# Special Quantum PCP and/or Quantum Codes: Simplicial Complexes and Locally Testable CodesDay

### room B-220, 2nd floor, Rothberg B Building

On Thursday, the 24th of July we will host a SC-LTC (simplicial complexes and classical and quantum locally testable codes) at the Hebrew university, Rothberg building B room 202 (second floor) in the Givat Ram campus. Please join us, we are hoping for a fruitful and enjoyable day, with lots of interactions. Coffee and refreshments will be provided throughout the day, as well as free “tickets” for lunch on campus
There is no registration fee, but please email dorit.aharonov@gmail.com preferably by next Tuesday if there is a reasonable probability that you attend –  so that we have some estimation regarding the number of people, for food planning

Program:SC-LTC day – simplicial complexes and locally testable classical and quantum codes –Rothberg building B202
9:00 gathering: coffee and refreshments

9:30 Irit Dinur: Locally testable codes, a bird’s eye view

10:15: coffee break

10:45 Tali Kaufman, High dimensional expanders and property testing

11:30 15 minutes break

11:45 Dorit Aharonov, quantum codes and local testability

12:30 lunch break

2:00 Alex Lubotzky: Ramanujan complexes

2:50 coffee break

3:15 Lior Eldar: Open questions about quantum locally testable codes and quantum entanglement

3:45 Guy Kindler: direct sum testing and relations to simplicial complexes ( Based on David, Dinur, Goldenberg, Kindler, and Shinkar, 2014)

4:15-5 free discussion, fruit and coffee

# 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.

## 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

# 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

# My Quantum Debate with Aram Harrow: Timeline, Non-technical Highlights, and Flashbacks I

## How the debate came about

(Email from Aram Harrow, June 4,  2011) Dear Gil Kalai, I am a quantum computing researcher, and was wondering about a few points in your paper

(Aram’s email was detailed and thoughtful and at the end he proposed to continue the discussion privately or as part of a public discussion.)

(Gil to Aram) Thank you for your email and interest. Let me try to answer the points you raised. …   (I gave a detailed answer.) …  Right now, I don’t plan on initiating a public discussion. How useful such public discussions are (and how to make them useful) is also an interesting issue. Still they were useful for me, as two of my conjectures were raised first in a discussion on Dave Bacon’s blog and another one is partially motivated by a little discussion with Peter Shor on my blog. Continue reading

# A Discussion and a Debate

## Heavier than air flight of the 21 century?

The very first post on this blog entitled “Combinatorics, Mathematics, Academics, Polemics, …” asked the question “Are mathematical debates possible?” We also had posts devoted to debates and to controversies.

A few days ago, the first post in a discussion between Aram Harrow, a brilliant computer scientists and quantum information researcher (and a decorated debator), and myself on quantum error correction was launched in Dick Lipton and Ken Regan’s big-league blog, Gödel’s Lost Letter and P=NP.

The central question we would like to discuss is:

Are universal quantum computers based on quantum error correction possible.

In preparation for the public posts, Ken, Aram, Dick, and me are having very fruitful and interesting email discussion on this and related matters, and also the post itself have already led to very interesting comments. Ken is doing marvels in editing what we write.

Dick opened the post by talking on perpetual motion machines which is ingenious because it challenges both sides of the discussion. Perpetual motion turned out to be impossible: will quantum computers enjoy the same fate? On the other hand (and closer to the issue at hand), an argument against quantum mechanics based on the impossibility of perpetual motion by no other than Einstein turned out to be false, are skeptical ideas to quantum computers just as bogus? (The answer could be yes to both questions.) Some people claimed that heavier-than-air flight might is a better analogy. Sure, perhaps it is better.

But, of course, analogies with many human endeavors can be made, and for these endeavors, some went one way, and some went the other way, and for some we don’t know.

Although this event is declared as a debate, I would like to think about it as a discussion. In the time scale of such a discussion what we can hope for is to better understand each other positions, and, not less important, to better understand our own positions.  (Maybe I will comment here about some meta aspects of this developing discussion/debate.)

