Fault-Tolerant Logical Clifford Gates from Code Automorphisms

Mark Webster Sep 30 2024 09:58 UTCChia-Ying Lin Sep 27 2024 18:47 UTC

Hi! I found this paper is cool, and would like to know where to find the proof of the lemma 3.7 and 3.8?

Angelo Lucia Sep 21 2024 12:30 UTC

Aram is correct: we roughly prove that if you can show a slower than 1/n^2 lower bound to the gap, you can bootstrap it to a constant bound. But if the gap closes faster than you don't get any improvement.

Aram Harrow Sep 21 2024 12:02 UTC

The gap can vanish faster than 1/n^2. Their theorem just says it can’t vanish more slowly. See eq 8.

Varun Menon Sep 20 2024 14:03 UTC

...(continued)Nice result! I am wondering how to reconcile this result with this example of the 'area weighted Motzkin chains' https://arxiv.org/abs/1611.03147 where the model is frustration free, the Hamiltonian is a sum of projectors, and has short range interactions everywhere (even at the boundaries where the

Unme Sep 19 2024 14:17 UTC

Thanks for the interesting paper!

I would like to ask whether the condition in Theorem 6 should be $PE_i^{\dagger}E_jP=\alpha_{ij}P$?

Mark M. Wilde Sep 19 2024 06:23 UTC

Thanks for your reply and for your comment.

Indeed it is true that the mentioned papers do not use the phrase "superchannel", which could lead to one missing these references. They instead use the phrase "bipartite operation", which is a less fitting term.

D.-S. Wang Sep 19 2024 05:58 UTC

...(continued)Dear Mark, thank you for letting us know the papers and sorry for replying late. I managed to create an account. We will definitely cite them in future work if they are relevant on a technical level. My arXiv posting is random. Given my limited knowledge, probably I wouldn't encounter them in the co

Mark M. Wilde Sep 18 2024 11:27 UTC

...(continued)This kind of theory, thinking of codes in terms of superchannels, was developed previously in https://arxiv.org/abs/1406.7142 , which is not cited in the paper. See also the related work https://arxiv.org/abs/1210.4722 . The paper is now published before the arxiv posting appeared, which is unfortun

MariusK Sep 16 2024 14:30 UTC

...(continued)Your *Better Solution Probability* (BSP) metric reminds me of the concept of *advantage* used in reinforcement learning. There, given a *reward* (or better, a *discounted return*) $R$, the advantage is defined via $\mathrm{adv} = R - \mathrm{base}$. Here, $\mathrm{base}$ is a baseline. It could be a

John Martyn Sep 11 2024 19:05 UTC

...(continued)Hi Seiseki, thanks for the comment! Indeed, as our desired map is not trace-preserving, we prove Lemma 2 in order to apply randomized compiling to QSP. This offers a quadratic suppression of error in general.

As you mention, your lemmas 4.1 and 4.2 upper bound the error of a randomly-compiled chan

seiseki akibue Sep 09 2024 22:47 UTC

...(continued)Congratulations on successfully expanding the use case of randomized compiling. Thank you for citing our work in your research. It seems that you are applying randomized compiling to channels with projection. This requires the extended version of the mixing lemma, such as Lemma 2. Since your target

John Martyn Sep 09 2024 16:47 UTC

...(continued)Hey Yue, thanks for bringing this work to our attention! Indeed, the idea of using the mixing lemma to improve performance is quite similar. We’ll be sure to cite your work accordingly in an updated draft. It’s super neat that you were able to observe such good performance by simply mixing over two

Yue Wang Sep 08 2024 05:38 UTC

...(continued)I found this paper very interesting and would like to draw your attention to "Faster Quantum Algorithms with 'Fractional'-Truncated Series" (https://arxiv.org/abs/2402.05595). I noted some similarities, particularly in performing randomization on polynomials with modified coefficients. Both your and

Jake Xuereb Sep 06 2024 09:02 UTC

...(continued)Is SciRate really the place for these types of comments? Is it not more constructive to communicate such feedack privately over an email?

I always felt that SciRate comments were for public discussion that could benefit the authors and also the whole quantum community that might see the reply. Thi

Shi Jie Samuel Tan Sep 05 2024 21:12 UTC

Hi Jahan,

Thank you for informing us about the general opinion that the referees had regarding the partial dot product notation! We will be sure to update our paper with notation that is consistent with yours!

Jahan Claes Sep 05 2024 13:19 UTC

...(continued)This looks great, looking forward to reading it in more detail!

Just FYI, the partial dot product notation Argyris and I used in our proof was generally disliked by referees, so we've updated our paper to not use it (will be updated on arXiv at some point).

Instead of partial dot products, you

Victory Omole Sep 03 2024 02:13 UTC

The authors agreed with the above comment in a private correspondence.

