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Congratulations on this interesting work!

I've noticed a small typo in your Definition 2, Equation (17): it seems $\gamma_1$ should be $\gamma_2$.

Additionally, Appendix C: ``Code surgery on qLDPC codes'' in arXiv:2505.06981 presents a generalization of Definition 2, which involves simultaneou

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@Victory Omole and @Craig Gidney -- You might want to check out authors' updated manuscript (v3, Oct 5, 2025). It includes an example of factoring a number \( N > 10^6 \) using a maximum of only \( m_{\text{max}} = 4 \) phase qubits per block. It shows that Craig's concern about samples from small b

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Your claim that the block size must satisfy $m_i > \log_2 r'$ is clearly wrong and has been refuted by the examples included in the updated manuscript (v3, 5 Oct 2025).

The authors successfully demonstrated factoring a number $N > 10^6$ with order $r=3800$ using only $m_{\max}=4$ phase qubits. Th

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That's right, even for CSS codes things become non-trivial precisely because of the degeneracy from the stabilisers, and the CSS property doesn't really help with that. So you have to find another way around it, which I believe is something Ani is actively thinking about.

Oh, I see. Thanks for pointing this out. Is the translation to stabiliser codes more obvious if I only consider CSS codes? Naively I would expect that I can just consider each "half" of the code as a classical code on which CX acts analogously to classical CNOT, but maybe it is the equivalence of lo

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Hi Hari,

Thanks for the reply! I think I understand your confusion better now. You are correct in saying that
> It seems easier to just rotate and leave them in the original space.

This is precisely what we do :) We claim that we do not use any knowledge of the actual number $K_{\lambda,\mu}$

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Hi Dmitry,

Thanks very much for your comment. I’m still trying to understand this. It’s a little confusing because the vectors are first in a larger dimension (of $d_\lambda$). The isometry rotates them and essentially truncates the dimension to $K_{\lambda,\mu}$ but we still need to embed them in

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The authors will probably have a more insightful answer, but I'd maybe highlight that the scope of this preprint is classical fault tolerance and classical linear codes (where CZ gates are not defined). And translating the machinery to stabilizer codes, say, is not immediately obvious.

Is there an easy way to understand why the intuition and arguments used in this work don't generalise to other types of entangling gates? For example (to my understanding) https://arxiv.org/abs/2507.05392 gives a construction of asymptotically good codes where CZ and CCZ gates **are** addressable in

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banger

Hi Johnnie,
Thanks for your comment. We’ll be sure to check it out and update the citation with a clearer acknowledgement of your result.

Hi Siddhant and Yifan, nice work! I thought I'd mention that in https://arxiv.org/abs/2504.07344 (which you do already cite as [51]) we actually do use the cluster expansion rather than the series expansion. We use the counting numbers, $c_r$, derived from the 'region graph' defined as in generalize

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Dear Hari,

Thanks a lot for your feedback! We will do our best to make the presentation clearer in the next update :)

Regarding your question, indeed, as a mathematical operation $V_{\lambda,\mu}$ is an isometry with domain of dimension $K_{\lambda,\mu}$ and codomain of dimension $d_\lambda$ (dime

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I thought I should say something here. This looks like a really nice, thorough analysis of high-dimensional Schur transforms. There's a lot of detail here and it hasn’t been easy to digest (especially for me). In fact, it took me a few days to recover after I first saw this on the arxiv. Since then,

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A nice progress! It seems that the instance generation is really a bottleneck of the peaked circuits protocol. Using postselection, the success probability is exponentially small. Using the variational search, computing the loss function reduces to computing the zero-to-zero amplitude of a quantum c

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A welcome result!

Great, that does clear a few things up. I’m somewhat on board now.

Great, look forward to reading that.

Thanks! No, we purely focused on the decoding problem here without much regard for the associated dual optimization problem.

Hi, thanks for your interest in our paper and your question. The pseudo-thresholds for independent errors and erasures for the 72 qubit code are lower than the other codes presented. It also has logical error rate scaling suggestive of a lower distance code in the sub-threshold regime as expected. I

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Hi KdV,

To again relay a message for William Zhong.

"I've updated the paper with additional notes on the double-checking circuit in the appendix, and also rescaled the error parameter in all of our numerics to allow for better comparison with the original cultivation paper. Unfortunately,

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I wrote that one *basically* needs $m_i > \log_2 r$ above, and I then described that there are some caveats. In particular if the order is even. I have not thought it through in complete detail, but right off the bat it should always hold that you need $m_i > \log_2 r'$ for constructive interference

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@Martin: I was able to factor 161 using a maximum block size of 6 for a=3. Since the order here is r=66, your claim that each block size $m_i $ must be larger than $\log_2 r$ does not appear to hold. It seems the effect of the shift in the unitary for all blocks beyond the first was not accounte

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This is great progress on BPQM! Did you consider the difficulty of the optimization problem dual to decoding turbo codes?

