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What are the advantages of this method over rejection sampling, which also scales inversely in the filling ratio and requires one call to the oracle per attempt?

Ah, that makes sense. Thank you for the detailed answer. I will try Qermit out!

Hi Victory, many thanks for your message! Mitiq is great, and you're quite right that it breaks down the implementations of each protocol. In the case of Mitiq this could be seen as a result of the attentive development of the software. In the case of Qermit this breakdown of protocols into submodul

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Thanks, Lorenzo, for your reply. That makes sense, and I fully agree with everything you say :).

Maybe it's worth to stress that point then: your result applies to *families of states* that are defined by their amount of magic rather than individual states, correct?

Thanks for this package! The paper says

> Qermit is complementary to Mitiq [23], which is an opensource Python toolkit that implements an overlapping set of error mitigation schemes. Qermit takes a different approach that
breaks-down the implementation of each protocol into standalone modular un

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Dear Dominik, happy to hear from you. We are sorry for the late response.

The answer is yes and no. From the one hand, what you say is true. You can perform direct fidelity estimation on the state $(U_{1}\otimes\ldots U_{n})|\psi\rangle $ by expanding the fidelity between the theoretical state an

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What is referred to as "dual unitaries" was previously called "maximally entangled unitaries" in https://arxiv.org/abs/quant-ph/0306144, and more such operators were constructed there from biunimodular functions. They are maximally entangled in the sense of Mike Nielsen's operator schmidt decomposi

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Hi Lorenzo, Salvatore and Alioscia,

consider an $n$-qubit stabilizer state $| \psi \rangle$ with stabilizers $S_1,\ldots, S_n$. Then the state $U | \psi \rangle$ with $U = U_1 \otimes \cdots \otimes U_n$ for *arbitrary* single qubit unitaries $U_1, \ldots, U_n$ is stabilized by $U S_1 U^\dagger

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*I've removed the text of this comment, because it addressed a prior comment that has since been deleted. Absent the prior comment, there is no need for my words to appear here. However, I've chosen not to delete the comment entirely in order to leave a record that there was once a discussion here

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Comments are welcome!

This is an interesting development!

Up until this point, it has seemed to me that while the RQM literature endorsed up front a strong form of relationalism, when one dug into the details, the writing backed away from it. For example, measurement outcomes were treated as relative to an observer, b

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Hey Hong-Ye, yes thank you for the reply!

Hi Jerome,

Thank you so much for the additional references! We will include these works in the next update.

Best regards,

Robert (Hsin-Yuan Huang)

Very interesting! Regarding Section III.G about variational quantum-classical algorithms, it could be worth mentioning grouping methods that are not directly related to randomized measurements. Indeed, these methods seem to outperform classical shadow methods in several examples. See Fig. 3 in http

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I enjoyed this paper! One question, on equation (3) and the immediately preceding definition of P_n(U), is the a_n supposed to be a_{n+1}?

Hi Tom, thanks for your interest. In the second case you mentioned, it is still a $[[N=nk,k]]$ error correction code.

If one uses global Clifford group for shadow tomography, the sample complexity for predicting O scales as the rank of operator, here it would be P*O, where P is the projection op

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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!