysis and Computer Science. It is taking place next week on September 9-13 and has three lecture series. All lecture series are related to the topic of the course and especially:

Analysis of Boolean Functions – week 1

Home page of the course.

In the first lecture I defined the discrete n-dimensional cube and  Boolean functions. Then I moved to discuss five problems in extremal combinatorics dealing with intersecting families of sets.

1) The largest possible intersecting family of subsets of [n];

2) The largest possible intersecting family of subsets of [n] so that the family of complements is also intersecting;

3) The largest possible family of graphs on v vertices such that each two graphs in the family contains a common triangle;

4) Chvatal’s conjecture regarding the maximum size of an intersecting family of sets contained in an ideal of sets;

5) Erdos-Ko-Rado Theorem.

Exercise: Prove one of the following

a) The Harris-Kleitman’s inequality

b) (from the H-K inequality) Every family of subsets of [n] with the property that every two sets have non-empty intersection and no full union contains at most 2^{n-2} sets.

More reading: this post :”Extremal combinatorics I: extremal problems on set systems“. Spoiler: The formulation of Chvatal’s conjecture but also the answer to the second exercise can be found on this post: Extremal combinatorics III: some basic theorems. (See also peoblen 25 in the 1972 paper Selected combinatorial research problems by Chvatal, Klarner and Knuth.)


I moved to discuss the problem of collective coin flipping and the notion of influence as defined by Ben-Or and Linial. I mentioned the Baton-passing protocol, the Alon-Naor result, and Feige’s two-rooms protocol.

More reading: this post :” Nati’s influence“. The original paper of Ben-Or Linial:  Collective coin flipping, M. Ben-Or  and N. Linial, in “Randomness and    Computation” (S. Micali ed.) Academic Press, New York, 1989, pp.    91-115.

BosonSampling and (BKS) Noise Sensitivity

Update (Nov 2014): Noise sensitivity of BosonSampling and computational complexity of noisy BosonSampling are studied in this paper by Guy Kindler and me. Some of my predictions from this post turned out to be false. In particular the noisy BosonSampling is not  flat and it does depend on the input matrix.  However when the noise level is a constant BosonSampling is in P, and when it is above 1 over the number of bosons, we cannot expect robust experimental outcomes.




Following are some preliminary observations connecting BosonSampling, an interesting  computational task that quantum computers can perform (that we discussed in this post), and noise-sensitivity in the sense of Benjamini, Schramm, and myself (that we discussed here and here.)

BosonSampling and computational-complexity hierarchy-collapse

Suppose that you start with n bosons each can have m locations. The i-th boson is in superposition and occupies the j-th location with complex weight a_{ij}. The bosons are indistinguishable which makes the weight for a certain occupation pattern proportional to the permanent of a certain n by n submatrix of the n by m matrix of weights.

Boson Sampling is a task that a quantum computer can perform. As a matter of fact, it only requires a “boson machine” which represents only a fragment of quantum computation. A boson machine is a quantum computer which only manipulates indistinguishable bosons with gated described by phaseshifters and beamsplitters.

BosonSampling and boson machines were studied in a recent paper The Computational Complexity of Linear Optics of Scott Aaronson and Alex Arkhipov (AA). They proved (Theorem 1 in the paper) that if (exact) BosonSampling can be performed by a classical computer then this implies a collapse of the computational-complexity polynomial hierarchy (PH, for short). This result adds to a similar result achieved independently by Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd (BJS) in the paper entitled: “Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy,” and to older results by  Barbara Terhal and David DiVincenzo (TD) in the paper Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games, Quant. Inf. Comp. 4, 134-145 (2004).

Since universal quantum computers can achieve BosonSampling (and the other related computational tasks considered by TD and BJS), this is a very strong indication for the computational complexity advantage of quantum computers which arguably brings us with quantum computers to the “cozy neighborhood” of NP-hardness.

Noisy quantum computers with quantum fault-tolerance are also capable of exact BosonSampling and this strong computational-complexity quantum-superiority applies to them as well.

Realistic BosonSampling and Gaussian Permanent Estimation (GPE)

Aaronson an Arkhipov considered the following question that they referred to as Gaussian Permanent Approximation.

Problem (Problem 2 from AA’s paper): (|GPE|_{\pm}^2): Given as imput a matrix {\cal N}(0,1)_{\bf C}^{n \times n} of i.i.d Gaussians,together with error bounds ε, δ > o, estimate to within additive error \pm \epsilon n! with probability at leat 1-δ over X, in poly(n,1/\epsilon,1/\delta) time.