## A real debate

A real emerging debate is if we (scientists) should boycott Elsevier. I tend to be against such an action, and especially against including refereeing papers for journals published by Elsevier as part of the boycott. I liked, generally speaking,  Gowers’s critical post on Elsevier, but the winds of war and associated rhetoric are not to my liking.  The universities are quite powerful, and they deal, negotiate and struggle with scientific publishers, and other similar bodies, on a regular  basis. I tend to think that the community of scientists should not be part of such struggles and that such involvement will harm the community and science. This is a real debate! But it looks almost over.  Many scientists joined the boycott and not many opposing opinions were made. It looks that we will have a little war and see some action. Exciting, as ever.

# When Noise Accumulates

I wrote a short paper entitled “when noise accumulates” that contains the main conceptual points (described rather formally) of my work regarding noisy quantum computers.  Here is the paper. (Update: Here is a new version, Dec 2010.) The new exciting innovation in computer science conference in  Beijing seemed tailor made for this kind of work, but the paper did not make it that far. Let me quote the first few paragraphs. As always, remarks are welcome!

From the introduction: Quantum computers were offered by Feynman and others and formally described by Deutsch, who also suggested that they can outperform classical computers. The idea was that since computations in quantum physics require an exponential number of steps on digital computers, computers based on quantum physics may outperform classical computers. A spectacular support for this idea came with Shor’s theorem that asserts that factoring is in BQP (the complexity class described by quantum computers).

The feasibility of computationally superior quantum computers is one of the most fascinating and clear-cut scientific problems of our time. The main concern regarding quantum-computer feasibility is that quantum systems are inherently noisy. (This concern was put forward in the mid-90s by Landauer, Unruh, and others.)

The theory of quantum error correction and fault-tolerant quantum computation (FTQC) and, in particular, the threshold theorem which asserts that under certain conditions FTQC is possible, provides strong support for the possibility of building quantum computers.

However, as far as we know, quantum error correction and quantum fault tolerance (and the highly entangled quantum states that enable them) are not experienced in natural quantum processes. It is therefore not clear if computationally superior quantum computation is necessary to describe natural quantum processes.

We will try to address two closely related questions. The first is, what are the properties of quantum processes that do not exhibit quantum fault tolerance and how to formally model such processes. The second is, what kind of noise models cause quantum error correction and FTQC to fail.

A main point we would like to make is that it is possible that there is  a systematic relation between the noise and the intended state of a quantum computer. Such a systematic relation does not violate linearity of quantum mechanics, and it is expected to occur in processes that do not exhibit fault tolerance.

Let me give an example: suppose that we want to simulate on a noisy quantum computer a certain bosonic state. The standard view of noisy quantum computers asserts that under certain conditions this can be done up to some error that is described by the computational basis. In contrast, the type of noise we expect amounts to having a mixed state between the intended bosonic state and other bosonic states (that represent the noise).

Criticism: A criticism expressed by several readers of an early version of this paper is that no attempt is made to motivate the conjectures from a physical point of view and that the suggestions seem “unphysical.” What can justify the assumption that a given error lasts for a constant fraction of the entire length of the process? If a noisy quantum computer at a highly entangled state has correlated noise between faraway qubits as  we suggest, wouldn’t it allow signaling faster than the speed of light?

I had sort of a mixed reaction toward this criticism. On the one hand I think that it is important and may be fruitful  to examine various models of noise while putting the physics aside. Nevertheless, I added a  brief discussion of  some physical aspects. Here it is:

# Detrimental Noise

### John Lennon

Disclaimer: It is a reasonable belief  (look here, and here), and an extremely reasonable working assumption (look  here) that computationally superior quantum computers can be built.