Zhongxia Shang Aug 28 2024 01:35 UTC

...(continued)Dear PPfeffer,

Thanks for your comment! Your summary is concise and accurate.For your first question: The entanglement of the state $|A\rangle$ only refers to the entanglement between the upper system and the lower system. Thus, the classical simulatability can be easily destroyed by entanglem

PPfeffer Aug 27 2024 09:03 UTC

...(continued)**Long story short:**

Use $Tr(U_B \rho_A)$ instead of (for instance) standard Hadamard test for $\langle B|A\rangle$ amplitude estimation.

**What do we know?**

$\vert B\rangle$, not $\vert A\rangle$.

**What do we need?**

Diagonal (Unitary) Block state Encoding (UBSE) for $\vert B\r

Shubham Jain Aug 26 2024 21:27 UTC

...(continued)Hi Anqi,

Thanks for your detailed comment! You are indeed right that the resultant 189 qubit code would not have all stabilizer weights divisible by 8 and not satisfy the triply even code definition mentioned in the paper. This was an oversight on our part and we will clarify this in an updated ver

Anqi Gong Aug 26 2024 08:39 UTC

...(continued)Dear authors, please correct me if I am wrong. When applying code doubling using a doubly-even code of length m and a triply-even code of length n, from Eq. 60 of arXiv:1509.03239, for the last row to have weight divisible by 8, don't you need (m+n) to be divisible by 8? For the first code you const

Josu Etxezarreta Martinez Aug 21 2024 21:29 UTC

...(continued)Nice paper! I do miss some references on coherence time drift and fluctuations which seems an important part of it though, including a couple of our team. I leave them here as a reference:

- https://www.nature.com/articles/s41534-021-00448-5

- https://journals.aps.org/prresearch/abstract/10.1103

Roy J Garcia Aug 20 2024 21:39 UTC

...(continued)Hi Lorenzo,

Thank you for pointing us to your paper on pseudomagic quantum states. We’re glad to see that there has been progress made in this direction! We’ve updated the manuscript and have mentioned your results. Thank you again and we look forward to your future work!

Best,

Roy on behal

Nobuyuki Yoshioka Aug 15 2024 01:02 UTC

...(continued)Just for the record (as I have already mentioned to Aleksei via email), a similar algorithm was presented in [arXiv:2311.01362][1], specifically in Appendices A and B, while we were also unable to find any prior literature explicitly discussing this approach.

[1]: https://arxiv.org/abs/2311.0

Aleksei Ivanov Aug 14 2024 09:23 UTC

...(continued)Hi Craig, thanks for the comment. We were searching the literature for the simple and explicit equation for Pauli decomposition but to our surprise we could not find any. This is why we derived and proved Eqs.(6)-(9) ourselves. Algorithm is just a direct implementation of those equations. I will inc

Craig Gidney Aug 13 2024 18:24 UTC

...(continued)I can't find the original citation, and probably that's a sign that it's useful to have it pointed out to the community another time, but this transformation *is* known. For example, although I don't remember where I learned it, I described it and gave working source code for a stack exchange answer

David Gosset Aug 12 2024 13:44 UTC

Yes exactly, thanks Oliver for your response!

Lorenzo Leone Aug 06 2024 23:35 UTC

...(continued)Hi Roy and coauthors,

Congratulations on this interesting piece of work on magic-state resource theory!

I would like to comment a bit on Conjecture 1, bringing to your attention some new results from our team that you may not be aware of. The observation that a magic monotone cannot be accurat

Oliver Reardon-Smith Aug 01 2024 11:04 UTC

...(continued)Others can probably answer this better than I can, but quoting from the introduction

> A series of recent works [15–22] have provided unconditional results

> addressing a more limited notion of quantum advantage. In these

> works, a computational problem is introduced that can be solved by

> g

Tom Scruby Aug 01 2024 05:55 UTC

...(continued)Apologies for what I'm sure is a very naive question, but what exactly is meant by the claim of "unconditional quantum advantage" in cases such as this? As someone who is fairly ignorant of complexity theory I would have thought that an unconditional proof that a certain problem can be efficiently s

Zhihao Ma Aug 01 2024 03:29 UTC

...(continued)I am one of the authors, I think it is a very interesting work, in fact I think it is my best work in these years, so I am excited, and want to share this paper with my friends (they are all studying quantum information), and some of them give comments about it. I wish more people to pay atten

Evan Peters Jul 31 2024 21:47 UTC

note that a large number of the scites (and most comments) for this preprint come from otherwise inactive accounts that were created within the last month...

Joshua Cudby Jul 29 2024 13:04 UTC

Dear Michal,

Thank you for your pointing out this work, which we were not aware of. We have noted that efficient learning has already been established in an updated listing of our paper.