They conjectured that such Gaussian Permanent Approximation is computationally hard and showed (Theorem 3) that this would imply that sampling w.r.t. states achievable by boson machines cannot even be approximated by classical computing (unless PH collapses). They regarded questions about approximation more realistic in the context of decoherence where we cannot expect exact sampling.

Scott Aaronson also expressed guarded optimism that even without quantum fault-tolerance BosonSampling can be demonstrated by boson machines for 20-30 bosons, leading to strong experimental evidence for computational advantage of quantum computers (or, if you wish, against the efficient Church-Turing thesis).

Is it so?

More realistic BosonSampling and Noisy Gaussian Permanent Estimation (NGPE)

Let us consider the following variation that we refer to as Noisy Gaussian Permanent Estimation:

Problem 2′: (|NGPE|_{\pm}^2): Given as imput a matrix M= {\cal N}(0,1)_{\bf C}^{n \times n} of i.i.d Gaussians, and a parameter t>0 let NPER (M),  be the expected value of the permanent for \sqrt {1-t^2}M+E where E= {\cal N}(0,t)_{\bf C}^{n \times n}.  Given the input matrix M together with error bounds ε, δ > o, estimate NPER(M) to within additive error \pm \epsilon n! with probability at leat 1-δ over X, in poly(n,1/\epsilon,1/\delta) time.

Problem 2′ seems more relevant for noisy boson machines (without fault-tolerance). The noisy state of the computer is based on every single boson  being slightly mixed, and the permanent computation will average these individual mixtures. We can consider the relevant value for t to be a small constant. Can we expect Problem 2′ to be hard?

The answer for Question 2′ is surprising. I expect that even when t is very very tiny, namely t=n^{-\beta} for \beta <1, the expected value of NPER(M) (essentially) does not depend at all on M!

Noise Sensitivity a la Benjamini, Kalai and Schramm

Noise sensitivity for the sense described here for Boolean functions was studied in a paper by Benjamini Schramm and me in 1999.  (A related notion was studied by Tsirelson and Vershik.) Lectures on noise sensitivity and percolation is a new beautiful monograph by Christophe Garban and Jeff Steif which contains a description of noise sensitivity. The setting extends easily to the Gaussian case. See this paper by Kindler and O’donnell for the Gaussian case. In 2007, Ofer Zeituni and I studied the noise sensitivity in the Gaussian model of the maximal eigenvalue of random Gaussian matrices (but did not write it up).


Noise sensitivity depends on the degree of the support of the Fourier expansion. For determinants or permanents of an n by n matrices the basic formulas as sums of generalized diagonals describe the Fourier expansion,  so the Fourier coefficients are supported on degree-n monomials. This implies that the determinant and the permanent are very noise sensitive.

Noisy Gaussian Permanent Estimation is easy

Noisy Gaussian Permanent Estimation is easy, even for very small amount of noise, because the outcome does not depend at all on the input. It is an interesting question what is the hardness of NGPE is when the noise is below the level of noise sensitivity.

Update (March, 2014) Exploring the connection between BosonSampling and BKS-noise sensitivity shows that the picture drawn here is incorrect. Indeed, the square of the permanent is not noise stable even when the noise is fairly small and this shows that the noisy distribution does not approximate the noiseless distribution. However the noisy distribution does depend on the input!


AA’s protocol and experimental BosonSampling

Scott and Alex proposed a simple experiment described as follows : “An important motivation for our results is that they immediately suggest a linear-optics experiment, which would use simple optical elements (beamsplitters and phaseshifters) to induce a Haar-random m \times m unitary transformation U on an input state of n photons, and would then check that the probabilities of various final states of the photons correspond to the permanents of n \times n submatrices, as predicted by quantum mechanics.”

Recently, four groups carried out interesting BosonSampling experiments with 3 bosons (thus for permanents of 3 by 3 matrices.) (See this post on Scott’s blog.)

BKS-noise sensitivity is giving  simple predictions on how things will behave as a function of the number of bosons and this can be tested even with experiments with very small number of bosons. When you increase the number of bosons the distribution will quickly become uniform for the various final states. The correlation between the probabilities and the value corresponding to permanents will rapidly go to zero.