(This post and the draft will be freely updated) I am struggling to meet the deadline of a week ago for a chapter regarding adversarial noise models for quantum error correction. (Update Nov 20: here is the draft; comments are welcomed. Update April 23: Here is the arxived paper, comments are welcome! ) My hypothetical model is called “detrimental.” (This is a reason substantial math postings are a bit slow recently; but I hope a few will come soon.) This project is quite central to my research in the last three years, and it often feels like running, over my head, after my own tail that might not be there. So this effort may well be a CDM (“career damaging move”) but I like it nevertheless. It is related to various exciting conceptual and foundational issues.

I do have occasionally a sense of progress, (often followed by a backtrack) and for this chapter, rather than describing detrimental noise by various (counterintuitive) properties as I always did, I think I have an honest definition of detrimental noise. Let me tell you about it. (Here is a recent useful guide: a zoo of quantum algorithms, produced by Stephen Jordan.)

## Detrimental noise

Consider a quantum memory with $n$ qubits at a state $\rho_0$. Suppose that $\rho_0$ is a tensor product state. The noise affecting the memory in a short time interval can be described by a quantum operation $E_0$. Lets suppose that $E_0$ acts independently on different qubits and, for qubit $i$ with some small probability $p_i$$E_0$ changes it state to the maximum entropy state $\tau_i$.

This is a very simple form of noise that can be regarded as basic to understanding the standard models of noise as well as of detrimental noise.

In the standard model of noise, $E_0$ describes the noise of the quantum memory regardless of the state $\rho$ stored in the memory. This is a quite natural and indeed expected form of noise.

A detrimental noise will correspond to a scenario in which, when the quantum memory is at a state $\rho$ and $\rho= U \rho_0$, the noise $E$ will be $U E_0 U^{-1}$. Such noise is the effect of first applying $E_0$ to $\rho_0$ and then applying $U$ to the outcome noiselessly.

Of course, in reality we cannot perform $U$ instantly and noiselessly and the most we can hope for is that $\rho$ will be the result of a process. The conjecture is that a noisy process leading to $\rho$ will be subject to noise of the form we have just described. A weaker weaker conjecture is that detrimental noise is present in every natural noisy quantum process. I also conjecture that damaging effects of the detrimental noise cannot be canceled or healed by other components of the overall noise.When we model a noisy quantum system either by a the qubits/gates description or in other ways we make a distinction between “fresh” errors which are introduced in a single computer cycle (or infinitesimally when the evolution is described by a continuous model) and the cumulative errors along the process. The basic insight of fault tolerant quantum computing is that if the incremental errors are standard and sufficiently small then we can make sure that the cumulated errors are as well. The conjecture applies to fresh errors.

(Updated: Nov 19; sorry guys, the blue part is over-simplified and incorrect; But an emergency quantifier replacement seemed to have helped; it seems ok now)  The definition of detrimental noise for general quantum systems that we propose is as follows:

A detrimental noise of a quantum system at a state $\rho$ commutes with every some non-identity quantum operation which stabilizes $\rho$.

Note that this description,

Just like for the standard model of noise, we do not specify a single noise operation but rather gives an envelope for a family of noise operations.

In the standard model of noise the envelope $\cal D_{\rho}$ of noise operations when the computer is at state $\rho$ does not depend on $\rho$. For detrimental noise there is a systematic relation between the envelope of noise operations ${\cal D}_\rho$ and the state $\rho$ of the computer. Namely,

${\cal D}_{U\rho} = U {\cal D}_\rho U^{-1}$.

## Why is it detrimental?

Detrimental noise leads to highly correlated errors when the state of the quantum memory is highly entangled. This is quite bad for quantum error-correction, but an even more devastating property of detrimental noise is that the notion of “expected number of qubit errors” becomes sharply different from the rate of noise as measured by fidelity or trace distance. Since conjugation by a unitary operator preserves fidelity-metric, the expected number of qubit errors increases linearly with the number of qubits for highly entangled states.

Here is another little thing from my paper that I’d like to try on you:

## A riddle: Can noise remember the future?

Suppose we plan a process and carry it out up to a small amount of errors. Can there be a systematic relation between the errors at some time and the planned process at a later time? Continue reading