> a common drawback for concatenated codes is their syndrome check weight,
which can grow exponentially with the number of layers `L`

Is this still a drawback in the wake of [blocklet concatentation][1] which claim to only rely on the syndrome measurements of the base code?

[1]: https://s

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Very interesting. We also worked with a single-shot inference QML model in our paper https://arxiv.org/pdf/2501.02148 and found it to be pretty effective (see discussion at top of page 4)!

Very cool result! This is exactly the sort of idea that seems promising for getting DQI around some of the challenges associated with speedups in unstructured settings.

Any updates on Oscar Higgott's comment from 26 days ago?

Thanks! We're working on optimising the code and will share it.

Congratulations on this very wonderful paper! I would like to ask whether there is a GitHub repo or any code available for this paper ?

Sorry, do you mean you want references about the importance of free energy for thermalization? Or did I misunderstand your question?

The stability of ordered phases at finite temperature is controlled by the free energy, which captures the competition between the energy and entropy of excitations

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Thanks a lot for the feedback! I'd be more than happy to correct that and add references to previous works on the topic, if there are some you'd recommend?

The author has made a significant update of the manuscript in V3: https://arxiv.org/pdf/2501.19375. One major update is the new Section VI discussing how to map the high-dimensional manifold back to a CW complex via deformation retraction, which gives rise to a non-topological code defined on a hid

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Can you comment on why the [[72, 12, 6]] code yields the largest area in the last plot of your [python notebook][1], when considering psuedo-threholds?

[1]: https://github.com/susanxschen/qldpc-fusion-lattices/blob/main/Phenomenological%20noise%20simulations/Get%20BB%20lattice%20error%20rate

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Did you catch the discussion by Gidney, VicQ, Pattinson, and squiggles on quantum computing stackexchange, which highlights the substantial overestimation of stabilizer decomposition costs? Your work appears to significantly surpass the numbers discussed [there][1]. Gidney appears to deprioritize ex

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Very nice work!

A quick semantic comment: it is a bit odd to say in the abstract that you "introduce the more general notion of a free energy barrier, whose absence is also demonstrated to guarantee fast thermalisation". The concept of a free-energy barrier, and its connection to slow dynamics has

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Wonderful, I look forward to reading it.

Hi KdV,

I have uploaded a new version of the manuscript, addressing your main concerns.

Please refer to arXiv version 2 for additional simulations and strategies to further reduce runtime, for example using Pauli webs.

Best,
Zhenghao

A few points related to the above:

1. When I briefly reviewed this pre-print as it appeared on the ArXiv I interpreted the authors to say that they perform a number of independent runs, and that the length of the control register (before reycling) is $m_i$ qubits in the $i$:th run.

If so, Crai

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Hi Noah,

Sorry for the late reply, I've been traveling.

For QAOA and AMP we can prove explicit bounds for any $k$ and $\lambda$ someone gives us. [This has actually already been done for depth-1 QAOA][1], all we do that builds on this is pick some explicit choice of parameters and look at the scal

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Each block’s measurement is not just a random sample, it is an estimate of a specific segment of a true eigenphase. While it’s true that Shor’s algorithm can involve as many as 2^{2048} eigenphases (“colors”), there is no need to see the same color twice. Because two different colors can still share

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And does it continue to work when there are 2^2048 colors, instead of 2 colors? Because when there are 2^2048 colors, you'll de-facto never see the same color twice.

Dear Namit,

Thank you for the quick review of our preprint. We appreciate your feedback and hope our reply will address any misunderstandings or shortcomings in our work.

‘Utility’ is usually defined as a demonstration of the ability to solve problems at a scale beyond brute force classica

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Are you able to prove any explicit bounds for finite k?
Eg, for the (3,6) Gallager ensemble, what is the maximum achievable satisfaction fraction of DQI, AMP, QAOA?

Do you have any results on stability when the value of \ell is beyond half the minimum distance?

Also, have you considered bound

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@Craig Gidney - It seems we agree that the authors’ method works when the given state is an eigenstate. The authors then extend their method to the case of a superposition of two eigenstates in https://arxiv.org/abs/2508.05805
and use it for modular amplitude estimation. That part looks correct to

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Really nice work!!

I think this would be a great tool to have in the community. Do you have any plans to release a public version of your software?

@Q_cat_1729 It looks their proof only considers the case where the given state is an eigenstate of the operator, which isn't true in Shor's algorithm. When the state isn't an eigenstate, step 13 of algorithm 1 will fail to converge.

Thank you very much for the answer and the references. I was not aware of those works.

Cheers,

Raf

@Craig Gidney - You may want to check out this paper https://arxiv.org/abs/2507.22460 where the authors provided a method for performing phase estimation with fewer qubits. The paper contains the mathematical proof and the quantum circuits for verification. They have used the same ideas in the Shor'

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