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I am a little confused by this. As I use the term, lines are equiangular if and only if the "trace of pairwise product of (distinct) projectors is constant". You seem to be using the word in a different sense. It might be helpful if you were to explain exactly what is that sense.

To define the complex angle, we used the (cyclotomic) field norm to the power one over the degree of the field, as stated in the introduction. It recovers the particular case of angles for SICs. In this sense "equiangular" means that all pairs of distinct lines make the same angle.

This appears to be an odd and nonstandard definition of "equiangular", unless I'm missing something? Most references I'm aware of, including [Wikipedia][1] and [Renes et al 2004][2] agree that "a set of lines is called equiangular if every pair of lines makes the same angle". For unit vectors (ray

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The trace of pairwise product of (distinct) projectors is not constant. For example, with the state $(0,1,-1,-1,1)$, one gets an equiangular IC-POVM in which the trace is trivalued: it is either $1/16$, or $(7 \pm 3\sqrt{5})/32$. For the state (0,1,i,-i,-1), there are five values of the trace.
We s

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This is why I am confused (it is probably just a reading comprehension error on my part): If the POVM is IC, it must have at least $d^2$ elements. If it is a minimal IC-POVM, it must have exactly $d^2$ elements. But if it is minimal, IC and equiangular, then the angle is fixed by the requirement tha

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Yes, the IC-POVMs under consideration are minimal. The IC-POVM in dimension 5 is equiangular but is also not a SIC. In particular the trace product relation of a SIC is not satisfied. For the equiangular IC-POVM in dimension 7, we have a similar result.

Clarification request: Are all the IC-POVMs in this paper minimal? That is, does the number of elements in each POVM equal the square of the dimension? If so, I am confused about the quoted value of the inner product between projectors for the equiangular IC-POVM in dimension 5.

Hey Noon,

thanks for the feedback! I'm happy to share the code and will send it to you via mail until monday.

Zak, David: thanks! So (I think) this is a relation problem, not a decision problem (or even a partial function). Which is fine -- I'm happier with relation problems than with sampling problems, and the quantum part of Shor's algorithm is solving a relation problem, which is a pretty good pedigre

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Nice work! Are you planning on sharing the code you wrote to run this in the IBM quantum experience system?

However, one should note that I_3322 may be able to do something that this paper doesn't. William's work leaves open the question of whether there are games with infinite-dimensional tensor product strategies but no finite-dimensional ones. Some of us might expect that I_3322 has this property.

Thanks Zak, that's exactly right-- for each instance there is a set of possible solutions. Like in the Bernstein-Vazirani problem, a solution is a bit string. It can't just be a single bit since then we would have the problem you describe, Robin.

You are completely correct that in order to check whether a give output is "correct" for the input, we would require an additional log-depth classical circuit, but this is not how the problem is defined. In particular, for each input there is a set of "accepting" outputs, and we only need to guaran

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Is it okay to be a quantum supremacist? I thought I was, but maybe if it's "tainted" I should reconsider.

On a more serious note... a question for somebody who has read (or written) the paper. If the computation is performed on Poly(n) qubits, and all of them are relevant, and you are only allo

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This is interesting work.

Did the authors happen to make their code available? I think there might be a few other fun experiments to run, and in particular I'd be interested to know how to use this framework for picking a network that does best at _both_ tasks (from the experiments section). That

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I would like to publicly thank the authors for using the term "advantage" instead of the tainted word "supremacy" that makes me cringe every time I hear it.

Also, great result!

Excellent!

A provable separation between analogous quantum and classical circuit classes!

Regarding the pre-2017 state of play, I think experimentalists knew there was a problem for large errors and theorists had known there was a potential problem (as shown by grant proposals for the QCVV program) but nobody had really sat down and thought about how everything behaved for small errors.

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I agree that we pretty much agree on all these points! For the record, though... when you describe the pre-2017 state of play as "...*someone wants to use this theory, but they can't match the sufficient conditions, so they appeal to heuristics to argue that they can use the theory anyway*," this i

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Yes, I did indeed mean that the results of the previous derivations are correct and that predictions from experiments lie within the stated error bounds. To me, it is a different issue if someone derives something with a theoretical guarantee that might have sufficient conditions that are too strong

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I agree with much of your comment. But, the assertion you're disagreeing with isn't really mine. I was trying to summarize the content of the present paper (and 1702.01853, hereafter referred to as [PRYSB]). I'll quote a few passages from the present paper to support my interpretation:

1. "[T

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I disagree with the assertion (1) that the previous theory didn't give "the right answers." The previous theory was sound; no one is claiming that there are any mistakes in any of the proofs. However, there were nonetheless some issues.