Best wishes,

Josh

Joshua Cudby Jul 29 2024 13:03 UTC

...(continued)Dear Barbara,

Thank you for your comment. Our work is indeed related to the paper you linked above, with both concerning learning fermionic Gaussian objects. While we learn the whole unitary rather than just the state, we do so only in the "undoped" regime, whereas the authors of the other work a

Eric Aspling Jul 23 2024 18:41 UTC

...(continued)Very interesting paper. I have a novice question specific to fluxonium. Fluxonium's low frequency 01 transition is highly subjected to thermal noise, even at 10mK. This noise comes from many factors including nontrivial phonon lifetimes in sample substrates, 'antennization' of wires and other compo

Thomas Schuster Jul 21 2024 03:41 UTC

...(continued)Hi Yuchen, thank you for the question! For 1D circuits, the intuition you outline is basically correct. (We discuss this in more detail in the "Comparison to existing results" section of our Supp Info.) However, for general circuits, large amounts of entanglement can form much more quickly, so this

Yuchen Guo Jul 21 2024 02:39 UTC

...(continued)Great paper! Can I understand the physical intuition behind your results in this way? For a noisy quantum circuit with error rate $\gamma$, the maximal entanglement of such a circuit will be $S=O(1/\gamma)$, hence we could use a classical representation for the quantum states, such as matrix product

Zhenyu Cai Jul 18 2024 16:58 UTC

...(continued)Hi Thomas, I understand what you mean. Let me put it in another way: your statement means to be “any quantum circuit for which error mitigation is efficient ‘in the limit of very large n’ must be classically simulable.” While your current sentence can be misinterpreted as “a given quantum circuit of

Thomas Schuster Jul 18 2024 14:34 UTC

...(continued)Hi Zhenyu, thank you so much for the comment and suggestions! Our statement holds for any arbitrarily small O(1) noise rate independent of the number of qubits n. For even smaller noise rates, e.g. scaling with the inverse of the number of qubits O(1/n), it is possible for both error mitigation to b

Michal Oszmaniec Jul 18 2024 11:55 UTC

...(continued)I wanted to remark that efficient learning of unknown FLO unitaries has been already established in https://arxiv.org/pdf/2012.15825 (Part IX, Theorem 8). Therein, we gave an efficient reconstruction method that approximates unknown FLO transformation to additive precision in diamond norm. The meth

Yichen Huang Jul 18 2024 11:33 UTC

...(continued)This paper is published in New Journal of Physics, an open access journal with an article publication charge (APC) for authors. The journal offers various discounts:

1. The current APC is £1660/€1890/\$2485. According to the rate today, £1660=\$2157 and €1890=$2067. Everyone should choose € to sav

Zhenyu Cai Jul 18 2024 09:36 UTC

...(continued)Congratulations on a very interesting paper! A quick comment on the possible misinterpretation of the last sentence. I think a better statement would be ''any quantum circuit for which error mitigation scales efficiently with the amount of circuit noise must be classically simulable’’. Or ``any quan

Barbara Terhal Jul 18 2024 07:17 UTC

What is the overlap of this paper with https://arxiv.org/abs/2402.18665?

Victory Omole Jul 05 2024 02:07 UTC

Ah. Right. Thanks!

Anonymous Observer Jul 04 2024 23:22 UTC

I believe it's correct as written (it looks like you missed an inverse?), with:

$$Q \geq \beta^{-1} \ln 2 = ((k_B T)^{-1})^{-1}\ln 2 = k_B T \ln 2$$

Victory Omole Jul 04 2024 21:39 UTC

...(continued)Equation 1: $Q \geq B^{-1} \ln 2$, which can written as $Q \geq \frac{\ln 2}{(k_{B} T)}$, contradicts [Wikipedia's statement of Landauer's principle][1] ($Q \geq k_{B} T \ln 2 $).

Equation 1 claims that the theoretical lower bound energy dissipation of resetting a qubit is below the one for rese

PPfeffer Jun 28 2024 08:25 UTC

...(continued)The problem of measurement/sampling is not discussed in this article. At one point, it is said that "$N_{shot}$ *is the number of shots/iteration (= 10,000)*" and only in the paragraph before the conclusion, it is mentioned that "*While our estimate [..] appears daunting at first sight, it [..] iden

Michael Biercuk Jun 21 2024 01:58 UTC

...(continued)Hi Pau. Thanks for your comment.

In Figure 4a-c of the manuscript we have removed all elements of the QAOA pipeline to exclusively focus on the role that error suppression plays in executing a "plain vanilla" instance of QAOA .

"In all executions, the equal weights superposition state ser

Pau F Jun 20 2024 20:00 UTC

...(continued)This manuscript shows that Q-CTRL’s implementation of QAOA with initial angle tuning attains a solution quality that strongly overlaps with random sampling (Figure 4). That is significantly poorer than the included alternatives Local Solver and D-Wave (without initial solution preparation).

The m