Some follow-up questions

Here are a few interesting questions that deserve further study.

1) Does problem 2′ capture the general behavior of noisy boson machines? To what generality noise sensitivity applies for general functions described by Boson sampling distributions?

(There are several versions for photons-based quantum computers including even an important  model by Knill, Laflamme, and Milburn that support universal quantum computation. The relevance of BKS noise-sensitivity should be explored carefully for the various versions.)

2) Is the connection with noise sensitivity relevant to the possibility to have boson machines with fault tolerance?

3) What is the Gaussian-quantum analog for BKS’s theorem asserting that noise sensitivity is the law unless  we have substantial correlation with the majority function?

4) What can be said about noise-sensitivity of measurements for other quantum codes?

A few more remarks:

More regarding noisy boson machines and quantum fault tolerance

Noisy boson machines and BosonSampling are related to various other issues regarding quantum fault-tolerance. See my added recent remarks (about the issue of synchronization, and possible modeling using smoothed Lindblad evolutions) to my old post on AA’s work.

Noise sensitivity and the special role of the majority function


The main result of Itai, Oded, and me was that a Boolean function which is not noise sensitive must have a substantial correlation with the majority function. Noise sensitivity and the special role of majority for it gave me some motivation to look at quantum fault-tolerance in 2005  (see also this post,) and this is briefly discussed in my first paper on the subject, but until now I didn’t find an actual connection between quantum fault-tolerance and BKS-noise-sensitivity.


It is an interesting question which bosonic states are realistic, and it came up in some of my papers and in the discussion with Aram Harrow and Steve Flammia (and their paper on my “Conjecture C”).

A sort of conclusion

BosonSampling was offered as a way to prove that quantum mechanics allows a computational advantage without using the computational advantage for actual algorithmic purpose. If you wish, the AA’s protocol is offered as a sort of zero-knowledge proof that the efficient Church-Turing thesis is false.  It is a beautiful idea that attracted interest and good subsequent work from theoreticians and experimentalists. If the ideas described here are correct, BosonSampling and boson machines may give a clear understanding based on BKS noise-sensitivity for why quantum mechanics does not offer computational superiority (at least not without the magic of quantum fault-tolerance).

Avi’s joke and common sense

Here is a quote from AA referring to a joke by Avi Wigderson: “Besides bosons, the other basic particles in the universe are fermions; these include matter particles such as quarks and electrons. Remarkably, the amplitudes for n-fermion processes are given not by permanents but by determinants of n×n matrices. Despite the similarity of their definitions, it is well-known that the permanent and determinant differ dramatically in their computational properties; the former is #P-complete while the latter is in P. In a lecture in 2000, Wigderson called attention to this striking connection between the boson and fermion dichotomy of physics and the permanent-determinant dichotomy of computer science. He joked that, between bosons and fermions, ‘the bosons got the harder job.’ One could view this paper as a formalization of Wigderson’s joke.”

While sharing the admiration to Avi in general and Avi’s jokes in particular, if we do want to take Avi’s joke seriously (as we always should), then the common-sense approach would be first to try to understand why is it that nature treats bosons and fermions quite equally and the dramatic computational distinction is not manifested at all. The answer is that a crucial ingredient for a computational model is the modeling of noise/errors, and that noise-sensitivity makes bosons and fermions quite similar physically and computationally.

Eigenvalues, determinants, and Yuval Filmus

It is an interesting question (that I asked over Mathoverflow) to understand the Fourier expansion of the set of eigenvalues, the maximum eigenvalue and related functions. At a later point,  last May,  I was curious about the Fourier expansion of the determinant, and for the Boolean case I noticed remarkable properties of its Fourier expansion. So I decided to ask Yuval Filmus about it:

Dear Yuval

 I am curious about the following. Let D be the function defined on {-1,1}^n^2
which associates to every +1/1 matrix its determinant.
What can be said about the Fourier transform of D? It looks to me that easy arguments shows that the Fourier transform is supported only on subsets of the entries
so that in every raw and columns there are odd number of entries. Likely there are even further restrictions that I would be interested to explore.
Do you know anything about it?
best Gil

Yuval’s answer came a couple of hours later like a cold shower:

Hi Gil,

The determinant is a sum of products of generalized diagonals.
Each generalized diagonal is just a Fourier character, and they are all different.

In other words, the usual formula for the determinant *is* its Fourier transform

This reminded me of a lovely story of how I introduced Moni Naor to himself that I should tell sometime.