The first issue is that the previous analysis of gate-depe

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That's a hard question to answer. I suspect that on any questions that aren't precisely stated (and technical), there's going to be some disagreement between the authors of the two papers. After one read-through, my tentative view is that each of the two papers addresses three topics which are pre

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So what is the deal?

Does this negate all the problems with https://scirate.com/arxiv/1702.01853 ?

Great result!

For those familiar with I_3322, William here gives an example of a nonlocal game exhibiting a behaviour that many of us suspected (but couldn't prove) to be possessed by I_3322.

I feel that while the proliferation of GUNs is unquestionable a good idea, there are many unsupervised networks out there that might use this technology in dangerous ways. Do you think Indifferential-Privacy networks are the answer? Also I fear that the extremist binary networks should be banned ent

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"To get the videos and their labels, we used a YouTube video annotation system, which labels videos with their main topics."
Can anyone explain a bit about this?

A good paper for learning about exRec's is this one https://arxiv.org/abs/quant-ph/0504218. Also, rigorous threshold lower bounds are obtained using an adversarial noise model approach.

Good point, I wish I knew more about ExRecs.

I totally agree -- that part is confusing. It's not clear whether "arbitrary good precision ... using a limited amount of hardware" is supposed to mean that arbitrarily low error rates can be achieved with codes of fixed size (clearly wrong) or just that the resources required to achieve arbitraril

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I think I was mostly reacting to where he tries to sell the importance of the work.

>Fault tolerant theorems show that an arbitrary good precision can be obtained using a limited amount of hardware...we unveil the role of an implicit assumption made in these mathematical theorems: the ability to

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@Chris: as Ben says, the model for measurement errors is "You measure in a basis that's off by a small rotation".

@Ben: I don't think either of the techniques you mention will directly resolve the paper's concern/confusion. That concern is with the post-QEC state of the system. That state isn't

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Do any fault-tolerance theorems claim to hold for small codes without repeated measurement, as is the case in these supposed counter examples?

The assumption that no-one ever thought about this noise before is the faulty one here.

It seems like the problem is that the measurement basis is unknown (the actual operator being measured isn't exactly Z, for example, but some other Hermitian operator close to Z). However, this seems like it can be re-expressed using an unknown operation that occurs immediately before measurement of

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Could you be more specific by what you mean when you say "the ability to perform quantum measurements with infinite precision"? Several circuit level noise thresholds have been computed where measurement errors are taken into account. Even with measurement errors, thresholds as high as 10^-2 have be

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I'm wondering if this result could have any interesting consequences for Hamiltonian complexity. The LCL problem sounds very much like a local Hamiltonian problem, with the run-time of an LCL algorithm corresponding to the range of local interactions in the Hamiltonian.

Maybe one caveat is that thi

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Dear Authors,

This is in reference of your preprint arxiv 1702.0638.

Above all I must say that I am puzzled with the level of publicity your work has got at http://www.nature.com/news/long-awaited-mathematics-proof-could-help-scan-earth-s-innards-1.21439. Is this a new way for mathematicians t

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A **GREAT** paper. Where you learn how to extract work from the measurement of a qubit coupled to a drive. The authors build an engine with very unusual and interesting features such as efficiency of 1 (no entropy creation) arising for conditions where the power extrated is maximum! This maximum dep

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You are correct, that is a mistake -- it should be $\\{0,1\\}^n\rightarrow\\{0,1\\}$. Thank you for spotting it!

In the abstract, should the domain of $f$ be $\lbrace0,1\rbrace^n$ instead of just $\lbrace0,1\rbrace$?

Regarding Mark's above comment on the role of the stabilizer states: Yes, all previous works on the subject have used the stabilizer states and Clifford gates as the classical backbone. This is due to the Gottesman-Knill theorem and related results. But is it a given that the free sector in quantum

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Are you sure? Since we do not propose a conjecture, there is nothing wrong. A class of strange states underlie the pentagons in question. The motivation is to put the magic of computation in the permutation frame, one needs more work to check its relevance.

It seems interesting at first sight, but after reading it the motivation is very muddled. It boils down to finding pentagons (which enable KCBS-style proofs of contextuality) within sets of projectors, some of which are stabilizer states and some of which are non-stabilizer states (called magic stat

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Here is a nice description, with additional links, about the importance of this work if it turns out to be flawless (thanks a lot to Martin Schwarz for this link): [dichotomy conjecture][1].

[1]: http://processalgebra.blogspot.com/2017/01/has-feder-vardi-dichotomy-conjecture.html

This is a cool paper!

What a nice book! And it's available for free on arXiv!

Indeed, Schur complement is the answer to the ultimate question!

Very good Insight on android security problems and malware. Nice Work !

Hi, How can i get it??