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Really nice! One thing I'm not sure about after reading is whether the $k$ in the $4^k$ in the bound for sample complexity refers to either:

1. the $k$ in the stabiliser code definition i.e. $[n,k]$ indicates you can encode $k$ logical qubits in $n$ physical qubits.
2. The total number of logi

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Hi,

I found your work quite interesting and original in the way you change the storage of an operator. Maybe you would be interested in this somewhat similar work, that also works for weakly dissipative bosonic systems https://arxiv.org/abs/2102.04265 (see appendix I.5 and figure 6)

Best

This was an important conversation to start, thank you for doing it.

This looks like a nice result, but it might be worth referencing the main result of "[Efficient Method for Computing the Maximum-Likelihood Quantum State from Measurements with Additive Gaussian Noise][1]" (PRL 108, 070502 (2012)). It seems closely related, albeit focused on removing the negative p

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Hey Anthony, no worries! The connection is not obvious and we do not mention local testability in our paper.

Hi Niko.
Really sorry for that! We cited your paper when we mentioned recent constructions of good quantum LDPC codes, but it's completely true that we should have mentioned that your balanced product codes were also giving the codes of Dinur et al. We'll acknowledge that in a revision of the paper

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I do not want to be *that guy*, but I would like to point out that the codes of Dinur et al. are the balanced product codes we conjectured to be good LDPC quantum codes in https://arxiv.org/abs/2012.09271 .
Dinur et al. point this out in a revised version of their paper https://arxiv.org/abs/2111.0

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Daochen Wang pointed out that Lemma 4.3 is false. In fact, discarding registers 1 and 2 yields a mixture of |z(f(a))> for uniformly random $a\in A$

Interesting work! I was under the impression that any imaginary time evolution algorithm had to fail for some local Hamiltonians, as finding the ground state of gapped local Hamiltonians is NP complete. If you try to do the imaginary time evolution with LCU, you could see the success probability of

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I also would like to point out that Gregory Rosenthal also has a paper from November that shows how to construct arbitrary n-qubit states in depth O(n) with exponentially many ancillas. (https://arxiv.org/pdf/2111.07992.pdf).

The result of this Theorem 1 was actually a special case of a more general result in arXiv:2108.06150v2 in November of last year, where the authors showed how to achieve the optimal depth for an anbitrary number of ancillary qubits, among other results.

Indeed, that's what Theorem 1 states:

Theorem 1 (Arbitrary quantum state preparation). With only
single- and two-qubit gates, an arbitrary 𝑛-qubit quantum state
can be deterministically prepared with a circuit depth Θ(𝑛)
and 𝑂(𝑁) ancillary qubits.

where N = 2^n.

I haven't read the paper but the abstract says they use ancilla, presumably exponentially many in the worst case.

Arbitrary quantum states cannot be prepared with poly sized quantum circuits in the sense of https://arxiv.org/abs/1102.1360. How is this compatible with these results?

Parikh and Verlinde's "De Sitter Holography with a Finite Number of States" is cited three times in the bibliography (references 38, 39 and 40 are isomorphic). In addition, reference 45 replicates reference 44.

Apropos: Does anyone know the *first* time that the Bekenstein bound was reinterpreted

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Thanks for clarification David! In my thinking I was "fixated" on density matrices and channels and it was harder to see this.

I have been wondering about this 2018 ( https://quantumcomputing.stackexchange.com/q/4912/119 ) and finally someone did it.

Hi, nice results!

You a small typo on your proof of Theorem 17:
To minimize over P.S.D operators you need to switch to
$A=\rho^{1/4}B\rho^{1/4}$ and not $A=\rho^{1/2}B$ which is generally not positive.

Is the Appendix currently available anywhere? :)

Thanks, Michal! To answer your question, say we are interested in the marginal probability for obtaining $x\in \{0,1\}^{|A|}$ when measuring a subset $A\subseteq [n]$ of the qubits in the computational basis, starting from a state $U|0^n\rangle$ where $U$ is a depth-$d$ circuit. We can write this m

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We have made another update to our paper. This is a complete rewrite with simplified proofs and extended discussions. We have added the two-qubit case as a minimal version of our main result with shorter proofs. Additionally, we now give a normal form for the Kraus representation of stabiliser opera

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Magnificent paper!

I have one small question. On page 3, when comparing gate-by-gate and qubit-by-qubit simulations you state that the estimated cost of computing the marginal probability is f(n,2d), where f(n,d) is the cost of computing an amplitude of depth d circuit acting on n qubits. Why is

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I received some criticisms of this work on generalized conditional expectations (GCEs), most likely from someone who has a fundamentalist attitude about the Belavkin nondemolition principle. I think there may be other people who have similar questions and it's now very difficult to travel or attend

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In this connection, see also section 6 ("Decoupling of the cyclotomic field") in [SIC-POVMs from Stark units](https://scirate.com/arxiv/2112.05552).