What else can a quantum computer sample?

The ability of quantum computers to sample (exactly) random complex Gaussian matrices according to the value of their permanents is truly amazing! If you are not too impressed by efficient factoring but still do not believe that QC can reach the neighborhood of NP-hard problems you may find this possibility too good to be true.

I am curious if sharp P reductions give us further results of this nature. For example,  can a QC sample random 3-SAT formulas (by a uniform distribution or by a certain other distribution coming from sharp-P reductions) according to the number of their satisfying assignments?

Can QC sample integer polytopes by their volume or by the number of integer points in them? Graphs by the number of 4-colorings?

Lawler-Kozdron-Richards-Stroock’s combined Proof for the Matrix-Tree theorem and Wilson’s Theorem

wilson  curvature

David Wilson and a cover of Shlomo’s recent book “Curvature in mathematics and physics”

A few weeks ago, in David Kazhdan’s basic notion seminar, Shlomo Sternberg gave a lovely presentation Kirchho ff and Wilson via Kozdron and Stroock. The lecture is based on the work presented in the very recent paper by Michael J. Kozdron,  Larissa M. Richards, and Daniel W. Stroock: Determinants, their applications to Markov processes, and a random walk proof of Kirchhoff’s matrix tree theorem. Preprint, 2013. Available online at arXiv:1306.2059.

Here is the abstract:

Kirchhoff’s formula for the number of spanning trees in a connected graph  is over 150 years old. For example, it says that if c_2, \dots, c_n are the nonzero  eigenvalues of the Laplacian, then the number k of spanning trees is k= (1/n)c_2\cdots c_n. There are many proofs.  An algorithm due to Wilson via loop erased random walks produces such a tree, and Wilson’s theorem is that all spanning trees are produced by his algorithm with equal probability. Hence,  after the fact, we know that Wilson’s algorithm produces any given tree with probability 1/k.  A proof due to Lawler, using the Green’s function, shows directly that Wilson’s algorithm has the probability 1/k  of producing any given spanning tree, thus simultaneously proving Wilson’s theorem and Kirchhoff’s formula. Lawler’s proof has been considerably simplified by Kozdron and Stroock. I plan to explain their proof. The lecture will be completely self-contained, using only Cramer’s rule from linear algebra.

(Here are also lecture notes of the lecture by Ron Rosenthal.)

Here is some background.

The matrix-tree theorem

The matrix tree theorem asserts that the number of rooted spanning trees of a connected graph G  is the product of the non-zero eigenvalues of L(G), the Laplacian of G.

Suppose that G has n vertices. The Laplacian of G is the matrix whose (i,i)-entry is the degree of the ith vertex, and its (i,j) entry for i \ne j is 0 if the ith vertex is not adjacent to the jth vertex, and -1 if they are adjacent. So  L(G)=D-A(G) where A(G) is the adjacency matrix of G, and D is a diagonal matrix whose entries are the degrees of the vertices.  An equivalent formulation of the matrix-tree theorem is that the number of spanning trees is the determinant of a matrix obtained from the Laplacian by deleting the j th row and j th column.

We considered a high dimensional generalization of the matrix tree theorem in these posts (I, II, III, IV).

How to generate a random spanning tree for a graph G?

Using the matrix-tree theorem

Method A: Start with an edge e \in G, use the matrix-tree theorem to compute the probability p_e that e belongs to a random spanning tree of G, take e with probability p_e. If e is taken consider the contraction G/e and if G is not taken consider the deletion G \backslash e and continue.

This is an efficient method to generate a random spanning tree according to the uniform probability distribution. You can extend it by assigning each edge a weight and chosing a tree with probability proportional to the product of its weights.

Random weights and greedy

Method B: Assign each edge a random real number between 0 and 1 and chose the spanning tree which minimizes the sum of weights via the greedy algorithm.

This is a wonderful method but it leads to a different probability distribution on random spanning trees which is very interesting!

The Aldous-Broder random walk method

Method C: The Aldous-Broder theorem. Start a simple random walk from a vertex of the graph until reaching all vertices, and take each edge that did not form a cycle with earlier edges. (Or, in other words, take every edge that reduced the number of connected components of the graph on the whole vertex set and visited edges.)

Amazingly, this leads to a random uniform spanning tree. The next method is also very amazing and important for many applications.

David Wilson’s algorithm

Method D: Wilson’s algorithm. Fix a vertex as a root. (Later the root will be a whole set of vertices, and a tree on them.) Start from an arbitrary vertex u not in the root and take a simple random walk until you reach the root. Next, erase all edges in cycles of the path created by the random walk so you will left with a simple path from  u to the root. Add this path to the root and continue!

Here is a link to Wilson’s paper! Here is a nice presentation by Chatterji  and Gulwani.

The Kadison-Singer Conjecture has beed Proved by Adam Marcus, Dan Spielman, and Nikhil Srivastava

…while we keep discussing why mathematics is possible…

The news

Adam Marcus, Dan Spielman, and Nikhil Srivastava posted a paper entitled “Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem,” where they prove the 1959 Kadison-Singer conjecture.

(We discussed part I of “interlacing families” in this post about new Ramanujan graphs.  Looks like a nice series.)

The Kadison-Singer Conjecture

The Kadison-Singer conjecture refers to a positive answer to a question posed by Kadison and Singer: “They asked ‘whether or not each pure state of \cal B is the extension of some pure state of some maximal abelian algebra’ (where \cal B is the collection of bounded linear transformations on a Hilbert space.”) I heard about this question in a different formulation known as the “Bourgain-Tzafriri conjecture” (I will state it below) and the paper addresses a related well known discrepancy formulation by Weaver. (See also Weaver’s comment on the appropriate “quantum” formulation of the conjecture.)

Updates: A very nice post on the relation of the Kadison-Singer Conjecture  to foundation of quantum mechanics is in this post in  Bryan Roberts‘ blog Soul Physics. Here is a very nice post on the mathematics of the conjecture with ten interesting comments on the proof by Orr Shalit, and another nice post on Yemon Choi’s blog and how could I miss the very nice post on James Lee’s blog.. Nov 4, 2013: A new post with essentially the whole proof appeared on Terry tao’s blog, Real stable polynomials and the Kadison Singer Problem.

Update: A very nice blog post on the new result was written by  Nikhil Srivastava on “Windows on theory.” It emphasizes the discrapancy-theoretic nature of the new result, and explains the application for partitioning graphs into expanders.

The Bourgain-Tzafriri theorem and conjecture

Let me tell again (see this post about Lior, Michael, and Aryeh where I told it first) about a theorem of Bourgain and Tzafriri related to the Kadison-Singer conjecture.

Jean Bourgain and Lior Tzafriri considered the following scenario: Let a > 0 be a real number. Let A be a n \times n matrix with norm 1 and with zeroes on the diagonal. An s by s principal minor M is “good” if the norm of M is less than a.

Consider the following hypergraph F:

The vertices correspond to indices {}[n]=\{1,2,\dots,n\}. A set S \subset [n] belongs to F if the S \times S sub-matrix of M is good.

Bourgain and Tzafriri showed that for every a > 0 there is C(a) > 0 so that for every matrix A we can find S \in F so that |S| \ge C(a)n.

Moreover, they showed that for every nonnegative weights w_1,w_2,\dots w_n there is S \in F so that the sum of the weights in S is at least C(a) times the total weight. In other words, (by LP duality,) the vertices of the hypergraph can be fractionally covered by C(a) edges.

The “big question” is if there a real number C'(a) > 0 so that for every matrix M, {}[n] can be covered by C'(a) good sets. Or, in other words, if the vertices of F can be covered by C'(a) edges. This question is known to be equivalent to an old conjecture by Kadison and Singer (it is also known as the “paving conjecture”). In view of what was already proved by Bourgain and Tzafriri what is needed is to show that the covering number is bounded from above by a function of the fractional covering number. So if you wish, the Kadison-Singer conjecture had become a statement about bounded integrality gap. Before proving the full result, Marcus, Spielman and Srivastava gave a new proof of the Bourgain-Tzafriti theorem.

Additional references:

KADISON-SINGER MEETS BOURGAIN-TZAFRIRI by PETER G. CASAZZA, ROMAN VERSHYNIN,  The Kadison-Singer Problem in Mathematics and Engineering: A Detailed Account pdf, and many other recent publications by Pete Casazza.

Why is mathematics possible?