It works now! :)

Here is a binder link if you want to try the notebooks: https://mybinder.org/v2/gh/vsiddhu/SDP-Quantum-OR/HEAD (this will launch an interactive jupyter-lab environment in the cloud)

GitHub repo is here: https://github.com/vsiddhu/SDP-Quantum-OR

I've been informed that our information-theoretic bounds might overlap with the results presented in this paper:
https://arxiv.org/abs/1304.2336.
I'll go through it and then update the preprint with the correct references.

On the other hand, the applications to learning theory and crypto should

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It's definitely a universal problem, but I'm pretty sure I know what's causing it and it *should* be fixed with an update tomorrow. Thanks for spotting it!

I was able to replicate this, no pdf available for me

I don't know if this is a universal problem, but for some reason, no PDF is generated. Here or on the arXiv.

Dear authors, the paper looks quite interesting to us. Personally, we quite believe in applying distributed learning techniques into the variational quantum algorithms. Thanks for pushing forward this idea! What a coincidence, our preprint "Accelerating variational quantum algorithms with multiple q

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In this paper, Madsen, Valdetaro, and Moelmer wrote

> in [12–14] probing of time dependent perturbations with Gaussian
> noise correlations by quantum systems restricted to Gaussian states
> was shown to closely follow the classical theory of
> Mayne-Fraser-Potter two-filter smoothers [15]. Here, w

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Upvoted! Is this going to be a trend now, where we each post older and older theses? :-) (Kidding of course, always good to have it on the public record.)

Thanks Hsin-Yuan Huang!

Thank you so much for pointing out the typo in Eqs. (83)-(87)! Also, the rank condition in Eq. (78) should be replaced by "All eigenvalues of $O_i$ are $+1$ or $-1$". This is satisfied by all the explicit constructions we consider later in the section.

Now, the relation between Eq. (81) and the hyp

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Thanks for pointing point.

Interesting work! The separation between separable and entangled measurements is an old and underappreciated topic in quantum estimation theory (Gill and Massar https://arxiv.org/abs/quant-ph/9902063; see also Belliardo and Giovannetti https://doi.org/10.1088/1367-2630/ac04ca). People in that area h

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Dear Changhao Yi, for the purpose of solving linear systems with the quantum walk, we need to show that there is no phase error either, see our appendix G. This is because we need to keep track of two states in a superposition, and there should not be a phase factor introduced between them.

Dear authors：Thanks for mentioning our work in [27]. I think we are looking at a similar question. But our contribution is, sometimes even when ||U_T - U_T^A|| is large, in terms of fidelity, the discrete adiabatic process still works as the error can accumulate in the phase. The detail is in Sec II

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How can the authors deduce hope for a lack of barren plateaus from demonstrating non-vanishing gradients for 4-qubit examples? Am I missing something?

Dear Alexander, thank you for your interest in our work! This is indeed a typo, there should be no restriction on the parameter a here—I will update the arXiv version shortly, thanks for spotting this. In particular, there is no restriction on the functions that may be represented by the factorizati

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Dear authors,
today we talked about your work in a group meeting and stumbled across the HW factorization theorem (equation 6). We thought that there are holomorphic functions satisfying equation 5 which can not be represented in the way you proposed. Specifically, consider the function f(z) = z-

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Would it make sense to just make entanglement a free resource in quantum information theory? From a quantum field theory standpoint (entanglement harvesting), one could argue this is physically motivated. This would make a lot of things more natural, such as reversibility in entanglement manipulatio

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Ah -- that motivation is interesting! I've spent a lot of hours trying to do those integrals too. To me, that framing -- i.e., around understanding and demonstrating obstacles to doing the Bayesian integrals for pure-state or low-rank priors -- is more compelling than the current framing. That's

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Very true! And this is something that a couple others have also suggested, that I perhaps should make more clear in a revision. This paper was born from me initially trying to find an efficient closed-form calculation for Bayesian computations, and then determining that this was (modulo complexity a

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It is worth noting that finding the MLE or Bayesian mean/MAP estimate of a d-dimensional quantum state is only hard when the candidate states are restricted to a nonconvex prior. Finding the most likely *pure* state is indeed hard for arbitrary measurement records, but the hardness goes away if you

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Yes, we deal with entanglement in the usual way, by assuming that the underlying Hilbert space is bipartite -- separable but possibly infinite dimensional. I'm not sure how to define the theory of entanglement manipulation, and in particular the problem we are tackling here, without these basic ingr

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Just wondering, it seems like this work deals with infinite Hilbert spaces by assuming that they have a tensor product structure. Have you thought about what would happen if instead the Hilbert spaces were communitive? I'm not too familiar with how entanglement in these systems work, but this seem

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Indeed, thanks for catching that!

Great work! I think there is a typo; before Eq.(17), "||X_3||_1=2" should be "||X_3||_∞=2", I guess.