Spectacular advances in number theory

Last weeks we heard about two spectacular results in number theory.  As announced in Nature, Yitang Zhang proved that there are infinitely many pairs of consecutive primes (p_n, p_{n+1}) which are at most 70 million apart! This is a sensational achievement. Pushing 70 million to 2 will settle the ancient conjecture on twin primes, but this is already an extremely amazing breakthrough.  An earlier breakthrough came in 2005 when Daniel Goldston, János Pintz, and Cem Yıldırım proved that the gaps between consecutive primes p_{n+1}-p_n is infinitely often smaller than \sqrt {\log p_n} \log \log ^2 p_n.

Update: A description of Zhang’s work and a link to the paper can be found on Emmanuel Kowalski’s bloog Further update: A description of Zhang’s work and related questions and results can be found now in Terry Tao’s blog. Terry Tao also proposed a new polymath project aimed to reading Zhang’s paper and attempting to improve the bounds.

Harald Helfgott proved that every integer is the sum of three primes.  Here the story starts with Vinogradov who proved it for sufficiently large integers, but pushing down what “sufficiently large” is, and pushing up the computerized methods needed to take care of “small” integers required much work and ingenuity.

Why is Mathematics possible?

The recent news, and a little exchange of views I had with Boaz Barak, bring us back to the question: “Why is mathematics possible?” This is an old question that David Kazhdan outlined in a lovely 1999 essay “Reflection on the development of mathematics in the twentieth century.” The point (from modern view) is this: We know that mathematical statements can, in general, be undecidable.  We also know that a proof for a short mathematical statement can be extremely long. And we also know that even if a mathematical statement admits a short proof, finding such a proof can be computationally intractable. Given all that, what are the reasons that mathematics is at all possible?

It is popular to associate “human creativity” with an answer. The problem with incorrect (or, at least, incomplete) answers is that they often represent missed opportunities for better answers. I think that for the question “why is mathematics possible” there are opportunities (even using computational complexity thinking) to offer better answers.

Please offer your answers.



qstart poster


Physics, Computer Science, Mathematics, and Foundations’
views on quantum information

Inauguration conference for the Quantum Information Science Center (QISC),
Hebrew university of Jerusalem

Update: The news of our conference have made it to a big-league blog.

Update (July 2013): QStart was a very nice event- there were many interesting talks, and the speakers made the effort to have lectures accessible to the wide audience while discussing the cutting edge and at times technical matters.Streaming video of the talks is now available.

My Quantum Debate with Aram III

This is the third and last post giving a timeline and some non technical highlights from my debate with Aram Harrow.  

Where were we

After Aram Harrow and I got in touch in June 2011, and decided to have a blog debate towards the end of 2011, the first post in our debate describing my point of view was launched on January, 2012 and was followed by three posts by Aram. The discussion was intensive and interesting.  Here is a link to my 2011 paper that initiated the debate and to a recent post-debate presentation at MIT.

 Happy_Passover  Happy passover, readers!

Back to the debate: Conjecture C is shot down!


In addition to his three posts, Aram and Steve Flammia wrote a paper refuting one of my Conjectures (Conjecture C).  We decided to devote a post to this conjecture.

Quantum refutations and reproofs

Post 5, May 12, 2012. One of Gil Kalai’s conjectures refuted but refurbished

Niels Henrik Abel was the patron saint this time

The first version of the post started with this heartbreaking eulogy for Conjecture C. At the end most of it was cut away. But the part about Aram’s grandchildren was left in the post.

Eulogy for Conjecture C

(Gil; old version:) When Aram wrote to me, inn June 2011, and expressed willingness to publicly discuss my paper, my first reaction was to decline and propose having just private discussions. Even without knowing Aram’s superb track record in debates, I knew that I put my beloved conjectures on the line. Some of them, perhaps even all of them, will not last. Later, last December, I changed my mind and Aram and I started planning our debate. My conjectures and I were fully aware of the risks. And it was Conjecture C that did not make it.

A few words about Conjecture C

Conjecture C, while rooted in quantum computers skepticism, was a uniter and not a divider! It expressed our united aim to find a dividing line between the pre- and post- universal quantum computer eras.

Aram’s grandchildren and the world before quantum computers

When Aram’s grandchildren will ask him: “
Grandpa, how was the world before quantum computers?” he could have replied: “I hardly remember, but thanks to Gil we have some conjectures recording the old days, and then he will state to the grandchildren Conjectures 1-4 and the clear dividing line in terms of Conjecture C, and the grandchildren will burst in laughter about the old days of difficult entanglements.” Continue reading

Mittag-Leffler Institute and Yale, Winter 2005; Test your intuition: Who Played the Piano?

This is a little “flashback” intermission in my posts about my debate with Aram Harrow. This time I try to refer to Cris Moore’s question regarding  the motivation for my study. For the readers it gives an opportunity to win a $50 prize! 

Let me also bring to your attention an interesting discussion (starting here) between Peter Shor and me regarding smoothed Lindblad evolutions.

(Cris Moore, the debate’s very first comment!) I am also a little confused by Gil’s motivation for his conjectures.  (My response:)  To the best of my memory, my main motivation for skeptically studying quantum fault-tolerance was that I thought that this is a direction worth pursuing and that I had a shot at it.

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While listening with Ravi Kannan to this 2002 lecture by Michel Devoret at Yale, I wondered if there are enough scientists working on the “mirage” side.

Flashback: Mittag-Leffler 2005

I started systematically thinking about quantum fault-tolerance in February 2005. There were several things that triggered my interest to the question in the previous fall and I decided to spend some time learning and thinking about it in our winter break.  One of those triggers was something Dorit Aharonov told me a few months earlier: she said that once, when she was telling her students about quantum computers, she suddenly had a feeling that maybe it was all just nonsense. Another trigger came from a former student who told me about a Polish scientist (whose name he could not remember) who wrote an article about impossibility of quantum error-correction. I thought that the lack of a quantum analog of the repetition code, and the unique properties of the majority function  in terms of sensitivity to noise that I studied with Itai Benjamini and Oded Schramm earlier could be a good starting point for looking skeptically at quantum computers.  

In our 2005 winter break, I spent two weeks at Yale and then additional two weeks at the Mittag-Leffler institute near Stockholm.  At Yale, I only had little time to think about quantum computers. I had to finish a survey article with Muli Safra about threshold phenomena (To a volume that Cris Moore and Allon Perkus were among the editors).  One of the last days in Yale we went to dinner with two guests, Chris Skinner who gave the colloquium talk, and Andrei Okounkov who visited me and gave a talk about partition enumeration and mirror symmetry. At the dinner Andrew Casson asked, out of the blue, if we think that quantum computers can be built and it almost seemed as if that Andrew was reading my mind on what I plan to work on the weeks to come. My answer there was the same as my answer now, that I tend to find it implausible.

Mittag-Leffler Institute February 2005 with Xavier Viennot and Alain Lascoux

In Sweden I spent most of my time on quantum fault-tolerance. I was jet-lagged and being jet-lagged in the Mittag-Leffler institute already worked for me once, when finding my subexponential randomized variant of the simplex algorithm was a substitute for sleeping some night in fall 1991 . In 2005 it was not as bad, I just came to my office very early in the morning and started working. And very early in the morning somebody was already playing the piano.

And who was playing the piano at the institute in the cold Swedish mornings of February 2005? The first reader to guess correctly, and convince me in a comment that she or he knows the answer without revealing it to everybody else will get $50. Continue reading

My Quantum Debate with Aram II

This is the second of three posts giving few of the non-technical highlights of my debate with Aram Harrow. (part I)

After Aram Harrow and I got in touch in June 2011, and decided to have a blog debate about quantum fault-tolerance towards the end of 2011, the first post in our debate was launched on January 30, 2012.  The first post mainly presented my point of view and it led to lovely intensive discussions. It was time for Aram’s reply and some people started to lose their patience.

(rrtucky) Is Aram, the other “debater”, writing a dissertation in Greek, as a reply?

Flying machines of the 21st century

Post II, February 6, 2011. First of three responses by Aram Harrow

Dave Bacon was the patron saint for Aram’s first post.

(Aram) There are many reasons why quantum computers may never be built…  The one thing I am confident of is that we are unlikely to find any obstacle in principle to building a quantum computer.

(Aram) If you want to prove that 3-SAT requires exponential time, then you need an argument that somehow doesn’t apply to 2-SAT or XOR-SAT. If you want to prove that the permanent requires super-polynomial circuits, you need an argument that doesn’t apply to the determinant. And if you want to disprove fault-tolerant quantum computing, you need an argument that doesn’t also refute fault-tolerant classical computing.

From the discussion

Why not yet? Boaz set a deadline


(Boaz Barak could [you] explain a bit about the reasons why people haven’t been able to build quantum computers with more than a handful of qubits yet? Continue reading